# 3.E: Systems of ODEs (Exercises)

These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.

## 3.1 Introduction to Systems of ODEs

Exercise 3.1.2: Find the general solution of ( x'_1 = x_2 - x_1 + t, x'_2 = x_2).

Exercise 3.1.3: Find the general solution of ( x'_1 = 3x_1 - x_2 + e^t, x'_2 = x_1 ).

Exercise 3.1.4: Write ( ay'' + by' + cy = f(x) ) as a first order system of ODEs.

Exercise 3.1.5: Write ( x'' + y^2y' - x^3 = sin (t), y'' + {(x' + y')}^2 - x = 0) as a first order system of ODEs.

Exercise 3.1.101: Find the general solution to ( y'_1 = 3y_1, y'_2 = y_1 + y_2, y'_3 = y_1 + y_3 ).

Exercise 3.1.102: Solve ( y' = 2x, x' = x + y, x(0) = 1, y(0) = 3).

Exercise 3.1.103: Write ( x''' = x + t ) as a first order system.

Exercise 3.1.104: Write ( y''_1 + y_1 + y_2 = t y''_2 + y_1 - y_2 = t^2 ) as a first order system.

## 3.2: Matrices and linear systems

Exercise 3.2.2: Solve ( egin {bmatrix} 1 & 2 3 & 4 end {bmatrix} vec {x} = egin {bmatrix} 5 6 end {bmatrix} ) by using matrix inverse.

Exercise 3.2.3: Compute determinant of ( egin {bmatrix} 9 & -2 & -6 -8 & 3 & 6 10 & -2 & -6 end {bmatrix}).

Exercise 3.2.4: Compute determinant of ( egin {bmatrix} 1 & 2 & 3 & 1 4 & 0 & 5 & 0 6 & 0 & 7 & 0 8 & 0 & 10 & 1 end {bmatrix} ). Hint: Expand along the proper row or column to make the calculations simpler.

Exercise 3.2.5: Compute inverse of ( egin {bmatrix} 1 & 2 & 3 1 & 1 & 1 0 & 1 & 0 end {bmatrix} ).

Exercise 3.2.6: For which (h) is ( egin {bmatrix} 1 & 2 & 3 4 & 5 & 6 7 & 8 & h end {bmatrix} ) not invertible? Is there only one such (h)? Are there several? Infinitely many?

Exercise 3.2.7: For which (h) is ( egin {bmatrix} h & 1 & 1 0 & h & 0 1 & 1 & h end {bmatrix} ) not invertible? Find all such (h).

Exercise 3.2.8: Solve ( egin {bmatrix} 9 & -2 & -6 -8 & 3 & 6 10 & -2 & -6 end {bmatrix} vec {x} = egin {bmatrix} 1 2 3 end {bmatrix} ).

Exercise 3.2.9: Solve ( egin {bmatrix} 5 & 3 & 7 8 & 4 & 4 6 & 3 & 3 end {bmatrix} vec{x} = egin {bmatrix} 2 0 0 end {bmatrix} ).

Exercise 3.2.10: Solve ( egin {bmatrix} 3 & 2 & 3 & 0 3 & 3 & 3 & 3 0 & 2 & 4 & 2 2 & 3 & 4 & 3 end {bmatrix} vec {x} = egin {bmatrix} 2 0 4 1 end {bmatrix} ).

Exercise 3.2.11: Find 3 nonzero ( 2 imes 2 ) matrices ( A, B, ) and (C) such that ( AB = AC ) but ( B e C ).

Exercise 3.2.101: Compute determinant of ( egin {bmatrix} 1 & 1 & 1 2 & 3 & -5 1 & -1 & 0 end {bmatrix} )

Exercise 3.2.102: Find ( t ) such that ( egin {bmatrix} 1 & t -1 & 2 end {bmatrix} ) is not invertible.

Exercise 3.2.103: Solve ( egin {bmatrix} 1 & 1 1 & -1 end {bmatrix} vec{x} = egin {bmatrix} 10 20 end {bmatrix} ).

Exercise 3.2.104: Suppose ( a, b, c ) are nonzero numbers. Let ( M = egin {bmatrix} a & 0 0 & b end {bmatrix}, N = egin {bmatrix} a & 0 & 0 0 & b & 0 0 & 0 & c end {bmatrix} ). a) Compute (M^{-1} ). b) Compute ( N^{-1} ).

Exercise 3.3.1: Write the system ( x'_1 = 2x_1-3tx_2+sin t) and (x'_2=e^t,x_1 - 3x_2 + cos t) in the form (vec{x}'=P(t), vec{x} + vec{f}(t)).

Exercise 3.3.2: a) Verify that the system ( vec{x}' = egin{bmatrix} 1&3 3&1 end{bmatrix} vec{x} ) has the two solutions ( egin{bmatrix} 11 end{bmatrix}e^{4t} ) and .(egin{bmatrix} 1-1end{bmatrix} e^{-2t} ) b) Write down the general solution. c) Write down the general solution in the form (x_1 = ?) ,(x_2 = ? ) (i.e. write down a formula for each element of the solution).

Exercise 3.3.3: Verify that ( egin{bmatrix} 11 end{bmatrix}e^t) and (egin{bmatrix} 1-1 end{bmatrix} e^t ) are linearly independent. Hint: Just plug in (t=0).

Exercise 3.3.4: Verify that (egin{bmatrix} 11 end{bmatrix}e^t) and (egin{bmatrix}1-11 end{bmatrix}e^t ) and (egin{bmatrix} 1-11 end{bmatrix}e^{2t} ) are linearly independent. Hint: You must be a bit more tricky than in the previous exercise.

Exercise 3.3.5: Verify that(egin{bmatrix}t ^2 end{bmatrix} ) and (egin{bmatrix} t^3 ^4 end{bmatrix} ) are linearly independent.

Exercise 3.3.101: Are ( egin{bmatrix} e^{2t}e^t end{bmatrix} ) and ( egin{bmatrix} e^te^{2t} end{bmatrix} ) linearly independent? Justify.

Exercise 3.3.102: Are ( egin{bmatrix} cosh(t) 1 end{bmatrix} ), (egin{bmatrix} e^t 1 end{bmatrix} ) and ( egin{bmatrix} e^{-t} 1 end{bmatrix} )linearly independent? Justify.

Exercise 3.3.103: Write (x' = 3x -y+e^t) and (y'=tx) in matrix notation.

Exercise 3.3.104: a) Write (x'_1=2t,x_2) and (x'_2=2t,x_2) in matrix notation. b) Solve and write the solution in matrix notation.

Exercise 3.4.5 (easy): Let (A) be a (3 imes 3) matrix with an eigenvalue of (3) and a corresponding eigenvector (vec{v}= left[ egin{array}{c} 1 -1 3 end{array} ight]). Find (A vec{v}).

Exercise 3.4.6: a) Find the general solution of ( x'_1=2x_1, x'_2=3x_2) using the eigenvalue method (first write the system in the form (vec{x}'=Avec{x}) ). b) Solve the system by solving each equation separately and verify you get the same general solution.

Exercise 3.4.7: Find the general solution of (x'_1 = 3x_1 +x_2, x'_2 =2x_1+4x_2) using the eigenvalue method.

Exercise 3.4.8: Find the general solution of (x'_1 =x_1-2x_2, x'_2=2x_1 x_2) using the eigenvalue method. Do not use complex exponentials in your solution.

Exercise 3.4.9: a) Compute eigenvalues and eigenvectors of (A= left[ egin{array}{ccc} 9 & -2 & -6 -8 & 3 & 6 10 & -2 & -6 end{array} ight] ). b) Find the general solution of (vec{x}'=Avec{x}).

Exercise 3.4.10: Compute eigenvalues and eigenvectors of ( left[ egin{array}{ccc} -2 & -1 & -1 3 & 2 & 1 -3 & -1 & 0 end{array} ight].)

