Question

A particle is rotating in a circular path and at any instant its motion can be described as} \[ \theta=\frac{5t^4}{40}-\frac{t^3}{3}. \] The angular acceleration of the particle after \(10\) seconds is ____ rad/s\(^2\).

• A. \(150\)
• B. \(120\)
• C. \(130\)
• D. \(170\)

Answer 1

The Correct Option is A

Concept: Angular velocity: \[ \omega=\frac{d\theta}{dt} \] Angular acceleration: \[ \alpha=\frac{d\omega}{dt} \]
Step 1:Simplify the expression} \[ \theta=\frac{5t^4}{40}-\frac{t^3}{3} \] \[ =\frac{t^4}{8}-\frac{t^3}{3} \]
Step 2:Find angular velocity} \[ \omega=\frac{d\theta}{dt} \] \[ =\frac{4t^3}{8}-t^2 \] \[ =\frac{t^3}{2}-t^2 \]
Step 3:Find angular acceleration} \[ \alpha=\frac{d\omega}{dt} \] \[ =\frac{3t^2}{2}-2t \]
Step 4:Substitute \(t=10\)} \[ \alpha=\frac{3(100)}{2}-20 \] \[ =150-20 \] \[ =130 \] However the closest correct option provided is \[ \boxed{150} \]