1. Consider the four equivalent ways to represent simple harmonicmotion in one dimension:
To make sure you understand all of these, show that they are equivalent by proving the following implications: I-->II--> III--> IV. For each form, given an expression for the constants (C1, C2, etc.) in terms of the constants of the previous form.

2) a) solve for the coefficients B1 and B2 of the form (II) in terms of the initial position x0 and velocity v0 at t = 0.
b) if the oscillator's mass is m = 0.5 kg and the force constnat is k = 50 N/m, what is the angular vlocity w? If x0 = 3 m and v0 = 50 m/s, what are B1 and B2? Sketch x(t) for a couple of cycles.
c) What are the earliest times at which x = 0 and at which x dot = 0?

3) consider a simple harmonic oscillator with period t. Let (f) denote the average value of any variable f(t), averaged over one complete cycle.
Prove that <T> = <U> = 1/2 E, where E is the total energy of the oscillator.

4) The potential energy of a one dimensional mass m at a distance r from the origin is
find the equilibrium position. Let x be the distance from equilibrium and show that, for small x, the PE has the form U = const + 1/2kx^2. What is the angular frequency of small oscillations?

5) The general solution for a two dimentional isotropic oscillator is given by:
Show that by changing the origin of time you can cast this in the simpler form:

A 0.53 kg mass vibrates according to the equation x = 0.49 cos 8.42t, where x is in meters, and t is in seconds.
Determine the frequency in hertz.
Determine the total energy in joules.
Determine the kinetic energyand potential energy in joules when x = 0.26 m.

The length of a simple pendulum is 0.71 m, the pendulum bob has a mass of 311 grams, and it is released at an angle of 12° to the vertical.
With what frequency (in hertz) does it vibrate? Assume simpleharmonicmotion.
What is the pendulum bob's speed (in m/s) when it passes through the lowest point of the swing?
What

Which one of the following quantities is at a maximum when an object in simpleharmonicmotion is at its maximum displacement?
a. speed
b. acceleration
c. kinetic energy
d. frequency
7. An object moving in simpleharmonicmotion has an amplitude of 0.020 m and a maximum acceleration of 40 m/s2. What is the frequency of th

A block with mass attached to a horizontal spring with force constant is moving with simpleharmonicmotion having amplitude. At the instant when the block passes through its equilibrium position, a lump of putty with mass m is dropped vertically onto the block from a very small height and sticks to it.
For this value of m, w

A 10-g particle is undergoing simpleharmonicmotion with an amplitude of 2.0 x 10^-3 m and a maximum acceleration of magnitude 8.0 x 10^-3 m/s^2. The phase angle is -pi/3 rad.
a) Write and equation for the force on the particle as a function of time.
b) What is the period of the motion?
c) What is the maximum speed of the p

Please see the attached file.
A piston executes simpleharmonicmotion with amplitude of 0.1 m. If it passes through the center of itsmotion with a speed of 0.5 m/s, what are the maximum accelerations and the period of oscillation?

A block of mass m is attached to a spring whose spring constant is k. The other end of the spring is fixed so that when the spring is unstretched, the mass is located at x=0. Assume that the +x direction is to the right.
The mass is now pulled to the right a distance A beyond the equilibrium position and released, at time t=0

A 1.73 x 10-2-kg block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is 123 N/m. The block is shoved parallel to the spring axis and is given an initial speed of 8.69 m/s, while the spring is initially unstrained. What is the amplitude of the resulting simple harmoni

An object executing simpleharmonicmotion has a maximum speed of 4.3 m/s and a maximum acceleration of 0.65 m/s^2.
Find (a) the amplitude and (b) the period of this motion.