Exercise 3.4.11: Let (a, b, c, d, e, f) be numbers. Find the eigenvalues of ( left[ egin{array}{ccc} a & b & c 0 & d & e 0 & 0 & f end{array} ight].)

Exercise 3.4.101: a) Compute eigenvalues and eigenvectors of (A= left[ egin{array}{ccc} 1 & 0 & 3 -1 & 0 & 1 2 & 0 & 2 end{array} ight] ). b) Solve the system (vec{x}'=Avec{x}).

Exercise 3.4.102: a) Compute eigenvalues and eigenvectors of (A= left[ egin{array}{cc} 1 & 1 -1 & 0 end{array} ight].) b) Solve the system (vec{x}'=Avec{x}).

Exercise 3.4.103: Solve ( x'_1=x_2, x'_2=x_1) using the eigenvalue method.

Exercise 3.4.104: Solve ( x'_1=x_2, x'_2=-x_1) using the eigenvalue method.

## 3.5: Two dimensional systems and their vector fields

Exercise 3.5.1: Take the equation (m{x}'' + c{x}'+kx =0 ), with ( m>0, cgeq 0, k>0 ) for the mass-spring system.

1. Convert this to a system of first order equations.
2. Classify for what m, c, k do you get which behavior.
3. Can you explain from physical intuition why you do not get all the different kinds of behavior here?

Exercise 3.5.2: Can you find what happens in the case when ( P =egin{bmatrix} 1&1 0&1 end{bmatrix} )? In this case the eigenvalue is repeated and there is only one eigenvector. What picture does this look like?

Exercise 3.5.3: Can you find what happens in the case when ( P = egin{bmatrix} 1&11&1 end{bmatrix} )? Does this look like any of the pictures we have drawn?

Exercise 3.5.101: Describe the behavior of the following systems without solving:

1. (x' = x + y, y' = x- y)
2. (x_1' = x_1 + x_2, x_2' = 2x_2 )
3. (x_1' = -2x_2, x_2' = 2x_1 )
4. (x' = x+ 3y, y' = -2x-4y )
5. (x' = x - 4y, y' = -4x+y )

Exercise 3.5.102: Suppose that (vec{x} = A vec{x} ) where (A) is a (2 imes 2) matrix with eigenvalues ( 2pm i). Describe the behavior.

Exercise 3.5.103: Take ( egin{bmatrix} xy end{bmatrix}' = egin{bmatrix} 0&1 &0 end{bmatrix} egin{bmatrix} x y end{bmatrix} ). Draw the vector field and describe the behavior. Is it one of the behaviours that we have seen before?

## 3.6: Second order systems and applications

Exercise 3.6.3: Find a particular solution to

[ vec{x}'' = left[ egin{array}{cc} -3 & 1 2 & -2 end{array} ight] vec{x} + left[ egin{array}{c} 0 2 end{array} ight] cos(2t). ]

Exercise 3.6.4 (challenging): Let us take the example in Figure 3.12 with the same parameters as before: (m_1=2, k_1=4,) and (k_2=2,) except for (m_2), which is unknown. Suppose that there is a force ( cos(5t)) acting on the first mass. Find an (m_2) such that there exists a particular solution where the first mass does not move.

Note: This idea is called dynamic damping. In practice there will be a small amount of damping and so any transient solution will disappear and after long enough time, the first mass will always come to a stop.

Exercise 3.6.5: Let us take the Example 3.6.2, but that at time of impact, cart 2 is moving to the left at the speed of 3 m/s. a) Find the behavior of the system after linkup. b) Will the second car hit the wall, or will it be moving away from the wall as time goes on? c) At what speed would the first car have to be traveling for the system to essentially stay in place after linkup?

Exercise 3.6.6: Let us take the example in Figure 3.12 with parameters (m_1=m_2=1, k_1=k_2=1). Does there exist a set of initial conditions for which the first cart moves but the second cart does not? If so, find those conditions. If not, argue why not.

Exercise 3.6.101: Find the general solution to

[ left[ egin{array}{ccc} 1 & 0 & 0 0 & 2 & 0 0 & 0 & 3 end{array} ight] vec{x}''=left[ egin{array}{ccc} -3 & 0 & 0 2 & -4 & 0 0 & 6 & -3 end{array} ight] vec{x}+left[ egin{array}{c} cos(2t) 0 0 end{array} ight].]

Exercise 3.6.102: Suppose there are three carts of equal mass (m) and connected by two springs of constant (k) (and no connections to walls). Set up the system and find its general solution.

Exercise 3.6.103: Suppose a cart of mass 2 kg is attached by a spring of constant (k=1) to a cart of mass 3 kg, which is attached to the wall by a spring also of constant (k=1). Suppose that the initial position of the first cart is 1 meter in the positive direction from the rest position, and the second mass starts at the rest position. The masses are not moving and are let go. Find the position of the second mass as a function of time.

## 3.7: Multiple Eigenvalues

Exercise 3.7.2: Let ( A = egin{bmatrix} 5&-3 3&-1 end{bmatrix} ). Find the general solution of (vec{x} = Avec{x}).

Exercise 3.7.3: Let (A =egin{bmatrix} 5&-4&4 0&3&0 -2&4&-1end{bmatrix}. )

a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (lambda = Avec{x} ).

Exercise 3.7.4: Let ( A = egin{bmatrix} 2&1&0&2&0&0&2end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (vec{x} = Avec{x} ) in two different ways and verify you get the same answer.

Exercise 3.7.5: Let (A = egin{bmatrix}0&1&2 -1&-2&-2-4&4&7 end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (vec{x} = Avec{x} ).

Exercise 3.7.6: Let (A = egin{bmatrix} 0&4&-2-1&-4&1&0&-2 end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (vec{x} = A vec{x} ).

Exercise 3.7.7: Let (egin{bmatrix} 2&1&-1 -1&0&2-1&-2&4 end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (vec{x} = A vec{x}).

Exercise 3.7.8: Suppose that A is a (2 imes 2 ) matrix with a repeated eigenvalue (lambda). Suppose that there are two linearly independent eigenvectors. Show that (A = lambda I).

Exercise 3.7.101: Let (A = egin{bmatrix} 1&1&11&1&11&1&1 end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (vec{lambda} = A vec{lambda} ).

Exercise 3.7.102: Let (A = egin{bmatrix} 1&3&3 1&1&0 -1&1&2end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of (vec{x} = A vec{x} ).

Exercise 3.7.103: Let (A = egin{bmatrix} 2&0&0 -1&-1&9&-1&5end{bmatrix} ). a) What are the eigenvalues? b) What is/are the defect(s) of the eigenvalue(s)? c) Find the general solution of ( vec{x}= A vec{x}).

Exercise 3.7.104: Let (A = egin{bmatrix} a&a&cend{bmatrix} ), where (a), (b), and (c) are unknowns. Suppose that 5 is a doubled eigenvalue of defect 1, and suppose that (egin{bmatrix} 1 end{bmatrix} ) is the eigenvector. Find (A ) and show that there is only one solution.

## 3.8: Matrix Exponentials

Exercise 3.8.2: Using the matrix exponential, find a fundamental matrix solution for the system , (x' = 3x +y, y' = x+ 3y ).

Exercise 3.8.3: Find (e^{tA} ) for the matrix (A = egin{bmatrix} 2&3&2 end{bmatrix} ).

Exercise 3.8.4: Find a fundamental matrix solution for the system , (x'_1 =7x_1 +4x_2 +12x_3,~~~ x'_2= x_1 + 2x_2 +x_3,~~~x'_3 = -3x_1 - 2x_2 -5x_3 ). Then find the solution that satisfies ( vec{x} = egin{bmatrix}01-2end{bmatrix} ).

