1. If the amplitude of a simple harmonic oscillator is doubled, what would happe
ID: 1453977 • Letter: 1
Question
1. If the amplitude of a simple harmonic oscillator is doubled, what would happen to: (a) the period of oscillation, (b) the total energy, and (c) the maximum velocity of the mass oscillating?
2. Is the acceleration of a simple harmonic oscillator ever equal to zero? If so, where would this be located?
3. When an unknown mass is attached to the bottom of a vertically hung spring the spring stretches by 30 centimeters. Determine the period of which this mass will oscillate on this spring.
4. For a simple harmonic oscillator the kinetic energy is equal to ½ mv2 , and the potential energy is equal to ½ kx2 . When the displacement is equal to one-half the amplitude of oscillation what fraction of the total energy is kinetic and what fraction of the total energy is potential?
Explanation / Answer
1) If the amplitude of a simple harmonic oscillator is doubled a) Period remain unchanged
b) Total energy will change by a factor of 4 time.
c) Max. velocity will chage 2 times.
2) the idea is that the object moves back and forth with a regular frequency, and in the case of a simple harmonic oscillator the shape of the oscillation over time is sinusoidal (i.e. a cosine or sine wave).
If you assume that it oscillates back and forth over the origin, then the position changes sign from positive to negative. Since it moves in both directions, so does its velocity. Also, since it accelerates in both directions, it has both positive and negative accelerations. Since any sine/cosine function is continuous and so are all of its derivatives, this means that at some point the oscillator must have acceleration = 0.
So then think about a sine(or cosine) wave and its derivatives. If we imagine the position of the oscillator to be represented by a sine wave, then its derivative, velocity, will be represented by a cosine wave. The derivative of velocity, acceleration, will be a sine wave, just like position.
This should make the answer pretty transparent - the acceleration is equal to zero when position is equal to zero, i.e. the harmonic oscillator has zero acceleration as it passes through its equilibrium position.
4) note that the value of the total energy equals the value of the potential energy at the maximum amplitude.
The ratio of the potential energy to the total energy is therefore equal to x^2 / A^2, where x is the given displacement.
At x = A/2
PE / Total Energy = (A/2) ^ 2 / A^2 = 1/4
KE / Total Energy = (Total Energy - PE)/Total Energy = 1 - PE/Total Energy = 3/4
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