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

We are given the equation of motion: \[ x = a \sin \left( 50t + \frac{\pi}{3} \right). \] At time \( t_1 \), the particle comes to rest, which means the velocity is zero. The velocity is given by the time derivative of \( x \): \[ v = \frac{dx}{dt} = 50a \cos \left( 50t + \frac{\pi}{3} \right). \] For the particle to come to rest, we need \( v = 0 \), so: \[ \cos \left( 50t + \frac{\pi}{3} \right) = 0. \] This occurs when: \[ 50t + \frac{\pi}{3} = \frac{\pi}{2}, \quad \text{or} \quad t_1 = \frac{\pi}{300}. \] At time \( t_2 \), the particle has zero acceleration. The acceleration is the time derivative of the velocity: \[ a = \frac{dv}{dt} = -50^2 a \sin \left( 50t + \frac{\pi}{3} \right). \] For zero acceleration, we need: \[ \sin \left( 50t + \frac{\pi}{3} \right) = 0. \] This occurs when: \[ 50t + \frac{\pi}{3} = \pi, \quad \text{or} \quad t_2 = \frac{\pi}{75}. \]
Final Answer: \( \frac{\pi}{300} \, \text{s}, \, \frac{\pi}{75} \, \text{s} \)