# 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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#### Solution Preview

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.