Solved: A force of 400 newtons stretches a spring 2

A mass weighing 4 pounds is attached to a spring whose spring constant is 16 lb/ft. What is the period of simple harmonic motion?

A 20-kilogram mass is attached to a spring. If the frequency of simple harmonic motion is 2p cycles/s, what is the spring constant k? What is the frequency of simple harmonic motion if the original mass is replaced with an 80-kilogram mass?

A mass weighing 24 pounds, attached to the end of a spring, stretches it 4 inches. Initially, the mass is released from rest from a point 3 inches above the equilibrium position. Find the equation of motion

Determine the equation of motion if the mass in Problem 3 is initially released from the equilibrium position with a downward velocity of 2 ft/s.

A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 6 inches below the equilibrium position. (a) Find the position of the mass at the times t p12, p8, p6, p4, and 9p32 s. (b) What is the velocity of the mass when t 3p16 s? In which direction is the mass heading at this instant? (c) At what times does the mass pass through the equilibrium position?

A force of 400 newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of 10 m/s. Find the equation of motion

Another spring whose constant is 20 N/m is suspended from the same rigid support but parallel to the spring/mass system in Problem 6. A mass of 20 kilograms is attached to the second spring, and both masses are initially released from the equilibrium position with an upward velocity of 10 m/s. (a) Which mass exhibits the greater amplitude of motion? (b) Which mass is moving faster at t p4 s? At p2 s? (c) At what times are the two masses in the same position? Where are the masses at these times? In which directions are the masses moving?

A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of 2 ft/s .How many complete cycles will the mass have completed at the end of 4p seconds?

A mass weighing 8 pounds is attached to a spring. When set in motion, the spring/mass system exhibits simple harmonic motion. (a) Determine the equation of motion if the spring constant is 1 lb/ft and the mass is initially released from a point 6 inches below the equilibrium position with a downward velocity of (b) Express the equation of motion in the form given in (6). (c) Express the equation of motion in the form given in (6).

A mass weighing 10 pounds stretches a spring foot. This mass is removed and replaced with a mass of 1.6 slugs, which is initially released from a point foot above the equilibrium position with a downward velocity of . (a) Express the equation of motion in the form given in (6). (b) Express the equation of motion in the form given in (6) (c) Use one of the solutions obtained in parts (a) and (b) to determine the times the mass attains a displacement below the equilibrium position numerically equal to the amplitude of motion

A mass weighing 64 pounds stretches a spring 0.32 foot. The mass is initially released from a point 8 inches above the equilibrium position with a downward velocity of 5 ft/s. (a) Find the equation of motion. (b) What are the amplitude and period of motion? (c) How many complete cycles will the mass have completed at the end of 3p seconds? (d) At what time does the mass pass through the equilibrium position heading downward for the second time? (e) At what times does the mass attain its extreme displacements on either side of the equilibrium position? (f) What is the position of the mass at t 3 s? (g) What is the instantaneous velocity at t 3 s? (h) What is the acceleration at t 3 s? (i) What is the instantaneous velocity at the times when the mass passes through the equilibrium position? (j) At what times is the mass 5 inches below the equilibrium position? (k) At what times is the mass 5 inches below the equilibrium position heading in the upward direction? 1

A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of . Find the times at which the mass is heading downward at a velocity of 3 ft/s.

Under some circumstances when two parallel springs, with constants k1 and k2, support a single mass, the effective spring constant of the system is given by k 4k1k2 (k1 k2). A mass weighing 20 pounds stretches one spring 6 inches and another spring 2 inches. The springs are attached to a common rigid support and then to a metal plate. As shown in Figure 5.1.16, the mass is attached to the center of the plate in the double-spring arrangement. Determine the effective spring constant of this system. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 2 ft/s.

A certain mass stretches one spring foot and another spring foot. The two springs are attached to a common rigid support in the manner described in Problem 13 and Figure 5.1.16. The first mass is set aside, a mass weighing 8 pounds is attached to the double-spring arrangement, and the system is set in motion. If the period of motion is p15 second, determine how much the firs mass weighs.

A model of a spring/mass system is 4x e0.1t x 0. By inspection of the differential equation only, discuss the behavior of the system over a long period of time.

A model of a spring/mass system is 4x tx 0. By inspection of the differential equation only, discuss the behavior of the system over a long period of time.

