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Oscillations and Displacement

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Equation #1 x=A[cos(2pi(t)/T)

Question #29 If the displacement is represented by equation(1) and T=5 seconds,

A.)How long does it take the object to repeat its motion, i.e, what is the period.
B.)What are the first three times when the object is at x=+A?
C.)What are the first three times when the object is at x=-A?
D.) What are the first three times when the object is at x=0?

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Solution Preview

Please see the attached file.

Note that in the answers left T as a variable. You can substitute any number into it (in your case it should be 5)

We start with the equation for motion:

We shall show that the period of this motion is ...

Solution Summary

This solution contains step-by-step calculations to determine the period of the object as well as time the object reaches the maximum, minimum and zero displacement using trigonometric functions. A diagram for motion is included for further understanding.

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Rotational Simple Harmonic Oscillation: Rod with two springs

A 1-dimensional rod 2 meters long of mass 60 grams uniformly distributed has an axle through its midpoint allowing the rod to rotate in a vertical plane. At each end of the rod are identical, ideal springs which have spring constants of 40 N/cm. At equilibrium the rod lies horizontally in the vertical plane. The rod is given a slight rotational displacement of 5° about the axle through its center and then released. Assume the displacement is small enough such that the springs essentially stretch/compress in the vertical direction only and that the small-angle approximation can be used.

Determine the following:

a.) the angular frequency of the oscillation
b.) the period of the oscillation
c.) the kinetic energy of the rod 2 seconds after it is released
d.) the angle the rod is displaced from equilibrium 2 seconds after it is released

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