Download Circular Motion HW-1

AP Physics-1
Simple Harmonic Motion Review
Read Text Sections 8.3, 14.1, 14.2, 14.3, 14.4, & 14.5
Conceptual Questions (Answers to these concept questions will be posted on my website.)
1. A block of mass m and speed v collides with a spring, compressing it a distance x. How does x change if (a) v
is doubled, or (b) m is doubled?
2. A work W0 is required to stretch a certain spring 2 cm from its equilibrium position. (a) How much work is
required to stretch the spring 1 cm from equilibrium? (b) Suppose the spring is already stretched 2 cm from
equilibrium. How much additional work is required to stretch it to 3 cm from equilibrium?
3. A mass is attached to the bottom of a vertical spring. This causes the spring to stretch and the mass to move
downward. Does the potential energy of the spring increase or decrease? Does the gravitational potential energy
of the mass increase or decrease?
4. When a mass is placed on top of a vertical spring, the spring compresses and the mass moves downward.
Analyze this system in terms of mechanical energy.
5. A basketball player dribbles a ball with a steady period of T seconds. Is the motion of the ball periodic?
Is it simple harmonic motion? Explain.
6. A person rides on a Ferris wheel that rotates with constant angular speed. If the Sun is directly overhead, does
the person’s shadow on the ground undergo periodic motion? Does it undergo simple harmonic motion?
Explain.
7. An air-track cart bounces back and forth between the two ends of an air track. Is this motion periodic?
Is it simple harmonic motion? Explain.
8. A mass moves in simple harmonic motion with amplitude A and period T. How long does it take for the mass to
move a distance 2A? A distance 3A?
9. A mass moves in simple harmonic motion with amplitude A and period T. How far does it move in the time T?
In the time 5T/2?
10. A mass on a spring oscillates with simple harmonic motion of amplitude A. If the mass is doubled, but the
amplitude remains the same, does the total energy (i) increase, (ii) decrease, or (iii) stay the same? Explain.
11. A mass on a spring oscillates with simple harmonic motion of amplitude A about the equilibrium position x  0.
Its maximum speed is v max and its maximum acceleration is a max . (a) What is the speed and acceleration of the
mass at x  0? (b) What is the speed and acceleration of the mass at x  A ?
12. An object undergoes simple harmonic motion with a period T. In the time 3T/2 the object moves through a
distance 12D. What is the object’s amplitude of motion expressed in terms of D?
13. If the amplitude of a simple harmonic oscillator is doubled, by what factor do the following quantities change:
(a) angular frequency, (b) frequency, (c) period, (d) maximum speed, (e) maximum acceleration, (f) total
mechanical energy?
14. If a mass m and a mass 2m oscillate on identical springs with identical amplitudes, they both have the same
maximum kinetic energy. How can this be? Shouldn’t the larger mass have more kinetic energy? Explain.
15. An object oscillating with simple harmonic motion completes a cycle in a time T. If the object’s amplitude is
doubled, the time required for one cycle is still T, even though the object covers twice the distance. How can this
be? Explain.
16. An object moves with simple harmonic motion. If the period of motion is doubled (by quadrupling the mass),
how do the following quantities change: (a) angular frequency, (b) frequency, (c) maximum speed, (d)
maximum acceleration, (e) total mechanical energy.
17. The position of an object undergoing simple harmonic motion is given by x = Acos(Bt). Explain the physical
significance of the constants A and B. What is the frequency of this object’s motion in terms or A and/or B?
18. The velocity of an object undergoing simple harmonic motion is given by v = –Csin(Dt) Explain the physical
significance of the constants C and D. (a) What is the amplitude of this object’s motion in terms of C and/or D?
(b) What is the period of this object’s motion in terms of C and/or D?
19. An old car with worn-out shock absorbers oscillates with a given frequency when it hits a speed bump. If the
driver adds a passenger to the car, does its frequency of oscillation (i) increase, (ii) decrease, or (iii) remain the
same? Explain.
20. A grandfather clock keeps correct time at sea level. If the clock is taken to the top of a nearby mountain, would
you expect it to (i) keep correct time, (ii) run slow, or (iii) run fast? Explain.
21.
The pendulum bob in Figure 1 leaks sand onto the strip chart. What effect does this loss of sand have on the
period of the pendulum? Explain.
22.
23.
