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Question
Which of the following statements is/are true for a simple harmonic oscillator?
- Force acting is directly proportional to displacement from the mean position and opposite to it.
- Motion is periodic.
- Acceleration of the oscillator is constant.
- The velocity is periodic.
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Solution
a, b and d
Explanation:
Simple harmonic motion is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. When the system is displaced from its equilibrium position, a restoring force that obeys Hooke’s law tends to restore the system to equilibrium. As a result, it accelerates and starts going back to the equilibrium position.
An oscillation follows simple harmonic motion if it fulfils the following two rules:
- Acceleration is always in the opposite direction to the displacement from the equilibrium position.
- Acceleration is proportional to the displacement from the equilibrium position.
Let us write the equation for the SHM x = a sin(ωt + `phi`)
Clearly, it is a periodic motion as it involves since function.
Let us find velocity of the particle, `v = (dx)/(dt)`
= `d/(dt) (a sin(ωt + phi))`
= `aω cos(ωt + phi)`
Velocity is also periodic because it is a cosine function.
Now let us find acceleration, `A = (dv)/(dt)`
= `(d^2x)/(dt^2)`
= `- aω^2 sin(ωt + phi)`
The acceleration is a sine function, hence cannot be constant.
⇒ `A = - (ω^2a) sin(ωt + phi) = - ω^2x`
Force, F = Mass × Acceleration
= mA
= – mω2x
Hence, force acting is directly proportional to displacement from the mean position and opposite to it.
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