Exercise 3.8.5: Compute the matrix exponential (e^A) for (A = egin{bmatrix} 1&2&2end{bmatrix} ).

Exercise 3.8.6 (challenging): Suppose (AB=BA) . Show that under this assumption, (e^{A+B} = e^A e^B ).

Exercise 3.8.7: Use Exercise 3.8.6 to show that ((e^A)^{-1} = e^{-A} ). In particular this means that (e^A ) is invertible even if A is not.

Exercise 3.8.8: Suppose (A) is a matrix with eigenvalues -1, 1, and corresponding eigenvectors (egin{bmatrix} 11end{bmatrix} ), (egin{bmatrix}01end{bmatrix} ). a) Find matrix A with these properties. b) Find the fundamental matrix solution to (vec{x}' = A vec{x} ). c) Solve the system in with initial conditions (vec{x}(0) = egin{bmatrix} 23 end{bmatrix} ).

Exercise 3.8.9: Suppose that (A) is an (n imes n ) matrix with a repeated eigenvalue (lambda) of multiplicity n. Suppose that there are n linearly independent eigenvectors. Show that the matrix is diagonal, in particular (A = lambda mathit{I} ). Hint: Use diagonalization and the fact that the identity matrix commutes with every other matrix.

Exercise 3.8.10: Let (A =egin{bmatrix} -1&-11&-3 end{bmatrix} ). a) Find (e^{tA} ). b) Solve (vec{x}' = A vec{x} ), (vec{x}(0) = egin{bmatrix} 1-2 end{bmatrix} ).

Exercise 3.8.11: Let (A = egin{bmatrix} 1&23&4 end{bmatrix} ). Approximate (e^{tA} ) by expanding the power series up to the third order.

Exercise 3.8.101: Compute (e^{tA} ) where (A = egin{bmatrix} 1&-2-2&1end{bmatrix} ).

Exercise 3.8.102: Compute (e^{tA}) where (A=egin{bmatrix} 1&-3&2-2&1&2-1&-3&4 end{bmatrix} ).

Exercise 3.8.103: a) Compute (e^{tA} ) where (A = egin{bmatrix} 3& -11&1end{bmatrix} ). b) Solve (vec{x} ' = A vec{x} ) for ( vec{x}(0) = egin{bmatrix} 12 end{bmatrix} ).

Exercise 3.8.104: Compute the first 3 terms (up to the second degree) of the Taylor expansion of (e^{tA} ) where (A =egin{bmatrix} 2&32&2 end{bmatrix} ) (Write as a single matrix). Then use it to approximate (e^{0.1A} ).

## 3.9: Nonhomogeneous Systems

Exercise 3.9.4: Find a particular solution to ( x'=x+2y+2t, y'=3x+2y-4), a) using integrating factor method, b) using eigenvector decomposition, c) using undetermined coefficients.

Exercise 3.9.5: Find the general solution to ( x'=4x+y-1, y'=x+4y-e^t), a) using integrating factor method, b) using eigenvector decomposition, c) using undetermined coefficients.

Exercise 3.9.6: Find the general solution to ( x_1''=-6x_1+3x_2+ cos(t), x''_2=2x_1-7x_2+ 3 cos(t)), a) using eigenvector decomposition, b) using undetermined coefficients.

Exercise 3.9.7: Find the general solution to ( x''_1=-6x_1+3x_2 + cos(2t), x''_2=2x_1-7x_2+ cos(2t) ), a) using eigenvector decomposition, b) using undetermined coefficients.

Exercise 3.9.8: Take the equation

[ vec{x}' = left[ egin{array}{cc} frac{1}{t} & -1 1 & frac{1}{t} end{array} ight] vec{x}+ left[ egin{array}{c} t^2 -t end{array} ight]. ]

a) Check that

[ vec{x}_c = c_1left[ egin{array}{c} t sin t -t cos t end{array} ight] + c_2left[ egin{array}{c} t cos t t sin t end{array} ight]]

is the complementary solution. b) Use variation of parameters to find a particular solution.

Exercise 3.9.101: Find a particular solution to ( x' = 5x +4y+t,y'=x+8y-t), a) using integrating factor method, b) using eigenvector decomposition, c) using undetermined coefficients.

Exercise 3.9.102: Find a particular solution to ( x'=y+e^t, y'=x+e^t), a) using integrating factor method, b) using eigenvector decomposition, c) using undetermined coefficients.

Exercise 3.9.103: Solve ( x'_1=x_2+t, x'_2=x_1+t) with initial conditions ( x_1(0)=1, x_2(0)=2), using eigenvector decomposition.

Exercise 3.9.104: Solve ( x''_1=-3x_1+x_2+t, x''_2=9x_1+5x_2 +cos(t)) with initial conditions ( x_1(0)=0, x_2(0)=0, x'_1(0)=0, x'_2(0)=0), using eigenvector decomposition.

## 3.E: Energy (Exercises)

• Contributed by Timon Idema
• Associate Professor (Bionanoscience) at Delft University of Technology
• Sourced from TU Delft Open
1. Show that, if you ignore drag, a projectile fired at an initial velocity (v_0) and angle ( heta) has a range R given by
2. A target is situated 1.5 km away from a cannon across a flat field. Will the target be hit if the firing angle is (42^) and the cannonball is fired at an initial velocity of 121 m/s? (Cannonballs, as you know,do not bounce).
3. To increase the cannon&rsquos range, you put it on a tower of height (h_0). Find the maximum range in this case, as a function of the firing angle and velocity, assuming the land around is still flat.

3.2 You push a box of mass m up a slope with angle ( heta) and kinetic friction coefficient (mu). Find the minimum initial speed v you must give the box so that it reaches a height h.

3.3 A uniform board of length L and mass M lies near a boundary that separates two regions. In region 1, the coefficient of kinetic friction between the board and the surface is (mu _1), and in region 2, the coefficient is (mu _2). Our objective is to find the net work W done by friction in pulling the board directly from region 1 to region 2, under the assumption that the board moves at constant velocity.

1. Suppose that at some point during the process, the right edge of the board is a distance x from the boundary, as shown. When the board is at this position, what is the magnitude of the force of friction acting on the board, assuming that it&rsquos moving to the right? Express your answer in terms of all relevant variables (L, M, g, x, (mu _1), and (mu _2)).
2. As we&rsquove seen in Section 3.1, when the force is not constant, you can determine the work by integrating the force over the displacement, (W= int F(x) dx). Integrate your answer from (a) to get the net work you need to do to pull the board from region 1 to region 2.

3.4 The government wishes to secure votes from car-owners by increasing the speed limit on the highway from 120 to 140 km/h. The opposition points out that this is both more dangerous and will cause more pollution. Lobbyists from the car industry tell the government not to worry: the drag coefficients of the cars have gone down significantly and their construction is a lot more solid than in the time that the 120 km/h speed limit was set.

1. Suppose the 120 km/h limit was set with a Volkswagen Beetle ((=0.48)) in mind, and the lobbyist&rsquoscar has a drag coefficient of 0.19. Will the new car need to do more or less work to maintain a constant speed of 140 km/h than the Beetle at 120 km/h?
2. What is the ratio of the total kinetic energy released in a full head-on collision (resulting in an immediate standstill) between two cars both at 140 km/h and two cars both at 120 km/h?
3. The government dismisses the opposition&rsquos objections on safety by stating that on the highway, all cars move in the same direction (opposite direction lanes are well separated), so if they all move at 140 km/h, it would be just as safe as all at 120 km/h. The opposition then points out that running a Beetle (those are still around) at 120 km/h is already challenging, so there would be speed differences between newer and older cars. The government claims that the 20 km/h difference won&rsquot matter, as clearly even a Beetle can survive a 20 km/h collision. Explain why their argument is invalid.

