**Problems**

You can find the answers to problems marked with an asterisk at the end of the text. Note: Unless otherwise directed by your instructor, for each of the following problems select the easiest method (analytical or numerical) to solve the problem. Discuss the reasons for your choice. Your instructor might require you to use a

numerical method and might require you to check the numerical solution with the analytical solution if possible.

I> Sections 8.1, 8.2, and 8.3

1.* An object moves at a velocity vet) = 5 + 7t2 starting from the position x(2) = 5 at t = 2. Determine its position at t = 10.

2. The total distance D traveled by an object moving at velocity vet) from the time t = a to the time t = b is

D = lb Iv(t)1 dt The absolute value Iv(t)1 is used to account for the possibility that vet) might be negative. Suppose an object moves with a ‘velocity of vet) = – cosurr) for 0 ::s t ::s 1. Find the total distance traveled and the object’s location x(l) at t = 1 if x(O) = 2.

3. An object starts with an initial velocity of 3 mls at t = 0 and accelerates with an acceleration of aCt) = 51 mls2. Plot its velocity as a function of time for 0 ::s 1 ::s 5 and find the total distance the object travels in 5 s. 10. 11

4. The equation for the voltage v(t) across a capacitor as a function of time is v(t) = ~ (1′;(t)dt + Qo) •

where; (t) is the applied current and Qo is the initial charge. A certain capacitor initially holds no charge. Its’capacitance is C = 10-F. If a current ;(t) = 2[1 +sin(5/)]lO-4 A is applied to the capacitor, plot the voltage V(/) as a function of time rod> ::::I :::: 1.2 s ..

5. * Two electrons a distance x apart repel each other with a’ force k / x2, where k is a constant. Let one electron be fixed at x = O. Determine the work done by the force of repulsion in m6ving the second charge from x = 1 to x = 5.

6. A certain object’s position as a function of time is given by X(/) =61 sin 51. Plot its velocity and acceleration as functions of time for 0 ::::I ::::5.

7.* A ball was thrown vertically with a velocity v(O) mls. Its measured height as a function of time was determined to be h(/). = 61 – 4.9/2 m. Determine its initial velocity.

8. The volume of liquid in a spherical tank of radius r as a function of the liquid’s height h above the tank bottom is given by

h3 V(h) =Jrrh2 – Jr)

a. Determine the volume rate of change dV [dh with respect to height. b. Determine the volume rate of change dV [dt with respect to time.

9. A certain object moves with the velocity V(/) given in the following table. Determine the object’s position X(/) at t = 10 s if x(O) = 3.

Time (s) o 2 3 4 5 6 7 8 9 10 Velocity (m1s) 0 2 5 7 9 -I2 15 18 22 20 17

10.* A tank having vertical sides and a bottom area of 100 ft2 stores water. To fill the tank, water is pumped into the top at the rate given in the following table. Determine the water height h(t) at I = 10 min.

Time (min) o 2 3 4 5 6 7 8 9 10 Flow rate (ft3/min) 0 80 130 150 150 16Q 165 170 160 140 120

11. A cone-shaped paper drinking cup (the kind used at water fountains) has a radi us R and a height H. If the water height in the cup is h, the water volume is given by

**Suppose that the cup’s dimensions are R = 1.5 in**. and H = 4 in.

a. If the flow rate from the fountain into the cup is 2 in.3/sec, how long will it take to fill the cup to the brim?

b. If.the flow rate from the fountain into the cup is given by

2(1 – e-2t) in.3/sec, how long will it take to fill the cup to the brim?

12. A certain object has a mass of 100 kg and is acted on by a force f(t) = 500[2 – e:’ sin(57rt)] N. The mass is at rest at t = O.Determine the object’s velocity at t = 5 s.

13.* A rocket’s mass decreases as it burns fuel. The equation of motion for a rocket in vertical flight can be obtained from Newton’s law and is dv m(t) dt = T – m(t)g where T -is the rocket’s thrust and its mass as a function of time is given by m(t) = moO – rt jb). The rocket’s initial mass is mo, the burn time is b,

and r is the fraction of the total mass accounted for by the fuel.

Use the values T ;::::48,000N, mo =2200 kg, r =0.8, g =9.81 rnIs2,

and b = 40 s. Determine the rocket’s velocity at burnout.

14. The equation for the voltage v(t) across a capacitor as a function of time is v(t) = ~ (fot i(t)dt + Qo) where i(t) is the applied current and Qo is the initial charge. Suppose that C = 10-6 F and that Qo = O.Suppose the applied current is i(t) =

[0.01 +0.3e-St sin(257rt)]10-3 A. Plotthe voltage v(t)for 0::: t :::0.3 s.

