The ratio of maximum acceleration to the maximum velocity of a particle performing S.H.M. is equal to

Options

A. amplitude
B. angular velocity
C. square of amplitude
D. square of angular velocity

Answer

To find the ratio of maximum acceleration to maximum velocity of a particle performing Simple Harmonic Motion (S.H.M.), we can follow these steps:

### Step 1: Understand the equations of S.H.M.
The displacement \( y \) of a particle in S.H.M. can be expressed as:
\[ y = A \sin(\omega t) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time.

### Step 2: Find the maximum velocity.
The velocity \( v \) of the particle is the derivative of displacement with respect to time:
\[ v = \frac{dy}{dt} = \frac{d}{dt}(A \sin(\omega t)) = A \omega \cos(\omega t) \]
The maximum velocity \( v_{max} \) occurs when \( \cos(\omega t) = 1 \):
\[ v_{max} = A \omega \]

### Step 3: Find the maximum acceleration.
The acceleration \( a \) is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} = \frac{d}{dt}(A \omega \cos(\omega t)) = -A \omega^2 \sin(\omega t) \]
The maximum acceleration \( a_{max} \) occurs when \( \sin(\omega t) = 1 \):
\[ a_{max} = A \omega^2 \]

### Step 4: Calculate the ratio of maximum acceleration to maximum velocity.
Now, we can find the ratio of maximum acceleration to maximum velocity:
\[ \text{Ratio} = \frac{a_{max}}{v_{max}} = \frac{A \omega^2}{A \omega} = \frac{\omega^2}{\omega} = \omega \]

### Conclusion:
Thus, the ratio of maximum acceleration to maximum velocity of a particle performing S.H.M. is equal to \( \omega \).

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