Simple Harmonic Motion

Extracted from “Physics for Scientists and Engineerning”s. P.A. Tipler, G.P. Mosca, 5th edition

A common, very important, and very basic kind of oscillatory motion is simple harmonic motion such as the motion of an object attached to a spring.

In equilibrium, the spring exerts no force on the object. When the object is displaced an amount x from its equilibrium position, the spring exerts a force kx, as given by Hooke’s law

Fx= - kx

where k is the force constant of the spring, a measure of the spring’s stiffness. The minus sign indicates that the force is a restoring force; that is, it is opposite to the direction of the displacement from the equilibrium position. Combining  the previous  equation with Newton’s second law (F = ma) we have

-kx = max

 

or  ax= - (k/m)x

The acceleration is proportional to the displacement and is oppositely directed. This is the defining characteristic of simple harmonic motion and can be used to identify systems that will exhibit it:

Whenever the acceleration of an object is proportional to its displacement and is oppositely directed, the object will move with simple harmonic motion.

CONDITIONS FOR SIMPLE HARMONIC MOTION IN TERMS OF ACCELERATION

Because the acceleration is proportional to the net force, whenever the net force on an object is proportional to its displacement and is oppositely directed, the object will move with simple harmonic motion.

The time it takes for a displaced object to execute a complete cycle of oscillatory motion from one extreme to the other extreme and back—is called the period T. The reciprocal of the period is the frequency f, which is the number of cycles per second:

f= 1/T

The unit of frequency is the cycle per second (cy/s), which is called a hertz (Hz). For example, if the time for one complete cycle of oscillation is 0.25 s, the frequency is 4 Hz.

Next figure shows how we can experimentally obtain x versus t for a mass on a spring. The general equation for such a curve is

x=A cos(wt + d)   

POSITION IN SIMPLE HARMONIC MOTION

where A, w, and d are constants. The maximum displacement xmax from equilibrium is called the amplitude A. The argument of the cosine function, (wt + d), is called the phase of the motion, and the constant d  is called the phase constant, which is the phase at t = 0. (Note that cos(wt + d) = sin(wt + d + p/2); thus, whether the equation is expressed as a cosine function or a sine function simply depends on the phase of the oscillation at the moment we designate to be t = 0.) If we have just one oscillating system, we can always choose t = 0 at which d =0. If we have two systems oscillating with the same amplitude and frequency but different phase, we can choose d = 0 for one of them. The equations for the two systems are then:  x1 = A cos(wt)   and x2 = A cos(wt + d)

If the phase difference d is 0 or an integer times 2 p then x2 = x1 and the systems are said to be in phase. If the phase difference d is p or an odd integer times p, then x2 = -x1x and the systems are said to be 180º out of phase.

 

We can show that equation x=A cos(wt + d) is a solution of equation ax= - (k/m)x by differentiating x twice with respect to time. The first derivative of x gives the velocity v:

v= dx/dt= -wAsin(wt+ d)

VELOCITY IN SIMPLE HARMONIC MOTION

 

Differentiating velocity with respect to time gives the acceleration:

A= dv/dt= -w2Acos(wt+ d)

Substituting x for A cos(wt + d) gives: a= -w2x

                                               ACCELERATION IN SIMPLE HARMONIC MOTION

Comparing a= -w2x  with a= - (k/m)x, we see that x= A cos(wt + d) is a solution of the equation:

d2x/dt2 = - (k/m)x

The constant is called the angular frequency w. It has units of radians per second and dimensions of inverse time, the same as angular speed, which is also designed by w.

The cosine (and sine) function repeats in value when the phase increases by 2p, so:

wT = 2p ® w=2p /T

Substituting in the general equation: x= Acos(2p t/T +d)

We can see by inspection that each time t increases by T, the phase increases by 2p and one cycle of the motion is completed.

Because  , the frequency and period of an object on a spring are related to the force k and the mass m by:

FREQUENCY AND PERIOD FOR AN OBJECT ON A SPRING

 

The frequency and period of simple harmonic motion are independent of the amplitude

 

Simple Harmonic Motion and Circular Motion

There is a relation between simple harmonic motion and circular motion with constant speed. When a particle moves with constant speed in a circle, its projection onto a diameter of the circle moves with simple harmonic motion.

