Question

A record player turntable initially rotating at 3313 rev/min is braked to a stop at a constant rotational acceleration. The turntable has a rotational inertia of 0.031kg?m2 . When it is switched off, it slows down to 75% of its initial rotational speed in 4.0 s.A- How long does it take to come to rest?B- How much work has to be done on the turntable to bring it to rest?

Answer 1

Answer:

15.99861 seconds

1865.58658 Joules

Explanation:

$\alpha$ = Angular acceleration

$\theta$ = Angle of rotation

I = Moment of inertia of disc = 0.031 kgm²

t = Time taken

Initial angular speed

$\omega_i=3313\frac{2\pi}{60}\\\Rightarrow \omega_i=346.93\ rad/s$

Final angular velocity

$\omega_f=0.75\times \omega_i\\\Rightarrow \omega_f=0.75\times 346.93\\\Rightarrow 260.19\ rad/s$

From the equation of rotational motion

$\omega_f=\omega_i+\alpha t\\\Rightarrow \alpha=\frac{\omega_f-\omega_i}{t}\\\Rightarrow \alpha=\frac{260.19-346.93}{4}\\\Rightarrow \alpha=-21.685\ rad/s^2$

$\omega_f=\omega_i+\alpha t\\\Rightarrow t=\frac{\omega_f-\omega_i}{\alpha}\\\Rightarrow t=\frac{0-346.93}{-21.685}\\\Rightarrow t=15.99861\ s$

Time taken by the turntable to slow down is 15.99861 seconds

Rotational energy is given by

$K=\frac{1}{2}I\omega_i^2\\\Rightarrow K=\frac{1}{2}\times 0.031\times 346.93^2\\\Rightarrow K=1865.58658\ J$

The work has to be done on the turntable to bring it to rest is 1865.58658 Joules