15. Plot the estimate of the derivative dyjdx from the following data. Do this using forward, backward, and central differences. Compare the results. xO 2345678910 y 0 2 5 7 9 12 15 18 22 20 11

16. At a relative maximum of a curve y(x), the slope dy [dx is O.Use the following data to estimate the values of x and y that correspond to a maximum point. xO 2345678910 y0257910876810

17. Compare the performance of th-eforward, backward, and central

difference methods for estimating the derivative of the following function: y(x) = e-X sin(3x). Use 101 points from x = 0 to x = 4. Use,a random additive error of ±0.01. tions 8.4 and 8.S

**Plot the free and total response of the equation** 5y +Y = f(t)

if f(t) =0 for t < 0 and f(t) = 10 for t :::O.The initial condition is

y(O) = 5. ” 9. The equation for the voltage y across the capacitor of an RC circuit is dy ” RC dt +y = v(t) where v(t) is the applied voltage. Suppose that RC = 0.2 s and that the capacitor voltage is initially 2 V. Suppose also that the applied voltage goes {rom 0 to 10 V at t = O.Plot the voltage y(t) for 0 The following equation describes the temperature T(t) of a certain object immersed in a liquid bath of constant temperature Ti: dT lOTt + T = t;

Suppose the object’s temperature is initially T(O) = 70°F and the bath temperature is Tb = 170°C.

a. How long will it take for the object’s temperature T to reach the bath temperature? . b. How long will it take for the object’s temperature T to reach 168°F? . c. Plot the object’s temperature T(t) as a function of time.

22. * The equation of motion of a rocket-propelled sled is, from Newton’s law, mii = f -cv where m is the sled mass, f is the rocket thrust, and c is a air resistance coefficient. Suppose that m = 1000 kg and c = 500 N . slm. Suppose v(O) = 0 and f is constant for t :::O.

a. What is the form of the step response v(t)? ” b. Determine the final speed the sled will reach as a function of f. How long will it take to reach that speed?

22.* The following equation describes the motion of a mass connected to a spring with viscous friction on the surface.

my +cy +ky = f(t) where f(t) is an applied force.

a. What is the form of the free response if m = 3, c = 18, and k = 102? b. What is the formof the free response if m = 3, c = 39, and k = 120?

23. Theequation for the voltage y across the capacitor of an RC circuit is dy RC dt +y = v(t) where v(t) is the applied voltage. Suppose that RC = 0.2 s and that the capacitor voltage is initially 2 V. Suppose also that the applied voltage is vet) = 10[2 – e:’ sin(5JTt)]. Plot the voltage yet) for 0 S t S 5 s. Interpret the results in terms of the circuit’s time constant and the behavior of the applied voltage.

24. The equation describing the water height h in a spherical tank with a circular drain of area A at the bottom is ( 2)dh ~ 7T 2rh – h – = -CdAy2gh dt

Suppose the tank’s radius is r = 3 m and that the circular drain hole has a radius of 2 em. Assume that Cd =0.5 and that the initial water height is h(O) = 5 m. Use g = 9.81 mls2.

a. Use an approximation to estimate how long it takes for the tank to

empty.

b. Plot the water height as a function of time until h(t) = O.

25. The following equation describes a certain dilution process, where y(t) is the concentration of salt in a tank of fresh water to which salt brine is being added.

Suppose that y(O) = O.

a. Plot y(t) for 0 S t S 10.

b. Check your results by using an approximation that converts the

differential equation into one having constant coefficients.

**Sections 8.6, 8.7, and 8.8**

26. The following equation describes the motion of a certain mass connected to a spring with viscous friction on the surface

3y + 18y + 102y = f(t) where ftt) is an applied force. Suppose that f(t) = 0 for t < 0 and f(t) = 10 for t ~ O.

a. Plot y(t) for y(O) = y(O) = o.

b. Plot y(t) for y(O) =0 and y(O) =10. Discuss the effect of the nonzero initial velocity.

27. The following equation describes the motion of a certain mass connected to a spring with viscous friction on the surface

3y + 39y + 120y = f(t)

where f(t) is an applied force. Suppose that f(t) = 0 for t < 0 and

f(t) = 10 for t a. Plot yet) for y(O) = y(O) = o.

b. Plot y(t) for y(O) = 0 and y(O) = 10. Discuss the effect of the

nonzero initial velocity.

28. The following equation describes the motion of a certain mass connected to a spring with no friction

I f I 3y +75y = f(t) where f(t) is an applied force. The equation’s characteristic roots are s = ±5i, so the system’s natural frequency of oscillation is 5 radls. Suppose the applied force is sinusoidal with a frequency of to rad/s and an amplitude of 10 N: f(t) = 1Osin(wt).