A pendulum of length L has a period T. How long must the pendulum be if its period is to be 2T?
A pendulum of length L is suspended from the ceiling of an elevator. When the elevator is at rest, the period of
the pendulum is T. Does the period increase, decrease, or remain the same when the elevator (a) moves upward
with constant speed or (b) moves downward with constant speed? Explain.
A pendulum of length L is suspended from the ceiling of an elevator. When the elevator is at rest the period of
the pendulum is T. Does the period increase, decrease, or remain the same when the elevator (a) moves upward
with constant acceleration or (b) moves downward with constant acceleration? Explain.
A mass m is suspended from the ceiling of an elevator by a spring of force constant k. When the elevator is at
rest, the period of the mass is T. Does the period increase, decrease, or remain the same when the elevator (a)
moves upward with constant speed or (b) moves downward with constant speed? Explain.
A mass m is suspended from the ceiling of an elevator by a spring of force constant k. When the elevator is at
rest, the period of the mass is T. Does the period increase, decrease, or remain the same when the elevator (a)
moves upward with constant acceleration or (b) moves downward with constant acceleration? Explain.
The two blocks in Figure 2 have the same mass. When set into oscillation, is the period of block 1 (i) greater
than, (ii) less than, or (iii) the same as the period of block 2? Explain.
A mass oscillates on a vertical spring with a period T. If this system is taken to the Moon, does the period of
oscillation (i) increase, (ii) decrease, or (iii) stay the same? Explain.
A pendulum oscillates with a period T. If this system is taken to the Moon, does the period of oscillation (i)
increase, (ii) decrease, or (iii) stay the same? Explain.
Figure 1
24.
25.
26.
27.
28.
29.
Figure 2
Problems
1. It takes 130 J of work to compress a certain spring 0.10 m. (a) What is the force constant of this spring? (b) To
compress the spring an additional 0.10 m, does it take 130 J, more than 130 J, or less than 130 J? Verify your
answer with a calculation.
2. A 6.5-kg block rests on a rough, horizontal surface with a coefficient of static friction equal to 0.42. A spring
with a force constant k  1200 N / m is attached to one side of the block. (a) How much work does it take to
stretch the spring to the point where the block is on the verge of moving? (b) Once the block is in motion with
constant speed, do you expect the stretch of the spring to be more than, less than, or the same as it was just
before the block began to move? Explain.
3. A 2.2-kg mass is attached to a spring with a force constant of 560 N/m on a smooth, horizontal surface. (a) Find
the maximum compression and expansion of the spring if the maximum speed of the mass is 1.7 m/s. (b) By
what factor do your answers to part (a) change if the force constant of the spring is doubled?
4. A 1.4-kg block is pushed up against a spring whose force constant is 650 N/m, compressing it 6.1 cm. When the
block is released it moves horizontally without friction until it launches off the edge of a table 0.65 m above the
ground. Find the horizontal distance, d, covered by the block during its fall to the ground.
5. A backpack full of books weighing 52.0 N rests on a table in a physics laboratory classroom. A spring with a
force constant of 150 N/m is attached to the backpack and pulled horizontally, as indicated in Figure 3. (a) If the
spring is pulled until it stretches 2.00 cm and the pack remains at rest, what is the force of friction exerted on the
backpack by the table? (b) Does your answer to part (a) change if the mass of the backpack is doubled? Explain.
Figure 3
Figure 4
6. A spring with a force constant of 120 N/m is used to push a 0.27-kg block of wood against a wall, as shown in
Figure 4. (a) Find the minimum compression of the spring needed to keep the block from falling, given that the
coefficient of static friction between the block and the wall is 0.46. (b) Does your answer to part (a) change if the
mass of the block of wood is doubled? Explain.
7.
8.
The position of a mass on a spring is given by x  (6.5 cm) cos[2t /(0.88 s)]. (a) What is the period of this
motion? (b) Where is the mass at t  025
. s?
A mass attached to a spring oscillates with a period of 3.15 s. (a) If the mass starts from rest at x  00440
.
m
and time t  0, where is it at time t  637
. s ? (b) Is the mass moving in the positive or negative x direction at
t  637
. s ? Explain. (Hint: Calculate the velocity at t  637
. s? )
9.
A child rocks back and forth on a porch swing with an amplitude of 0.204 m and a period of 2.80 s. Assuming
the motion is approximately simple harmonic, find the child’s maximum speed.