3.5 Nuclear fusion, the process that powers the Sun, occurs when two low-mass atomic nuclei fuse together to make a larger nucleus, releasing substantial energy. Fusion is hard to achieve because atomic nuclei carry positive electric charge, and their electrical repulsion makes it difficult to get them close enough for the short-range nuclear force to bind them into a single nucleus. The figure below shows the potential-energy curve for fusion of two deuterons (heavy hydrogen nuclei, consisting of a proton and a neutron).The energy is measured in million electron volts ((MeV, 1 eV=1.6 cdot 10^ <-19>J)), a unit commonly used in nuclear physics, and the separation is in femtometers ((1 fm=10^ <-15>m)).

1. Find the position(s) (if any) at which the force between two deuterons is zero.
2. Find the kinetic energy two initially widely separated deuterons need to have to get close enough to fuse.
3. The energy available in fusion is the energy difference between that of widely separated deuterons and the bound deutrons after they&rsquove &lsquofallen&rsquo into the deep potential well shown in the figure. About how big is that energy?
4. Determine whether the force between two deuterons that are 4 fm apart is repulsive, attractive, or zero.

3.6 A pigeon in flight experiences a drag force due to air resistance given approximately by (F=bv^2), where v is the flight speed and b is a constant.

1. What are the units of b?
2. What is the largest possible speed of the pigeon if its maximum power output is P?
3. By what factor does the largest possible speed increase if the maximum power output is doubled
1. For which value(s) of the parameters (alpha, eta, ext < and >gamma) is the force given by [oldsymbol=left(x^ <3>y^<3>+alpha z^<2>, eta x^ <4>y^<2>, gamma x z ight)] conservative?
2. Find the force for the potential energy given by (U(x,y,z)=frac-frac).

3.8 A point mass is connected to two opposite walls by two springs, as shown in the figure. The distance between the walls is 2L. The left spring has rest length (l_1=frac<2>) and spring constant (k_1=k), the right spring has rest length (l_2=frac<3L><4>) and spring constant (k_2=3k).

1. Determine the magnitude of the force acting on the point mass if it is at x=0.
2. Determine the equilibrium position of the point mass.
3. Find the potential energy of the point mass as a function of x. Use the equilibrium point from (b) as your point of reference.
4. If the point mass is displaced a small distance from its equilibrium position and then released, it will oscillate. By comparing the equation of the net force on the mass in this system with a simple harmonic oscillator, determine the frequency of that oscillation. (We&rsquoll return to systems oscillating about the minimum of a potential energy in Section 8.1.4, feel free to take a sneak peak ahead).

3.9 A block of mass m=3.50 kg slides from rest a distance d down a frictionless incline at angle ( heta=30.0^circ),where it runs into a spring of spring constant 450 N/m. When the block momentarily stops, it has com-pressed the spring by 25.0 cm.

1. Find d.
2. What is the distance between the first block-spring contact and the point at which the block&rsquos speed is greatest?

3.10 Playground slides frequently have sections of varying slope: steeper ones to pick up speed, less steep ones to lose speed, so kids (and students) arrive at the bottom safely. We consider a slide with two steep sections (angle (alpha)) and two less steep ones (angle (eta)). Each of the sections has a width L. The slide has a coefficient of kinetic friction (mu).

1. Kids start at the top of the slide with velocity zero. Calculate the velocity of a kid of mass m at the end of the first steep section.
2. Now calculate the velocity of the kid at the bottom of the entire slide.
3. If L=1.0 m, (alpha=30^circ) and (mu=0.5), find the minimum value (eta) must have so that kids up to 30 kg can enjoy the slide (Hint: what is the minimum requirement for the slide to be functional)?
4. A given slide has (alpha=30^circ), (eta=20^circ), and (mu=0.5). A young child of 10 kg slides down, while its cousin of 20 kg sits at the bottom. When the sliding kid reaches the end, the two children collide, and together slide further over the ground. The coefficient of kinetic friction with the ground is 0.70. How far do the two children slide before they come to a full stop?

3.11 In this problem, we consider the anharmonic potential given b

where a, b, and (x_0) are positive constants.

1. Find the dimensions of a, b, and (x_0).
2. Determine whether the force on a particle at a position (x gt gt x_0) is attractive or repulsive (taking the origin as your point of reference).
3. Find the equilibrium point(s) (if any) of this potential, and determine their stability.
4. For b=0, the potential given in Equation (3.24) becomes harmonic (i.e., the potential of a harmonic oscillator), in which case a particle that is initially located at a non-equilibrium point will oscillate. Are there initial values for x for which a particle in this anharmonic potential will oscillate? If so,find them,and find the approximate oscillation frequency if not, explain why not. (NB: As the problem involves a third order polynomial function, you may find yourself having to solve a third order problem. When that happens, for your answer you can simply say: the solution x to the problem X).

3.12 After you have successfully finished your mechanics course, you decide to launch the book into an orbit around the Earth. However, the teacher is not convinced that you do not need it anymore and asks the following question: What is the ratio between the kinetic energy and the potential energy of the book in its orbit?

Let m be the mass of the book, (M_ ext < and >R_) the mass and the radius of the Earth respectively. The gravitational pull at distance r from the center is given by Newton&rsquos law of gravitation (Equation 2.2.3):

1. Find the orbital velocity v of an object at height h above the surface of the Earth.
2. Express the work required to get the book at height h.
3. Calculate the ratio between the kinetic and the potential energy of the book in its orbit.
4. What requires more work, getting the book to the International Space Station (orbiting at h=400 km)or giving it the same speed as the ISS?

3.13 Using dimensional arguments, in Problem 1.4 we found the scaling relation of the escape velocity (the minimal initial velocity an object must have to escape the gravitational pull of the planet/moon/other object it&rsquos on completely) with the mass of the radius of the planet. Here, we&rsquoll re-derive the result, including the numerical factor that dimensional arguments cannot give us.

1. Derive the expression of the gravitational potential energy,Ug, of an object of mass m due to a gravitational force (F_g) given by Newton&rsquos law of gravitation (Equation 2.2.3) [F_>=-frac> hat] Set the value of the integration constant by ( ightarrow 0 ext < as >r ightarrow infty)
2. Find the escape velocity on the surface of a planet of mass M and radius R by equating the initial kinetic energy of your object (when launched from the surface of the planet) to the total gravitational potential energy it has there.

3.14 A cannonball is fired upwards from the surface of the Earth with just enough speed such that it reaches the Moon. Find the speed of the cannonball as it crashes on the Moon&rsquos surface, taking the gravity of both the Earth and the Moon into account. Table B.3 contains the necessary astronomical data.

3.15 The draw force F(x) of a Turkish bow as a function of the bowstring displacement x (for x gt 0) is approximately given by a quadrant of the ellipse [left(frac> ight)^<2>+left(frac ight)^<2>=1] In rest, the bowstring is at x=0 when pulled all the way back, it&rsquos at x=-d.

1. Calculate the work done by the bow in accelerating an arrow of mass m=37 g, for d=0.85 m, and Fmax=360 N.
2. Assuming that all of the work is converted to kinetic energy of the arrow, find the maximum distance the arrow can fly.Hint: which variable can you control when shooting? Maximize the distance with respect to that variable.
3. Compare the result of (b) with the range of a bow that acts like a simple (Hookean) spring with the same values of Fmax and d. How much further does the arrow shot from the Turkish bow fly than that of the simple spring bow?

3.16 A massive cylinder with mass M and radius R is connected to a wall by a spring at its center (see figure).The cylinder can roll back-and-forth without slipping.

1. Determine the total energy of the system consisting of the cylinder and the spring.
2. Differentiate the energy of problem (16a) to obtain the equation of motion of the cylinder and spring system.
3. Find the oscillation frequency of the cylinder by comparing the equation of motion at (16b) with that of a simple harmonic oscillator (a mass-spring system).