Energy in Simple Harmonic Motion

When an object on a spring undergoes simple harmonic motion, the system´s potential energy and kinetic energy vary with time. Their sum, the total mechanical energy: E= K + U, is constant.

Consider and object a distance x from equilibrium, acted on an restoring force – kx. The system´s potential energy is : U= (1/2) kx2

For simple harmonic motion, x = A cos(wt + d). Substituting gives

U= (1/2) k A2 cos2(wt + d)

POTENTIAL ENERGY IN SIMPLE HARMONIC MOTION

The kinetic energy of the system is: K= (1/2) mv2 where m is the object´s mass and v is its speed. For simple harmonic motion  v=-wAsin(wt+ d). Substituting gives:

K= (1/2) m w2A2sin(wt+ d). Then using w2= k/m;

K= (1/2)k A2 sin2(wt+ d)

KINETIC ENERGY IN SIMPLE HARMONIC MOTION

 

The total mechanical energy is the sum of the potential and kinetic energies:

Etotal= (1/2) k A2

TOTAL MECHANICAL ENERGY IN SIMPLE HARMONIC MOTION

The total mechanical energy in simple harmonic motion is proportional to the square of the amplitude.

For an object at its maximum displacement, the total energy is all potential energy. As the object moves towards its equilibrium position, the kinetic energy of the system increases and its potential energy decreases. As it moves through its equilibrium position, the kinetic energy of the object is maximum, the potential energy of the system is zero, and the total energy is kinetic. As the object moves past the equilibrium point, its kinetic energy begins to decrease, and the potential energy of the system increases until the object again stops momentarily at its maximum displacement. At all times, the sum of the potential and kinetic energies is constant.

The Simple Pendulum

A simple pendulum consists of a string of length L and a bob of mass m. When the bob is released from an initial angle  j0  with the vertical, it swings back and forth with some period T.

The forces on the bob are its weight mg and the string tension T. At an angle j with the vertical, the weight has components mgcos j  along the string and mgsin j tangential to the circular arc in the direction of decreasing . Using tangential components, Newton´s second law (∑F= mat) gives:

- mgsin j=mat

For small j, sin j= j; so that, - mg j=mat

x = j L (arc= ang x radio; where L= Radio)

and  - mg j=- mgx/L

From the Hooke´s law: F= - kx

Where k= mw2= mg/L

Thus, the motion of a pendulum approximates simple harmonic motion for small angular displacements.

The period of the motion is thus:

 

 

Note that the mass m does not appear in the equation. The motion of a pendulum does not depend on its mass.

 

Large-Amplitude Oscillations

When the amplitude of a pendulum´s oscillation becomes large, its motion continues to be periodic, but it is no longer simple harmonic. A slight dependence on the amplitude must be accounted for when determining the period.

 ACTIVITIES

Conceptual Problems

1. What is the magnitude of the acceleration of an oscillator of amplitude A and frequency f when its speed is maximum? When its displacement from equilibrium is maximum?

 

2. Are the acceleration and the displacement (from equilibrium) of a simple harmonic oscillator ever in the same direction? The acceleration and the velocity ? The velocity and the displacement ? Explain.

 

3. True or false:

(a) For a simple harmonic oscillator, the period is proportional to the square of the amplitude.

(b) For a simple harmonic oscillator, the frequency does not depend on the amplitude.

(c) If the acceleration of a particle undergoing 1-dimensional motion is proportional to the displacement from equilibrium and oppositely directed, the motion is simple harmonic.

 

4. If the amplitude of a simple harmonic oscillator is tripled, by what factor is the energy changed?

 

5. An object attached to a spring has simple harmonic motion with an amplitude of 4.0 cm. When the object is 2.0 cm from the equilibrium position, what fraction of its total energy is potential energy?

(a) One-quarter. (b) One-third. (c) One-half. (d) Two-thirds. (e) Three-quarters.

 

6. True or false:

(a) For a given object on a given spring, the period is the same whether the spring is vertical or horizontal.

(b) For a given object oscillating with amplitude A on a given spring, the maximum speed is the same whether the spring is vertical or horizontal.

 

7. True or false: The motion of a simple pendulum is simple harmonic for any initial angular displacement.

 

8. True or false: The motion of a simple pendulum is periodic for any initial angular displacement.

 

9. The length of the string or wire supporting a pendulum bob increases slightly when its temperature is raised. How would this affect a clock operated by a simple pendulum?