10.
The pistons in an internal combustion engine undergo a motion that is approximately simple harmonic. If the
amplitude of motion is 3.5 cm, and the engine runs at 1700 rev/min, find (a) the maximum acceleration of the
pistons and (b) their maximum speed.
11.
When a 0.50-kg mass is attached to a vertical spring, the spring stretches by 15 cm. How much mass must be
attached to the spring to result in a 0.75-s period of oscillation?
12.
A spring with a force constant of 65 N/m is attached to a 0.50-kg mass. Assuming that the amplitude of motion
is 3.1 cm, determine the following quantities for this system: (a)  , (b) v max , (c) T.
13.
When a 0.213-kg mass is attached to a vertical spring it causes the spring to stretch a distance d, where it
reaches equilibrium. If the mass is now displaced slightly from equilibrium, it is found to make 102 oscillations
in 56.7 s. Find the stretch distance, d.
Energy Conservation in Oscillatory Motion
14. A 0.321-kg mass is attached to a spring with a force constant of 12.3 N/m. If the mass is displaced 0.256 m
from equilibrium and released, what is its speed when it is 0.128 m from equilibrium?
15.
A bunch of grapes is placed in a spring scale at a supermarket. The grapes oscillate up and down with a period
of 0.45 s, and the spring in the scale has a force constant of 650 N/m. What are (a) the mass and (b) the weight
of the grapes? (c) If the amplitude of oscillation is 2.3 cm, what is the maximum speed of the grapes?
16.
A 0.540-kg block slides on a frictionless, horizontal surface with a speed of 1.13 m/s. The block encounters an
unstretched spring and compresses it 25 cm before coming to rest. (a) What is the force constant of this spring?
(b) How long is the block in contact with the spring before it comes to rest?
17.
A 10.0-g bullet embeds itself in a 0.500-kg block, which is attached to a spring of force constant 36.0 N/m. If
the maximum compression of the spring is 1.50 cm, find (a) the initial speed of the bullet and (b) the time for
the bullet-block system to come to rest.
The Simple Pendulum
18. A large, simple pendulum is on display in the lobby of the United Nations building. If the pendulum is 10.0 m
in length, what is the least amount of time it takes for the bob to swing from a position of maximum
displacement to the equilibrium position of the pendulum?
(Assume that the acceleration of gravity is g = 9.81 m/s2 at the UN building.)
19.
Find the length of a simple pendulum that has a period of 1.00 s.
Assume that the acceleration of gravity is g = 9.81 m/s2.
20.
If the pendulum in the previous problem were to be taken to the Moon, where the acceleration of gravity is g/6,
(a) would its period increase, decrease, or stay the same? (b) Check your result in part (a) by calculating the
period of the pendulum on the Moon.
Answers
1. (a) k = 26 kN/m
2. (a) W = 0.30 J
3. (a) x = 0.11 m
(b) Three times more work: Wadditional = 390 J
(b) Less, because kinetic friction is usually smaller than static friction.
(b) x2  1
x1
2
4.
5.
6.
7.
8.
d = dx = 47.9 cm
(a) F = 3.0 N (b) No, the force is simply equal and opposite to the force exerted by the stretched spring.
(a) x = 4.79 cm (b) Yes, the spring displacement is proportional to the block’s mass.
(a) T = 0.88 s (b) x = –1.38 cm
(a) x = 0.0436 m (b) Negative x direction. At t = 6.30 s (= 2T) the mass was at its largest displacement in the positive x
direction. Then 0.07 s later it is headed toward the origin.
9. vmax = 0.458 m/s
10. (a) amax = 1100 m/s2
(b) vmax = 6.2 m/s
11. m = 0.465 kg
12. (a)  = 11.4 rad/s
(b) vmax = 0.353 m/s (c) T = 0.551 s
13. d = 7.67 cm
14. v = 1.37 m/s
15. (a) m = 3.33 kg (b) Weight = mg = 32.6 N (c) vmax = 0.321 m/s
16. (a) k = 11.0 N/m (b) t = 0.348 s
17. (a) v0 = 6.43 m/s (b) t = 0.187 s
18. t = 1.59 s
19. L = 24.8 cm
20. (a) Cutting g by a factor of
(b) TMoon = 2.45 s
1 means increasing T by a factor of
6 , so the period would increase.
6