# Harmonic oscillator, conservation of momentum, gravity

1. A simple harmonic oscillator has a total energy of E. a.) Determine the kinetic and potential energies when the displacement is one half the amplitude. B.)For what value of the displacement does the kinetic energy equal the potential energy?

2.A 5.00g bullet moving with an initial speed of 400m/s is fired into and passes through a 1.00kg block. The block, initially at rest on a frictionless horixontal surface, is connected to a spring with a spring constant of 900N/m. If the block moves 5.00cm to the right after impact, find a)the speed at which the bullet emerges from the block and b) the mechanical energy lost in the collision.

3.Assume that a hole is drilled through the center of Earth. It can be shown that an object of mass m at a distance r from Earth's center is pulled toward the center of the Earth. Assume Earth has a uniform desity p. Write down Newton's law of gravitation for an object at distance r from the center of Earth , and show that the force on it is of HOoke's law form, F=-kr, with an effective force constant of k=4/3(pie)pGm, where G is the gravitational constant.

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1. A simple harmonic oscillator has a total energy of E. a.) Determine the kinetic and potential energies when the displacement is one half the amplitude. B.)For what value of the displacement does the kinetic energy equal the potential energy?

E=1/2 K (Xm)^2

Xm= amplitude

a.) Determine the kinetic and potential energies when the displacement is one half the amplitude.

Potential energy=1/2 K (Xm/2)^2= ¼ {1/2 K (Xm)^2}

Therefore Kinetic energy= ¾ {1/2 K (Xm)^2}

B.)For what value of the displacement does the kinetic energy equal the potential energy?

KE=PE= 1/2 {1/2 K (Xm)^2}

=1/2 K (Xm/square root of 2)^2

Displacement= Xm/square root of 2

2.A 5.00g bullet moving with an initial speed ...

#### Solution Summary

Answers 3 questions on Simple harmonic oscillator, Conservation of momentum, Gravitaional Attraction.

A set of problems on circular and rotational motion, centre of mass, moment of inertia, simple harmonic motion.

1. A bicycle travels 141 m along a circular track of radius 15 m. What is the angular displacement in radians of the bicycle from its starting position?

a. 1.0 rad

b. 1.5 rad

c. 3.0 rad

d. 4.7 rad

e. 9.4 rad

2. Which equation is valid only when the angular measure is expressed in radians?

See the attachment

3. A grindstone, originally rotating at 126 rad/s undergoes a constant angular acceleration so that it makes 20.0 rev in the first 8.00 s. What is its angular acceleration?

a. 0.313 rad/s2

b. 0.625 rad/ s2

c. 2.50 rad/ s2

d. 1.97 rad/ s2

e. 13.79 rad/ s2

4. A 0.254-m diameter circular saw blade rotates at a constant angular speed of 117 rad/s. What is the tangential speed of the tip of a saw tooth at the edge of the blade?

a. 29.7 m/s

b. 14.9 m/s

c. 9.46 m/s

d. 7.45 m/s

e. 2.17 m/s

5. A ball attached to a string starts at rest and undergoes a constant angular acceleration as it travels in a horizontal circle of radius 0.30 m. After 0.65 sec, the angular speed of the ball is 9.7 rad/s, what is the tangential acceleration of the ball?

a. 4.5 m/s2

b. 0.32 m/s2

c. 15 m/s2

d. 7.6 m/s2

e. 28 m/s2

6. Two motorcycles are riding around a circular track at the same angular velocity. One motorcycle is at a radius of 15 m; and the second is at a radius of 18 m. What is the ratio of their linear speeds, v2/v1?

a. 1.0

b. 0.83

c. 1.4

d. 0.71

e. 1.2

7. Which statement concerning a wheel undergoing rolling motion is true?

a. The angular acceleration of the wheel must be zero.

b. The tangential velocity is the same for all points on the wheel.

c. The linear velocity for all points on the rim of the wheel is nonzero.

d. The tangential velocity is the same for all points on the rim of the wheel.

e. There is no slipping at the point where the wheel touches the surface on which it is rolling.

8. A car is moving along a horizontal road at a constant velocity that is directed 45 degree south of east. What is the direction of the angular velocity of the wheels of the cart?

a. 45 degree south of west

b. 45 degree north of west

c. 45 degree south of east

d. 45 degree north of east

e. due east

9. Complete the following statement: When a net torque is applied to a rigid object, it always produces a

a. constant acceleration.

b. rotational equilibrium.

c. constant angular velocity.

d. constant angular momentum.

e. change in angular velocity.

10. A meter stick is pivoted at the 0.50-m line. A 3.0-kg object is hung from the 0.10-m line. Where should a 5.0-kg object be hung to achieve equilibrium?

a. 0.06-m line

b. 0.24-m line

c. 0.56-m line

d. 0.74-m line

e. A 5.0-kg object cannot be used to sustain equilibrium in this system.