3.17 A small particle (blue dot) is placed atop the center of a hemispherical mount of ice of radius R (see figure). It slides down the side of the mount with negligible initial speed. Assuming no friction between the ice and the particle, find the height at which the particle loses contact with the ice.

Hint: To solve this problem, first draw a free body diagram, and combine what you know of energy and forces.

3.18 Pulling membrane tubes

The (potential) energy of a cylindrical membrane tube of length L and radius R is given by

Here (kappa) is the membrane&rsquos bending modulus and (sigma) its surface tension.

## Euler's method

A very simple ordinary differential equation (ODE) is the explicit scalar first-order initial value problem :

An analytic solution of an ODE is a formula , that we can evaluate, differentiate, or analyze in any way we want. Analytic solutions can only be determined for a small class of ODE's.

A numerical solution'' of an ODE is simply a table of abscissæ and approximate values that approximate the value of an analytic solution. This table is usually accompanied by some rule for interpolating solution values between the abscissæ. With rare exceptions, a numerical solution is always wrong the important question is, how wrong is it? One way to pose this question is to determine how close the computed values are to the analytic solution, which we might write as .

The simplest method for producing a numerical solution of an ODE is known as Euler's method . Given a solution value , we estimate the solution at the next abscissa by:

(The step size is denoted here. Sometimes it is denoted .) We can take as many steps as we want with this method, using the approximate answer from one step as the starting point for the next step.

Matlab note: In the following function, the name of the function that evaluates is arbitrary. Recall that if you do not know the actual name of a function, but it is contained in a Matlab variable (I often use the variable name f'') then you can evaluate the function using the Matlab function feval.'' Supposing you have a Matlab function m-file named my_ode.m and its signature line looks like and suppose this name is contained in the variable f in your m-file. Then you can call the function using the syntax

Some students have difficulty distinguishing when to enclose names in quotes and mistakenly use the syntax This syntax tells Matlab to use the function whose name is f and that can be found in the file named f.m . In contrast, the correct syntax, feval(f,x,y) , tells Matlab to find the function whose name is the value of the variable f . These are very different things.

Typically, Euler's method will be applied to systems of ODEs rather than a single ODE. This is because higher order ODEs can be written as systems of first order ODEs. The following Matlab function m-file implements Euler's method for a system of ODEs.

Warning: In the above code, the initial value is a column vector, and the function that gets called returns a column vector however, the values are returned in the rows of the matrix m ! The function transpose is used instead of a prime because a prime without a dot means Hermitian'' or adjoint'' or conjugate-transpose,'' but only a true transpose is needed here. The .' operator could have been used, but is harder to read.

In the following exercise, you will use euler.m to find the solution of the initial value problem

1. If you have not done so already, copy (use cut-and-paste) the above code into a file named euler.m or download euler.m .
2. Copy the following code into a Matlab m-file called expm_ode.m .
3. Now you can use Euler's method to march from y=yInit at x=0 : for each of the values of numSteps in the table below. Use at least four significant figures when you record your numbers, and you can use the first line as a check on the correctness of the code. In addition, compute the error as the difference between your approximate solution and the exact solution at x=2 , y=3*exp(-2)+2 , and compute the ratios of the error for each value of nstep divided by the error for the succeeding value of nstep . As the number of steps increases, your errors should become smaller and the ratios should tend to a limit.
4. You know the error is for some . There is a simple way to estimate the value of by successively halving . If the error were exactly , then by solving twice, once using and the second time using and taking the ratio of the errors, you would get . Since the error is only , the ratio is only approximately .

## How to Stimulate Parasympathetic Nervous System with Exercise

We have received an incredible response to our series on Balancing the Sympathetic and Parasympathetic Nervous Systems (if you haven’t read it, but sure and check it out – it’s a great primer for what we will discuss here!). Out of that series, we received several questions about how to stimulate the parasympathetic nervous system (PNS) with exercise. This is an interesting question, because exercise, especially when someone is just starting out or they are exercising intensely increases the sympathetic nervous system (SNS) rather than the PNS. However, the body is designed to spend most of its time in the PNS, and doing certain exercises can improve the PNS response and reduce the impact that stress has on your health.

High Intensity Training

High intensity exercise activates the SNS the more intense the exercise, the more stress the body can endure (assuming you are getting enough rest and adequate nutrition). Interestingly, studies have shown that repeated, intense exercise can decrease a person’s ability to reactivate the parasympathetic response and reestablish homeostasis. Thus, it appears that repeated, high intensity workouts can inhibit a balanced parasympathetic response.

Stimulating the Parasympathetic Nervous System

There are several ways to improve and strengthen the PNS response, which will help relax the mind and body. These include regular aerobic exercise, mind-body centered exercise (including yoga, tai chi and qi gong) and meditation.

Aerobic exercise

It has been shown that regular aerobic exercise can increase the activity of the parasympathetic nervous system and decrease sympathetic activity. Studies have shown that light to moderate intensity exercise for at least 30 minutes per day at least five days per week can improve the PNS response.

Resting and exercise heart rate are controlled by the sympathetic and parasympathetic nervous system. The SNS prepares the body for physical activity by increasing heart rate, increasing blood pressure, increasing the respiratory rate and releasing glucose from the liver to supply quick energy.

The PNS helps to slow down heart rate and breathing. At rest, the heart is controlled by the parasympathetic division, which is why the average resting heart rate of a person with a healthy PNS response is 60 beats per minute or less. With continued aerobic training, the PNS response increases and the resting heart rate decreases. This is one of the explanations as to why endurance athletes have such a low resting heart rate following long-term training.

Mind-body centered exercise

Mind-body centered exercises, like yoga, tai chi and qi gong, have been used for centuries to quiet the mind and strengthen the body’s reserves. The key is to focus on your breathing while deliberately/consciously moving in and through the poses without physical strain or mental chatter. In essence, in so doing you are getting the benefits of both light-intensity aerobic exercise as well as meditation.

Several studies have demonstrated that meditation can improve the parasympathetic response and delay the onset of SNS activation. A study done in Norway found that male runners who meditated for thirty minutes several times a week for six months had lower blood levels of lactic acid after exercise. This has at least two important implications. On the one hand, it indicates that mediation may give athletes a competitive edge by delaying the SNS activation by exercise, permitting them to exercise at higher levels of intensity for longer periods of time. On the other hand, for those that are solely looking to reestablish balance in their lives, it means that meditation may allow them to gain further PNS activation during and after their regular aerobic workouts.

We are often asked “What is the best time to meditate?” The answer is simple – whenever you can carve out 5-10 minutes. Some studies indicate that meditating after a workout can improve recovery and deepen the PNS response, so it may also be wise to end your workouts with a 5-minute guided meditation to speed recovery and promote a deeper relaxation response.

Getting Started

Many other techniques can also be used to help improve the PNS response, including deep breathing, Swedish and/or other light-touch massage and even walking. The key with any of these techniques is in their regular use. Incorporate one or two of these techniques into your daily regime every day and you’ll be experiencing the relaxation and calm that accompanies a healthy PNS response in no time at all.

Goldsmith RL, Bloomfield DM, Rosenwinkel ET. Exercise and autonomic function. Coron Artery Dis. 2000 Mar11(2):129-35.

Fu Q, Levine BD. Exercise and the autonomic nervous system. Handb Clin Neurol. 2013117:147-60.

James, D. V. B., Munson, S. C., Maldonado-Martin, S., & Croix, M. B. A. D. S. (2012). Heart rate variability: Effect of exercise intensity on post-exercise response. Research Quarterly for Exercise and Sport, 83(4), 533-9.

Sarang, P. S., & Telles, S. (2006). Oxygen consumption and respiration during and after two yoga relaxation techniques. Applied Psychophysiology and Biofeedback, 31(2), 143-53.

