Question

The equation of motion of a particle is given by \( x = a \sin \left( 50t + \frac{\pi}{3} \right) \, \text{cm}. \) The particle will come to rest at time \( t_1 \) and it will have zero acceleration at time \( t_2 \). The \( t_1 \) and \( t_2 \) respectively are _______.

• A. \( \frac{\pi}{300} \, \text{s}, \, \frac{\pi}{75} \, \text{s} \)
• B. \( \frac{\pi}{300} \, \text{s}, \, \frac{\pi}{100} \, \text{s} \)
• C. \( \frac{\pi}{300} \, \text{s}, \, \frac{\pi}{25} \, \text{s} \)
• D. \( \frac{\pi}{75} \, \text{s}, \, \frac{\pi}{25} \, \text{s} \)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The given equation describes Simple Harmonic Motion (SHM).
The particle comes to rest when its velocity is zero (at the extreme positions).
The particle has zero acceleration when it passes through the mean position.
We find velocity and acceleration by taking the first and second derivatives of the position function with respect to time.
Step 2: Key Formula or Approach:
Velocity \(v = \frac{dx}{dt}\).
Acceleration \(A = \frac{dv}{dt}\).
Set \(v = 0\) to find \(t_1\) and \(A = 0\) to find \(t_2\).
Step 3: Detailed Explanation:
Given \(x = a \sin(50t + \pi/3)\).
Differentiate to find velocity:
\[ v = \frac{d}{dt} [a \sin(50t + \pi/3)] = 50a \cos(50t + \pi/3) \] The particle comes to rest when \(v = 0\):
\[ 50a \cos(50t_1 + \pi/3) = 0 \implies \cos(50t_1 + \pi/3) = 0 \] The first positive time this occurs is when the phase equals \(\pi/2\):
\[ 50t_1 + \frac{\pi}{3} = \frac{\pi}{2} \] \[ 50t_1 = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} \] \[ t_1 = \frac{\pi}{6 \times 50} = \frac{\pi}{300} \text{ s} \] Now, differentiate velocity to find acceleration:
\[ A = \frac{d}{dt} [50a \cos(50t + \pi/3)] = -2500a \sin(50t + \pi/3) \] The acceleration is zero when \(A = 0\):
\[ -2500a \sin(50t_2 + \pi/3) = 0 \implies \sin(50t_2 + \pi/3) = 0 \] The first positive time this occurs is when the phase equals \(\pi\):
\[ 50t_2 + \frac{\pi}{3} = \pi \] \[ 50t_2 = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \] \[ t_2 = \frac{2\pi}{3 \times 50} = \frac{2\pi}{150} = \frac{\pi}{75} \text{ s} \] Step 4: Final Answer:
The times \(t_1\) and \(t_2\) respectively are \(\frac{\pi}{300}\text{ s}, \frac{\pi}{75}\text{ s}\).