11. Which of the following statements most accurately describes the center of gravity of an object?

a. It is the point where gravity acts on the object.

b. It is the point where all the mass is concentrated.

c. It must be experimentally determined for all objects.

d. It is the point on the object where all the weight is concentrated.

e. It is the point from which the torque produced by the weight of the object can be calculated.

12. Three objects are positioned along the x axis as follows: 4.4 kg at x = +1.1 m, 3.7 kg at x = -0.80 m, and 2.9 kg at x = -1.6 m. The acceleration due to gravity is the same everywhere. What is the distance from the location of the center of gravity to the location of the center of mass for this system?

a. zero meters

b. - 0.52 m

c. - 0.26 m

d. + 0.26 m

e. + 0.52 m

13. A 50 N?m torque acts on a wheel of moment of inertia 150 kg?m2. If the wheel starts from rest, how long will it take the wheel to make one revolution?

a. 0.33 s

b. 0.66 s

c. 2.4 s

d. 6.1 s

e. 10 s

14. Which one of the following statements concerning the moment of inertia I is false?

a. I may be expressed in units of kg?m2.

b. I depends on the angular acceleration of the object as it rotates.

c. I depends on the location of the rotation axis relative to the particles that make up the object.

d. I depends on the orientation of the rotation axis relative to the particles that make up the object.

e. e. Of the particles that make up an object, the particle with the smallest mass may contribute the greatest amount to I.

15. A string is wrapped around a pulley of radius 0.10 m and moment of inertia 0.15 kg?m2. The string is pulled with a force of 12 N. What is the magnitude of the resulting angular acceleration of the pulley?

a. 18 rad/s2

b. 0.13 rad/s2

c. 80 rad/s2

d. 0.055 rad/s2

e. 8.0 rad/s2

16. A 1.0-kg wheel in the form of a solid disk rolls along a horizontal surface with a speed of 6.0 m/s. What is the total kinetic energy of the wheel?

a. 9.0 J

b. 18 J

c. 27 J

d. 36 J

e. 54 J

17. A 60.0-kg skater begins a spin with an angular speed of 6.0 rad/s. By changing the position of her arms, the skater decreases her moment of inertia by 50%. What is the skater's final angular speed?

a. 3.0 rad/s

b. 4.5 rad/s

c. 9.0 rad/s

d. 12 rad/s

e. 18 rad/s

18. Gina's favorite exercise equipment at the gym consists of various springs. In one exercise, she pulls a handle grip attached to the free end of a spring to 0.80 m from the unstrained position. The other end of the spring (spring constant = 53 N/m) is held in place by the equipment frame. What is the magnitude of the force that Gina is applying to the handle grip?

a. 31 N

b. 36 N

c. 42 N

d. 54 N

e. 66 N

19. Which one of the following statements is true concerning an object executing simple harmonic motion?

a. Its velocity is never zero.

b. Its acceleration is never zero.

c. Its velocity and acceleration are simultaneously zero.

d. Its velocity is zero when its acceleration is a maximum.

e. Its maximum acceleration is equal to its maximum velocity.

20. The position of a simple harmonic oscillator is given by x(t) = (0.5m)cos(πt/3) where t is in seconds. What is the period of the oscillator?

a. 0.17 s

b. 0.67 s

c. 1.5 s

d. 3.0 s

e. 6.0 s

21. A spring required a force of 1.0 N to compress it 0.1 m. How much work is required to stretch the spring to 0.4 m?

a. 0.4 J

b. 0.6 J

c. 0.8 J

d. 2 J

e. 4 J

22. In a certain clock, a pendulum of length L1 has a period T1 = 0.95 s. The length of the pendulum is adjusted to a new value L2 such that T2 = 1.0 s. What is the ratio L2/L1?

a. 0.90

b. 0.95

c. 1.0

d. 1.1

e. 1.3

23. Complete the following sentence: Resonance occurs in harmonic motion when

a. the system is overdamped.

b. The system is critically damped.

c. The energy in the system is a minimum.

d. The driving frequency is the same as the natural frequency of the system.

e. The energy in the system is proportional to the square of the motion's amplitude.

24. A cable stretches by an amount d when it supports a crate of mass M. The cable is cut in half. What is the mass of the load that can be supported by either half of the cable if the cable stretches by an amount d?

a. M/4

b. M/2

c. M

d. 2M

e. 4M

25. A plastic box has an initial volume of 2.00 m3. It is then submerged below the surface of a liquid and its volume decreases to 1.96 m3. What is the volume strain on the box?

a. 0.02

b. 0.04

c. 0.2

d. 0.4

e. 0.98