 

10. The effect of the mass of a spring on the motion of an object attached to it is usually neglected. Describe qualitatively its effect when it is not neglected.

 

11. A lamp hanging from the ceiling of the club car in a train oscillates with period T when the train is at rest. The period will be (match left and right columns) with constant velocity.

1. greater than T0 when          A. the train moves horizontally constant velocity.

2. less than T0 when               B. the train rounds a curve of radius R with speed V. 

3. equal to T0 when                 C. the train climbs a hill of inclination q at constant speed.

D. the train goes over the crest of a hill of radius of curvature  R with constant speed.

12. Two mass—spring systems oscillate at frequencies fA and fB. If fA = 2fB and the spring constants of the two springs are equal, it follows that the masses are related by

(a) MA = 4MB ; (b) MA = MB/ √2 ; (c) MA = MB/2 ; (d) MA = MB/4

 

13. Two mass—spring systems A and B oscillate so that their energies are equal. If MA = 2 MB which formula relates the amplitudes of oscillation? (a) AA = AB/4 (b) AA = AB/ √2 ; (c) AA = AB ; (d) Not enough information is given to determine the ratio of the amplitudes.

 

14. Two mass—spring systems A and B oscillate so that their energies are equal. If kA = 2kB then which formula relates the amplitudes of oscillation? (a) AA = AB/4 (b) AA = AB/ √2 ; (c) AA = AB ; (d) Not enough information is given to determine the ratio of the amplitudes.

 

15. Pendulum A has a bob of mass MA and a length LA; pendulum B has a bob of mass MB and a length LB. If the period of A is twice that of B, then (a) LA = 2LB and MA = 2 MB; (b) LA = 4LB and MA = MB ; (c) LA = 4LB whatever the ratio MA/MB, (d) LA =√2 LB whatever the ratio MA/MB.

Numerical problems

 

1. You are on a boat, which is bobbing up and down. The boat´s vertical displacement y is given by:

y= 1.2 cos (t/2 + p/6)

(a) Find the amplitude, angular frequency, phase constant, frequency, and period of the motion. (b) Where is the boat at t=1s? (c) Find the velocity and acceleration as functions of time t.(d) Find the initial values of the position, velocity and acceleration of the boat.

 

2. An object oscillates with angular frequency w= 8.0 rad/s. At t=0, the object is at x=4 cm with an initial velocity of v= -25 cm/s. (a) Find the amplitude and phase constant for the motion. (b) Write x as a function of time.

 

3. Consider an object on a spring whose position is given by x= 5 cos (9.90t)  (cm). (a) What is the maximum speed of the object?. (b) When does this maximum speed first occur?. (c) What is the maximum of the acceleration of the object? (d) When does maximum acceleration first occur after t=0?

 

4. A 3 kg object attached to a spring oscillates with an amplitude of 4 cm and a period of 2s. (a) What is the total energy?. (b) What is the maximum speed of the object? (c) At what position x1 is the speed equal to half its maximum value?

 

5. The position of a particle is given by x = (7 cm) cos 6pt, where t is in seconds. What are (a) the frequency, (b) the period, and (c) the amplitude of the particle’s motion? (d) What is the first time after t=0 that the particle is at its equilibrium position? In what direction is it moving at that time?

 

6. What is the phase constant d in Equation x=A cos(wt + d) if the position of the oscillating particle at time t = 0 is (a) 0, (b) - A, (c) A, (d) A/2?

 

7.  A particle of mass m begins at rest from x = +25 cm and oscillates about its equilibrium position at x=0 with a period of 1.5 s. Write equations for (a) the position x as a function of t, (b) the velocity v as a function of t, and (c) the acceleration a as a function of t.

 

8.  Find (a) the maximum speed and (b) the maximum acceleration of the particle in Problem 5. (c) What is the first time that the particle is at x = 0 and moving to the right?

 

9. Work Problem 7 with the particle initially at x = 25 cm and moving with velocity y = +50 cm/s.

 

10.  The period of an oscillating particle is 8 s and its amplitude is 12 cm. At t = 0 it is at its equilibrium position. Find the distance travelled during the intervals (a) t = O to t = 2s, (b) t = 2s to t = 4s, (c) t = 0 to t= ls, and (d) t = is to t = 2 s.