. Suppose that the initial conditions are y(O) = y(O) =O. Plot yet) for 20 s. Do this for the following three cases. Compare the results

of each case.

a. w = 1 radls.

b. (J) = 5.1 rad/s. ….

c. w = 10 radls.

29. Van der Pol’s equation has been used to describe many oscillatory processes. It is

Y – JL(1 -l)y +y =0

Plot yet) for JL = 1 and 0 ::::t :::: 20, using the initial conditions y(O) = 2, y(O) =o.

30. The equation of motion for a pendulum whose base is’ accelerating horizontally with an acceleration aCt) is

LO +g sin 0 = a(t ) cos 0

Suppose that g = 9.81 rnIs2, L = 1 m, and 8(0) = O. Plot O(t) for 0

10 s for the following three cases:

a. The acceleration is constant: a = 5 rnIs2 and 0(0) = 0.5 rad.

b. The acceleration is constant: a = 5 rnIs2 and 0(0) = 3 rad.

c. The acceleration is linear with time: a = 0.5t rnIs2 and 0(0) = 3 rad.

31. The equations for an armature-controlled de motor are the following. The motor’s current is i, and its rotational velocity is ca.

di . L- = -Rl – K~w+ vet) dt dca ,..

/-dt = ”’-T’ .-cw (8.6-10)

where L. R. and I are the motor’s inductance, resistance, and inertia; K T and K; are the torque constant and back emf constant; c is a viscous damping constant; and v(t) is the applied voltage.

Use the values R = 0.8 n, L = 0.003 H, KT = 0.05 N· miA,

Ke =0.05 V . s/rad, c = 0, and I = 8 x 10-5 kg· m2.

a. Find the motor’s characteristic roots and time constants. If the applied voltage is constant, approximately how long will it take for the motor to reach a constant speed?

b. Suppose the applied voltage is 20 V. Plot the motor’s speed and

current versus time. Choose a final time large enough to show the

motor’s speed becoming constant. c. Suppose the applied voltage is trapezoidal as given here:

{ 400t 20 . v(t) = ~400(t – 0.2) +20

0::: t < 0.05

0.05 ::: t ::: 0.2

0.2 < t ::: 0.25

t > 0.25

Plot the motor’s speed versus time for 0 ::: t :::0.3 s. Also plot the

applied voltage versus time. Does the motor speed follow a:

trapezoidal profile?

32.* (Control System toolbox) Find the state-space form of the following model:

JOy + 3y + 7y = f(t)

33. (Control System toolbox) Find the state-space form of the following

model:

IOy+6y+2y=f+3j

34.* (Control System toolbox) Find the reduced form of the following state model in terms of Xl •

.[;~]= [-~ =j] [~~]+ [;] u(t)

35. (Control System toolbox) The following state model describes the motion of a certain mass connected to a spring with viscous friction on the surface, where m = 1. c = 2. and k = 5.

[~l] = [0 1][XI]+ [0]f(t)

X2 -5 -2 X2 1 ..

a. Use the ini t ial function to plot the position Xl of the mass if he

initial position is 5 and the initial velocity is 3.

b. Use the step function to plot the step response of the position and

velocity for zero initial conditions. where the magnitude of the step

input is 10. Compare your plot with that shown in Figure 8.~.

36. Consider the following equation:

5y + 2y + lOy = f(t)

a. Plot the free response for the-initial conditions y(O) = 10, )1(0)= -5.

b. Plot the unit-step response (for zero initial conditions).

c. The total response to a step input is the sum of the free response

and the step response. Demonstrate this fact for this equation by

plotting the sum of the solutions found in parts a and b and comparing the plot with that generated by solving for the total response with y(O) =10, y(O) = -5.

37. (Control System toolbox) Use the lsim function to solve the dc motor problem given in Example 8.6-2. Compare your results with those shown in Figure 8.6–6.

38. (Control System toolbox) The model for the RC circuit shown in

Figure P38 is

du.,

RCd! +vo = Vi

For RC = 0.1 s, plot the voltage response vo(t) for the case where the applied voltage is a single square pulse of height 10 V and duration 0.2 s, starting at t = O.The initial capacitor voltage is O.

39. Van der Pol’s equation is Y – JL(l -i)y +y =0

This equation is stiff for large values of the parameter JL. Compare the performance of ode4 5 and ode2 3s for this,equation. Use JL = 1000 and o ~ t ~ 3000 with the initial conditions y(O) = 2, y(O) = O.Plot y(t) versus t.

40. Use MATLAB to plot the trajectory of a ball thrown at an angle of 30° to the horizontal with a speed of 30 m/s. The ball bounces off the horizontal surface and loses 20 percent of its vertical speed with each bounce. Plot the trajectory showing three bounces.