Solberg, E.E., Ingjer, F., Holen, A., Sundgot-Borgen, J., Nilsson, S., & Holme, I. (2000). Stress reactivity to and recovery from a standardized exercise bout: a study of 31 runners practicing relaxation techniques. British Journal of Sports Medicine, 34: 268-272.

Telles, S., Reddy, S. K., & Nagendra, H. R. (2000). Oxygen consumption and respiration following two yoga relaxation techniques. Applied Psychophysiology and Biofeedback, 25(4), 221-7.

## Vector fields for autonomous systems of two first order ODEs

If the right hand side function g(t, y) does not depend on t, the problem is called autonomous. In this case the behavior of the differential equation can be visualized by plotting the vector g(t, y) at each point y = (y1,y2) in the y1,y2 plane (the so-called phase plane).

First save the files vectfield.m and vectfieldn.m into your home directory.

To plot the vector field for y1 going from a1 to b1 with a spacing of d1 and y2 going from a2 to b2 with a spacing of d2 use vectfield(g,a1:d1:b1,a2:d2:b2) . The command vectfieldn works in the same way, but produces arrows which all have the same length. This makes it easier to see the direction of the vector field.

Example: The pendulum problem from above without the driving force gives the first order system

We can plot the vector field and 10 trajectories with starting points (0,0), (0,0.3), . (0,2.7) in the phase plane as follows:

## 4 Answers 4

$egin x ' = x^2 + y^2 - 1 y'= x^2 - y^2 end$

As @RobertLewis has pointed out, we find the equilibrium points $(x,y)$ at the points where we have $x' = y' = 0$. We have

From $(4)$, we have $x^2 = y^2$. Substituting this back into $(3)$, yields $x=pm dfrac<1>>$. We can substitute this $x$ back into $(4)$, yielding $y = pm dfrac<1>>$. Thus, we have a total of critical points as:

You should validate that each of these four points gives you $x'= y' = 0$ by substituting them into the original system.

Your next step is to use linearization, find the Jacobian and evaluate the eigenvalues for those four critical points to determine stability. I am going to let you work that, but here are some nice notes with a summary, the gotchas with linearzation and examples (starts on page $4$). I am also not sure if you discussed nullclines, but the notes have those also and they can be superimposed on the phase portrait. Hint: there are three unstable and one stable critical point.

Here is a phase-portrait for the system showing the four critical points (which can be used to validate your critical points and stability analysis).

This answer posted in response to the modified system, given in (1) and (2) below:

Oy Gevalt! NOW there is work!

Where is Moshe now that we need him to lead us out from under Pharoah's toil?

Well, I ain't no $Moshe$ but I can cut some down some of the work, thus:

The system to be considered is now

the number of equilibria has jumped from none to four! To see this, note that now $x' = y' = 0$ implies, from (1), (2), that

$2x^2 = 1 Rightarrow x = pm dfrac<2> ag<5>$

we see from (4) that $y$ may take the same values the equilibria occur at the four points

It may also be seen, geoemtrically, that the equilibria are given by (6), since (3) is the equation of a circle, centered at the origin and of radius $1$, and (4) is the combined equation of the two lines $x pm y = 0$, since

$x^2 = y^2 Rightarrow x = pm y Rightarrow x pm y = 0 ag<7>$

the circle intersects these lines at the specified points (6). In any event, having the equilibria of the system (1)-(2) at hand, the next step is to linearize the equations about these four points, and see what we get. Linearizing requires computation of the Jacobian matrix $J(x, y)$ of the vector field $(x', y')^T = (x^2 + y^2 -1, x^2 - y^2)^T$ we have

and we next must evaluate and eigen-analyze $J(pm dfrac><2>, pm dfrac><2>)$ for all four possible combinations of $pm dfrac><2>$, i.e., at all four points $(pm dfrac><2>, pm dfrac><2>)$. Well, some the the Oy! Gevalt! can be mollified by lessening the amount of work by realizing that, due to certain symmetries of the problem, there are really only two matrices $J(x, y)$ which need to be considered, not four. This is most easily seen by breaking the situation up into quadrants:

in this way we only need perform the eigen-analysis on two matrices out of the four $J(pm dfrac><2>, pm dfrac><2>)$, so lets start with $J_I = J(dfrac ><2>, dfrac><2>)$ its characteristic polynomial, call it $p_I(lambda)$, is

$p_I(lambda) = det (egin sqrt <2>- lambda & sqrt <2> sqrt <2>& -sqrt <2>- lambda end) = lambda^2 - 4. ag<11>$

We see from (11) that the eigenvalues of $J_I$ are $lambda = pm 2$ since $J_I$ has a positive eigenvalue, the point $(dfrac><2>, dfrac><2>)$ is unstable since $J_I$ also has a negative eigenvalue, this point is a saddle by (9), $J_ = -J_I$ also has eigenvalues $pm 2$, and hence also unstable and a saddle. These facts are borne out by the excellent graphic contributed by Amzoti in his answer. We next turn to $J_$ since we are now in the second quadrant, we have

$p_(lambda) = det(J_ - lambda I) = det(egin -sqrt <2>- lambda & sqrt <2> -sqrt <2>& -sqrt <2>- lambda end)$ $= (lambda + sqrt<2>)^2 + 2 = lambda^2 + 2sqrt <2>lambda + 4 ag<13>$

the zeroes of $p_(lambda)$ are found via the quadratic formula:

$lambda = dfrac<1><2>(-2sqrt <2>pm sqrt<8 - 16>) = dfrac<1><2>(-2sqrt <2>pm 2isqrt<2>) = -sqrt <2>pm isqrt<2> ag<14>$

we we see that the eigenvalues of $J_$ are not real, but have negative real part thus $(-dfrac><2>, dfrac><2>)$ is a stable spiral. Furthermore, since $J_ = -J_$, the eigevalues of $J_$ are $lambda = sqrt <2>pm isqrt<2>$ thus the point $(-dfrac><2>, -dfrac><2>)$ is an unstable spiral. All these computations support and are supported by Amzoti's grahic of the phase portrait of (1)-(2).

The stable spiral point at $(-dfrac><2>, dfrac><2>)$ is in fact asymptotically stable this follows from the fact that $Re(lambda) < 0$ for each of the eigenvalues of $J_$ this fact is both well-known and well documented, for example in the excellent reference provided by Amzoti in his comment.

There appears to be a discrepancy in the calculation of eigenvalues by Monolinte and myself, but we agree on the qualitative features of the equilibrium points. As ever, abstract analysis is easy but arithmetic proves difficult! Until further notice, I'm standing by my calculations.

## 3.2: Structure of Alkyl Radicals: Hyperconjugation

### Problems

Indicate whether each Hydrogen present is primary, secondary, or tertiary. Why is quaternary not an option?

(a) ethane (b) 2,2-dimethylpropane (c) 1-methylcyclobutane

In each pair of radicals, determine whether the radical is primary, secondary, or tertiary and also decide which radical is more favorable. Give the general reason why for all of the cases.

(a) (b) (c)

Draw the electron pushing arrow mechanism for all possible pyrolysis radicals that can form from butane, assuming only C-C bonds are broken. What type of bond cleavage is this, heterolytic or homolytic?

Draw all the possible pyrolysis radical products for 2-methylbutane, and determine which bond is most likely to be broken.

Calculate the &DeltaH o (kJ/mol) of the following reactions using the given bond dissociation energies.