 

11. The period of an oscillating particle is 8 s. At t = 0, the particle is at rest at x = A = 10 cm. (a) Sketch x as a function of t. (b) Find the distance travelled in the first, second, third, and fourth second after t = 0.

 

12. The position of a particle is given by x = 2.5 cos pt, where x is in meters and t is in seconds. (a) Find the maximum speed and maximum acceleration of the particle.(b) Find the speed and acceleration of the particle when x= 1.5m.

 

13 A particle moves in a circle of radius 40 cm with a constant speed of 80 cm/s. Find  (a) the frequency of the motion and (b) the period of the motion. (c) Write an equation for the x component of the position of the particle as a function of time t, assuming that the particle is on the positive X axis at time t =0.

 

14. A particle moves in a circle of radius 15 cm, making 1 revolution every 3 s. (a) What  is the speed of the particle? (b) What is its angular velocity w? (c) Write an equation for the x component of the position of the particle as a function of time t, assuming that the particle is on the positive X axis at time t = 0.

 

15. A 2.4-kg object is attached to a horizontal spring of force constant k = 4.5 kN/m. The spring is stretched 10 cm from equilibrium and released. Find its total energy.

 

16. Find the total energy of a 3-kg object oscillating on a horizontal spring with an amplitude of 10 cm and a frequency of 2.4 Hz.

17. A l.5-kg object oscillates with simple harmonic motion on a spring of force constant k = 500 N/m. Its maximum speed is 70 cm/s. (a) What is the total mechanical energy? (b) What is the amplitude of the oscillation?

 

18. A 3-kg object oscillating on a spring of force constant 2 kN/m has a total energy of 0.9 J. (a) What is the amplitude of the motion? (b) What is the maximum speed?

 

19. An object oscillates on a spring with an amplitude of 4.5 cm. Its total energy is 1.4 J. What is the force constant of the spring?

 

20. A 3-kg object oscillates on a spring with an amplitude of 8 cm. Its maximum acceleration is 3.50 m/s Find the total energy.

 

21. A 2.4-kg object is attached to a horizontal spring of force constant k = 4.5 kN/rn. The spring is stretched 10 cm from equilibrium and released. What are (a) the frequency of the motion, (b the period, (c) the amplitude, (d) the maximum speed, and (e) the maximum acceleration? (f) When does the object first reach its equilibrium position? What is its acceleration at this time?

 

22. Answer the questions in Problem 21 for a 5-kg object attached to a spring of force constant k = 700 N/m when the spring is initially stretched 8 cm from equilibrium.

 

23. A 3-kg object attached to a horizontal spring oscillates with an amplitude A = 10 cm and a frequency f = 2.4 Hz. (a) What is the force constant of the spring? (b) What is the period of the motion? (c) What is the maximum speed of the object? (d) What is the maximum acceleration of the object?

 

24. An 85 kg person steps into a car of mass 2400 kg, causing it to sink 2.35 cm on its springs. Assuming no damping, with what frequency will the car and passenger vibrate on the springs?

 

25. A 4.5-kg object oscillates on a horizontal spring with an amplitude of 3.8 cm. Its maximum acceleration is 26 m/s2. Find (a) the force constant k, (b) the frequency, and (c) the period of the motion.

 

26. An object oscillates with an amplitude of 5.8 cm on a horizontal spring of force constant 1.8 kN/m. Its maximum speed is 2.20 m/s. Find (a) the mass of the object, (b) the frequency of the motion, and (c) the period of the motion.

 

27. A 0 4 kg block attached to a spring of force constant 12 N/m oscillates with an amplitude of 8 cm. Find (a) the maximum speed of the block, (b) the speed and acceleration of the block when it is at x = 4 cm from the equilibrium position, and (c) the time it takes the block to move from x = 0 to x = 4 cm.

 

28. An object of mass m is supported by a vertical spring of force constant 1800 N/m. When pulled down 2.5 cm from equilibrium and released from rest, the object oscillates at 5.5 Hz. (a) Find m. (bi) Find the amount the spring is stretched from its natural length when the object is in equilibrium. (c) Write expressions for the displacement x, the velocity v, and the acceleration a as functions of time t.