In Problems 1720 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward. x t x t x t x FIGURE 5.1.17 Graph for Problem 17

In Problems 1720 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward.tx

In Problems 1720 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward.t x

In Problems 1720 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward.t x

A mass weighing 4 pounds is attached to a spring whose constant is 2 lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 8 ft/s. Determine the time at which the mass passes through the equilibrium position. Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?

A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 5 ft/s. Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?

A 1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. Determine the equations of motion if (a) the mass is initially released from rest from a point 1 meter below the equilibrium position, and then (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 12 m/s.

In parts (a) and (b) of Problem 23 determine whether the mass passes through the equilibrium position. In each case find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?

A force of 2 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force that is numerically equal to 0.4 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. (b) Express the equation of motion in the form given in (23). (c) Find the first time at which the mass passes through the equilibrium position heading upward.

After a mass weighing 10 pounds is attached to a 5-foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire system is placed in a medium that offers a damping force that is numerically equal to the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from a point foot below the equilibrium position with a downward velocity of 1 ft/s. (b) Express the equation of motion in the form given in (23). (c) Find the times at which the mass passes through the equilibrium position heading downward. (d) Graph the equation of motion

A mass weighing 10 pounds stretches a spring 2 feet. The mass is attached to a dashpot device that offers a damping force numerically equal to b (b  0) times the instantaneous velocity. Determine the values of the damping constant b so that the subsequent motion is (a) overdamped, (b) critically damped, and (c) underdamped.

A mass weighing 24 pounds stretches a spring 4 feet. The subsequent motion takes place in medium that offers a damping force numerically equal to b (b  0) times the instantaneous velocity. If the mass is initially released from the equilibrium position with an upward velocity of 2 ft/s, show that when the equation of motion is

A mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to the instantaneous velocity. Find the equation of motion if the mass is driven by an external force equal to f(t) 10 cos 3t.

A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Find the equation of motion if the mass is driven by an external force equal to f(t) 12 cos 2t 3 sin 2t. (b) Graph the transient and steady-state solutions on the same coordinate axes. (c) Graph the equation of motion.

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t 0, an external force equal to f(t) 8 sin 4t is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity.

In Problem 31 determine the equation of motion if the external force is f(t) et sin 4t. Analyze the displacements for .

When a mass of 2 kilograms is attached to a spring whose constant is 32 N/m, it comes to rest in the equilibrium position. Starting at t 0, a force equal to f(t) 68e2t cos 4t is applied to the system. Find the equation of motion in the absence of damping.

In Problem 33 write the equation of motion in the form x(t) Asin(vt f) Be2t sin(4t u). What is the amplitude of vibrations after a very long time? t

A mass m is attached to the end of a spring whose constant is k. After the mass reaches equilibrium, its support begins to oscillate vertically about a horizontal line L according to a formula h(t). The value of h represents the distance in feet measured from L. See Figure 5.1.21. (a) Determine the differential equation of motion if the entire system moves through a medium offering a damping force that is numerically equal to b(dxdt). (b) Solve the differential equation in part (a) if the spring is stretched 4 feet by a mass weighing 16 pounds and b 2, h(t) 5 cos t, x(0) x(0) 0. (b

A mass of 100 grams is attached to a spring whose constant is 1600 dynes/cm. After the mass reaches equilibrium, its support oscillates according to the formula h(t) sin 8t, where h represents displacement from its original position. See Problem 35 and Figure 5.1.21. (a) In the absence of damping, determine the equation of motion if the mass starts from rest from the equilibrium position. (b) At what times does the mass pass through the equilibrium position? (c) At what times does the mass attain its extreme displacements? (d) What are the maximum and minimum displacements? (e) Graph the equation of motion.

In Problems 37 and 38 solve the given initial-value problem. d2x dt2 4x 5 sin 2t 3 cos 2t, x(0) 1, x(0) 1

In Problems 37 and 38 solve the given initial-value problem. d2 x dt2 9x 5 sin 3t, x(0) 2, x(0) 0 x(

(a) Show that the solution of the initial-value problem is x(t) . F0 2  2 (cos t  cos t) d2 x dt2 2 x F0 cos t, x(0) 0, x(0) 0 d2 (b) Evaluate

Compare the result obtained in part (b) of Problem 39 with the solution obtained using variation of parameters when the external force is F0 cos vt.

(a) Show that x(t) given in part (a) of Problem 39 can be written in the form . (b) If we define , show that when  is small an approximate solution is . When  is small, the frequency g2p of the impressed force is close to the frequency v2p of free vibrations. When this occurs, the motion is as indicated in Figure 5.1.22. Oscillations of this kind are called beats and are due to the fact that the frequency of sin t is quite small in comparison to the frequency of sin gt. The dashed curves, or envelope of the graph of x(t), are obtained from the graphs of (F0 2g) sin t. Use a graphing utility with various values of F0, , and g to verify the graph in Figure 5.1.22. x(t) F0 2 sin t sin t  1 2 (  ) x(t) 2F0 2  2 sin 1 2 (  )t sin 1 2 ( )t li

Can there be beats when a damping force is added to the model in part (a) of Problem 39? Defend your position with graphs obtained either from the explicit solution of the problem or from solution curves obtained using a numerical solver

(a) Show that the general solution of d2x dt2 2 dx dt 2 x F0 sin t d is where and the phase angles f and u are, respectively, defined by sin f c1 A, cos f c2 A and , . (b) The solution in part (a) has the form x(t) xc(t) xp(t). Inspection shows that xc(t) is transient, and hence for large values of time, the solution is approximated by xp(t) g(g) sin(gt u), where . Although the amplitude g(g) of xp(t) is bounded as show that the maximum oscillations will occur at the value . What is the maximum value of g? The number is said to be the resonance frequency of the system. (c) When F0 2, m 1, and k 4, g becomes . Construct a table of the values of g1 and g(g1) corresponding to the damping coefficients b 2, b 1, , and . Use a graphing utility to obtain the graphs of g corresponding to these damping coefficients. Use the same coordinate axes. This family of graphs is called the resonance curve or frequency response curve of the system. What is g1 approaching as ? What is happening to the resonance curve as ? 44. Consi

Consider a driven undamped spring/mass system described by the initial-value problem . (a) For n 2, discuss why there is a single frequency g12p at which the system is in pure resonance. (b) For n 3, discuss why there are two frequencies g12p and g2 2p at which the system is in pure resonance. (c) Suppose v 1 and F0 1. Use a numerical solver to obtain the graph of the solution of the initial-value problem for n 2 and g g1 in part (a). Obtain the graph of the solution of the initial-value problem for n 3 corresponding, in turn, to g g1 and g g2 in part (b). d2 x

Find the charge on the capacitor in an LRC-series circuit at t 0.01 s when L 0.05 h, R 2 , C 0.01 f, E(t) 0 V, q(0) 5 C, and i(0) 0 A. Determine the first time at which the charge on the capacitor is equal to zero. 46. F

Find the charge on the capacitor in an LRC-series circuit when , R 20 , , E(t) 0 V, q(0) 4 C, and i(0) 0 A. Is the charge on the capacitor ever equal to zero? In

In Problems 47 and 48 find the charge on the capacitor and the current in the given LRC-series circuit. Find the maximum charge on the capacitor. L f 5 3 h

In Problems 47 and 48 find the charge on the capacitor and the current in the given LRC-series circuit. Find the maximum charge on the capacitor. L 1 h, R 100 , C 0.0004 f, E(t) 30 V, q(0) 0 C, i(0) 2 A

Find the steady-state charge and the steady-state current in an LRC-series circuit when L 1 h, R 2 , C 0.25 f, and E(t) 50 cos t V. 5

Show that the amplitude of the steady-state current in the LRC-series circuit in Example 10 is given by E0 Z, where Z is the impedance of the circuit.

Use Problem 50 to show that the steady-state current in an LRC-series circuit when , R 20 , C 0.001 f, and E(t) 100 sin 60t V, is given by ip(t) 4.160 sin(60t  0.588). 52

Find the steady-state current in an LRC-series circuit when , R 20 , C 0.001 f, and E(t) 100 sin 60t 200 cos 40t V. 53

Find the charge on the capacitor in an LRC-series circuit when , R 10 , C 0.01 f, E(t) 150 V, q(0) 1 C, and i(0) 0 A. What is the charge on the capacitor after a long time? 54.

Show that if L, R, C, and E0 are constant, then the amplitude of the steady-state current in Example 10 is a maximum when . What is the maximum amplitude?

Show that if L, R, E0, and g are constant, then the amplitude of the steady-state current in Example 10 is a maximum when the capacitance is C 1Lg2

Find the charge on the capacitor and the current in an LC-series circuit when L 0.1 h, C 0.1 f, E(t) 100 sin gt V, q(0) 0 C, and i(0) 0 A. 57

Find the charge on the capacitor and the current in an LC-series circuit when E(t) E0 cos gt V, q(0) q0 C, and i(0) i0 A.

In Problem 57 find the current when the circuit is in resonance.