Bond Dissociation Energies (Homolytic) &DeltaH o (kJ/mol)
CH3-H (methane) 439
C2H5-H (ethane) 423
(CH3)3C-H 404
H-H 435
H-Cl 431
H-Br 364
CH3-Cl 356
CH3-Br 293
C2H5-Cl 352
C2H5-Br 293
(CH3)3C-Cl 356
(CH3)3C-Br 297
Cl-Cl 242
Br-Br 192

Predict all possible constitutional isomers possible if monohalogenation were performed on the molecules in problem 15 with Br2. Give the name of the haloalkane.

Given the following alkanes, draw the most likely product to form upon monohalogenation with Br2 (keep in mind that this may not be the only product to form though). If the reaction was performed with Cl2 would there be more or less selectivity in the desired product formation? Why?

(a) (b) (c)

Draw out the full mechanism of the monochlorination of ethane with electron-pushing arrows. Label the three overall steps of the mechanism.

Calculate the statistical probablity of monobromination of propane on each unique carbon, and discuss why this is not likely. Given that the experimental monobromination produces two products, 97% one and 3% the other, assign these percentages to the corresponding product.

Draw the mechanism of the bromination of propene using Br2, on the carbon adjacent to the double bond. Can you draw any resonance structures for the intermediates in the propagation steps? Would you expect this to make the radical more or less stable?

Based on the previous question, how do you think the ∆H o to form the propene radical would compare to that of a propane radical?

Similar to problem 21, describe the resemblence to products and reactants of an early and a late transition state. In a monohalogenation, what step is early what step is late (propagation and termination)?

Draw the line-bond structure of the major product for the following reaction, if a reaction occurs, assume monohalogenation.

(a) (b) (c)

(d) (e)

For problem 27.c, calculate the product ratios using the following information (hint use the number of hydrogens in each category present to calculate the ratios).

Chlorination: 1 o Reactivity=1 2 o Reactivity=4 3 o Reactivity=5

Halo alkanes are synthetically useful compounds as you'll learn later. However, the radical halogenation is not always selective as seen in problem 27. If any of the products of the reactions were to be used as subsequent reagents, which would be more useful, 27.c or 27.d?

What are potential problems of trying to brominate the second carbon in hexane? If however, this does form draw the Newman Projection of the most favorable staggered conformation looking down the C2-C3 axis.

Calculate the &DeltaH o value for the bromination and iodination of propene, on the secondary carbon, assume monohalogenation. The bond dissociation values are as follows. What is the major difference between the two values? (Note this problem uses kcal mol -1 )

Bond Bond Dissociation Energy (kcal mol -1 )
Br-Br 46
I-I 36
(CH3)2CH-H 98.5
H-Br 87
H-I 71
Secondary C-I 56
Secondary C-Br 71

Draw the bond polarity present of chloroethane, explain how this affects the electronics of the carbon attached to the chloro group.

Write a balanced combustion reaction for the following hydrocarbons, sugars, and alchols. Assume complete combustion, what is one typical product of incomplete combustion?

(a) butane (b) octane (c) glucose (C6H 12 O6) (d) methanol

At 25 o C the heat of combustion of 2-butanone, CH3CHOCH2CH3, is 2444.1 kJ mol -1 and the heat of combustion of butanal, CHOCH2CH2CH3, is 2470.3 kJ mol -1 . Which combustion is more endothermic? What does this tell us about the relative stability of the two compounds?

Often times NBS, N-bromosuccinimide, is substituted for Br2 in radical halogenation reactions to keep the concentration of Br2 low. This low concentration favors the radical halogenation and not other alkene reactions, that you will learn later on. Using NBS, drawn below, draw the radical bromination of propene.

N-bromosuccinimide

Peroxides are also good initiators for radical reactions. Given the peroxide, RO-OR, draw the initiation step of the general peroxide and the propagation of that radical to create bromine radicals with HBr.

One radical inhibitor BHT, butylated hydroxytoluene, is often added to diethylether, CH3CH2OCH2CH3, to prevent explosive peroxides from forming. Given BHT's structure below, draw the radical formation on the oxygen of BHT with a general radical, R . . The phenoxy radical is stabilized/made unreactive by steric hindrance of the tertbutyl groups and also resonance.

Draw the two radical halogenation products of 2-methylpropane, also known as isobutane, with Cl2. Given the percentages of the two products are 37% and 63%, calculate the reactivity of each hydrogen and the ratio between the two.

Given the following reactions and the relative &DeltaH o , what is most likely the rate limiting step of the radical halogenation mechanism?

1) X-X &rarr 2X . &DeltaH1 o >0

3) R-CH2 . + X . &rarr R-CH2X &DeltaH3 o <0

Explain or use a mechanism to show how BHT, or other radical inhibitors, could cause a radical halogenation to come to a stop?

Given the following heats of combustion propane &DeltaH= -2202 kJ/mol, gasoline &DeltaH=-44,000 kJ/kg, diesel &DeltaH=-45,000 kJ/kg what fuel could potentially provide the most miles per weight of fuel?

Write out the reactions for the complete combustion of propane and gasoline, assuming gasoline is completely C8H18 (gasoline is actually a mixture of smaller and larger hydrocarbons).

Draw the transition states of the reaction of Cl . and Br . with a secondary carbon, compare the radical character of that carbon between the two transition states.

CFC's, chlorofluorocarbons (CF2Cl2), saw widespread use as refrigerants and as aerosols. However, due to their low boiling point, when sprayed into the air they rise up to the atmosphere and are exposed to large amounts of uv radiation. This light, often seen as hv in chemical equations, provides the energy needed to break the CFC's into radicals. Chlorine radicals are formed and this degrades the protective layer of ozone into oxygen, O2. Draw the formation of the chlorine radical from a CFC.

N-bromosuccinimide, NBS, is a reagent with what purpose?

(a) a strong base (b) a strong acid (c) radical initiator (d) radical inhibitor

Name the following compound.

A polymer scientist is trying to perform a halogenation on the following polymer to perform a subsequent reaction (the structure repeats itself to form a length of n units). She/he has been using Cl2 to perform the radical halogenation, but cannot obtain a uniformly halogenated product. What suggestion would you give her/him to try and achieve a more uniformly halogenated product?

Given that a radical halogenation with chlorine yields an early transition state, as defined in Hammond's Postulate, is the halogenation likely to be endothermic or exothermic?

### Solutions

(a) (b) (c)

There can be no quaternary Hydrogen because a Carbon can only have four covalent bonds. In (b) the central carbon has no attached Hydrogen, but it is a quaternary carbon.

(a) left- tertiary right-secondary the left is more favorable (b) left-primary right-tertiary the right is more favorable (c) left- primary right- secondary the right is more favorable

For all of theses cases the more favorable radical is as follows 3 o >2 o >1 o . Since the radical is electron deficient, the more carbon substituents the more hyperconjugation that can occur. This stabilizes the radical, this "rule" is also true for carbocations as you will see later.

Make sure to draw single electron arrows. Also, this is homolytic bond cleavage because one electron goes to each atom involved in bonding.

The stability of radicals is as follows: 3 o >2 o >1 o >CH3, and so the second pair of radicals is most likely to form. A secondary and a primary radical are formed, compared to the other products that contain a methyl radical.

&DeltaH o =(&DeltaH o Bonds Broken)-(&DeltaH o Bonds Formed)

(g) H2 + Cl2 &rarr 2HCl &DeltaH o =(H-H + Cl-Cl)-2(H-Cl)=(435+242)-2(431)=-185 kJ mol -1

(a) ethane bromethane (b) 2,2-dimethylpropane 1-bromo-2,2-dimethylpropane (c) 1-methylcyclobutane (bromomethyl)cyclobutane, 1-bromo-1-methylcyclobutane, 1-bromo-2-methylcyclobutane, 1-bromo-3-methylcyclobutane

If the reaction were performed with Cl2 the product formation would not likely be as selective. This is due to the fact that the radical formation with chlorine is exothermic and the radical formation with bromine is endothermic. Hammond's postulate explains that the transition state of exothermic reaction will be more similar to the reactants, less like a radical, leading to a less selective radical formation. Whereas the transition state of an endothermic reaction will be more similar to the products, more radical like, leading to a selective radical formation.