 

29. An object of unknown mass 15 hung on the end of an unstretched spring and is released from rest. If the object falls 3.42 cm before first coming to rest, find the period of the motion.

 

30. Find the length of a simple pendulum if the period is 5 s at a location where g = 9.81 m/s

 

31. What would be the period of the pendulum in Problem 30 if the pendulum were on the moon, where the acceleration due to gravity is one-sixth that on earth?

 

32. If the period of a pendulum 70 cm long is 1.68 s, what is the value of g at the location of the pendulum?

 

33. A pendulum set up in the stairwell of a l0-story building consists of a heavy weight suspended on a 34.0-m wire. If g = 9.81 m/s what is the period of oscillation?

 

34. A particle has a displacement x = 0.4 cos (3t + p/4), where x is m meters and t is in seconds. (a) Find the frequency f and period T of the motion. (b) Where is the particle at t= 0? (c) Where is the particle at t=0.5 s?

 

35. Find an expression for the velocity of the particle whose position is given in Problem 34. (b) What is the velocity at time t = 0? (c) What is the maximum velocity? (d) At what time after t = O does this maximum velocity first occur?

 

36. An object on a horizontal spring oscillates with a period of 4.5 s. If the object is suspended from the spring vertically, by how much is the spring stretched from its natural length when the object is in equilibrium?

 

37. As your jet plane speeds down the run way on takeoff, you measure its acceleration by suspending your yo-yo as a simple pendulum and noting that when the bob (mass 40 g) is at rest relative to you, the string (length = 70 cm) makes an angle of 22C with the vertical. Find the period T for small oscillations of this pendulum.

 

38. A spring of force constant k = 250 N/rn is suspended from a rigid support. An object of mass 1 kg is attached to the unstretched spring and the object is released from rest. (a) How far below the starting point is the equilibrium position for the object? (b) How far down does the object move before it starts up again? (c) What is the period of osci lation? (d) What is the speed of the object when it first reaches its equilibrium position? (e) When does it first reach its equilibrium position?

 

39. A suitcase of mass 20 kg is hung from two bungie cords, as shown in Figure. Each cord is stretched

5 cm when the suitcase is in equilibrium. If the suitcase is pulled down a little and released, what will be its oscillation frequency?

40. A 0.12-kg block is suspended from a spring. When a small stone of mass 30 g is placed on the block, the spring stretches an additional 5 cm. With the stone on the block, the spring oscillates with an amplitude of 12 cm. (a) What is the frequency of the motion? (b) How long does the block take to travel from its lowest point to its highest point? (c) What is the net force on the stone when it is at the point of maximum up ward displacement?

 

41. In Problem 40, find the maximum amplitude of oscillation at which the stone will remain in contact with the block.

 

42. An object of mass 2.0 kg is attached to the top of a vertical spring that is anchored to the floor. The uncompressed length of the spring is 8.0 cm and the length of the spring when the object is in equilibrium is 5.0 cm. When the object is resting at its equilibrium position, it is given a down ward impulse with a hammer so that its initial speed is 0.3 m/s. (a) To what maximum height above the floor does the object eventually rise? (b) How long does it take for the object to reach its maximum height for the first time? (c) Does the spring ever become uncompressed? What minimum initial velocity must be given to the object for the spring to be uncompressed at some time?

 

43. A 2.5-kg object hanging from a vertical spring of force constant 600 N/m oscillates with an amplitude of 3 cm. When the object is at its maximum downward displacement, find (a) the total energy of the system, (b) the gravitational potential energy, and (c) the potential energy in the spring.

(d) What is the maximum kinetic energy of the object? (Choose U = O when the object is in equilibrium)

 

44. A 1.5-kg object that stretches a spring 2.8 cm from its natural length when hanging at rest oscillates with an amplitude of 2.2 cm. (a) Find the total energy of the system. (b) Find the gravitational potential energy at maximum downward displacement. (c) Find the potential energy in the spring at maximum downward displacement. (d) What is the maximum kinetic energy of the object?

 

45. A 1.2-kg object hanging from a spring of force constant 300 N/m oscillates with a maximum speed of 30 cm/s. (a) What is its maximum displacement? When the object is at its maximum displacement, find (b) the total en energy of the system, (c) the gravitational potential energy, and (d) the potential energy in the spring.