The statistical distribution is calculated from the following: Total Carbons=3, Two external carbons are not unique, so probability=2/3 x 100% The center carbon probability is 1/3 x 100%. However, this is not likely, because the secondary carbon forms a much more stable radical than a primary radical.

As you'll learn, this is not the only reaction that can occur. Alkene chemistry will be discussed later on.

Since the propene radical is more favorable than the propane radical, the ∆H o would be lower than that of propane.

An early transition state resembles the reactants, and a late transition state resembles the products (Hammond's postulate). The formation of the alkyl radical, propagation, is an early transition state (if you draw the reaction coordinate diagram for this step it should be endothermic), and the formation of the alkyl halide, termination, is a late transition state (if you draw the reaction coordinate diagram for this step it should be exothermic).

(a) (b)

(c) (d) Notice how the chlorination is not as selective as the bromination.

(e)

Assuming there is no primary chlorination, which in reality is likely not the case, only 2 o and 3 o hydrogen atoms are taken into account.

# 2 o =2 Multiply by reactivity=Relative Yield => 8 Ratio= 8/(5+8) x 100%=62% 2 o

# 3 o =1 Multiply by reactivity=Relative Yield => 5 Ratio=5/(5+8) x 100%=38% 3 o

27.d produces a much more pure mixture of products, and thus is more likely to be useful. Whereas, the chlorination, 27.c, of the same alkane potentially yields a mixture of products. This mixture would require a difficult separation, and is much less likely to be synthetically useful.

However, if a different haloalkane were desired, then this could be useful after purification. As you see later on, though, there are other methods of creating haloalkanes that are more selective.

The problem with this is that there are several secondary carbons, all about as equally likely to form a radical. There are three groups of unique hydrogens on the molecule, the two primary, the adjacent secondary, and the most interior secondary. Statistically speaking the inner two and second most outer two are equally likely to be brominated, there are equal number of hydrogens. If however, the second carbon is brominated, the following is the most favorable conformation.

Since the methyl group is much larger than the bromine, the rest of the carbon chain is adjacent to the bromine to reduce the steric interaction. It is also anti to the methyl group to further reduce steric interaction.

This problem could be broken up into the individual steps of the mechanism, and the &DeltaH o values for each step could be summed together to get the same value.

&DeltaH o =Bonds Broken - Bonds Made = (Br-Br + Secondary C-H) - (Secondary C-Br + H-Br) = (46 + 98.5) - (71 + 87)= -13.5 kcal mol -1

&DeltaH o =Bonds Broken - Bonds Made = (I-I + Secondary C-H) - (Secondary C-I + H-I) = (36 + 98.5) - (56 + 71)= +7.5 kcal mol - 1

The main difference between the two is that the bromination is exothermic, whereas the iodination is endothermic. This explains why no reaction occurs in 27.b

The connectivity to a chlorine atom produces a highly polarized bond. This chlorination has further reactivity as you will see later on, taking advantage of the electron deficient carbon.

With any complete combustion reaction the only products are CO2 and H2O.

If any incomplete combustion occurred, carbon monoxide, CO, would be produced.

Butanal's combustion is more exothermic than that of 2-butanone. The heats of combustion tell us that 2-butanone is more stable, as it has a lower heat of combustion an thus less potential combustion energy in the compound.

Total number of abstractable Hydrogens=(3x3)+1=10

Relative Reactivity=[(Percentage)/(Number of Hydrogens)]

Tertiary Reactivity = 63/1=63

The ratio between the two is 63/4.1=15

The most endothermic step, most postive value of &DeltaH o , is most likely to be the rate limiting step. So, (2) is the rate limiting step.

The inihibitor stops interrupts the propagation step, effectively terminating the reaction. If a carbon or halogen radical does form, it will react with the BHT to form the radical in problem 37. The phenoxy radical in BHT is unreactive, and the reaction stops there.

First, convert the 2202 kJ/mol to kJ/kg.

(-2202 kJ/mol) x (1 mol/44.1 g) x (1000g / 1kg) = -49,931 kJ/mol

Since propane has the largest, most negative, heat of combustion it is likely to yield the best mileage. It, and natural gas, is commonly used as a fuel in buses.

The transition state with the chlorine radical is considered an early transition state and more similar to the reactants. The transition state with the bromine radical is considered a late transition state and more similar to the products.

(c) NBS is a radical initiator.

As you may have seen in previous problems, radical halogenation with chlorine is less selective than with bromine. A good suggestion would be to use Br2, as it will almost exclusively be added to the tertiary carbon if used in a 1:1 stoichiometric ratio. Wheras, the chlorine will also be added to the secondary carbons, thus producing a non-uniform product.

From Hammond's Postulate, an early transition state indicates an endothermic reaction.

Review Insight on Society: Online Privacy: Facebook and the Age of Privacy on pages 40–41, and in particular, the types of personal data that Facebook collects. Then visit Yahoo!, and try to identify what information Yahoo can collect via its various online offerings. Prepare a short report or presentation summarizing your findings.

Review Insight on Business: Start-up Boot Camp on pages 33&ndash34. Then go online and pick a company mentioned in the case or find another Y Combinator-supported company of your choosing, and find out more about how that company has progressed since "graduating." Is the company publicly traded and if so, are their financials strong? If not, are they expected to go public soon? Describe the company’s business model, audience size, and growth prospects, and explain why the company would have been a good fit for an incubator like Y Combinator.

## Q3.41a

On what basis is the 0 value for standard enthalpy of formation assigned to a substance at a specific temperature? Base on that, identify which substances have &DeltafH° = 0 and which do not at 1 atm and 25°C (give explanation for those that do not).

### S3.41a

0 value for standard enthalpy of formation is assigned to elemental substances that are in their natural/standard physical state at that specific condition of temperature and pressure.

1. H2(g), Al(s), Cl2(g) are substances with &DeltafH° = 0 kJ/mol
2. O3(g) is an elemental substance but does not occur naturally at 25°C, O2(g) is.
3. Br2(g) is also not the standard state of bromine. At STP, the physical state of bromine is liquid
4. NaCl does not have a 0 value because it is not an elemental substance

### Q3.41b

Which of the following substances has a standard enthalpy of formation, at 298 K, of 0 K?

### Q3.41c

Why is the standard enthalpy of formation of O2 zero at a temperature of 298 K?

### S3.41c

The standard enthalpy of formation of O2 is zero at 298 K since O2 is the most stable allotropic form of oxygen at that particular temperature.

## Exercises: Set B

1. Plantwide Versus Department Allocations of Overhead: Service Company. Chan and Associates provides wetlands design and maintenance services for its customers, most of whom are developers. Billing is based on costs plus a 30 percent markup. Thus costs are allocated to customers rather than to products.

1. Assume Chan and Associates uses the plantwide approach to allocating overhead costs and direct labor costs as the allocation base. Calculate the predetermined overhead rate, and explain how this rate will be used to allocate overhead costs. Round results to the nearest cent.
2. Assume Chan and Associates uses the department approach for allocating overhead costs. Calculate the predetermined overhead rate for each department, and explain how these rates will be used to allocate overhead costs. Round results to the nearest cent.
3. What are two possible interpretations of the term costs in the following statement? &ldquoCustomers are billed based on costs plus a 30 percent markup.&rdquo
1. Computing Product Costs Using Activity-Based Costing. Petrov Company identified the following activities, estimated costs for each activity, and identified cost drivers for each activity for this coming year. (These are the first three steps of activity-based costing.)

The company produces two products, MX1 and MX2. Information about these products for the month of March follows:

Actual cost driver activity levels for the month of March are as follows: