Question

A turntable that spins at a constant 80.0 rpmrpm takes 3.50 ss to reach this angular speed after it is turned on. Find its angular acceleration (in rad/s2rad/s2), assuming it to be constant, and the number of degrees it turns through while speeding up.

Answer 1

Answer:

The angular acceleration is 2.39 rad/s².

The number of degrees it rotates is 841.68 degrees.

Explanation:

Given:

Initial angular speed (ω₀) = 0 rad/s

Final angular speed in rpm (N) = 80.0 rpm

Time taken (t) = 3.50 s

First, let us determine the final angular speed in radians per second.

We know that,

$\omega=\frac{2\pi N}{60}\ rad/s$

Plug in the values and find the final angular speed, 'ω'. This gives,

$\omega=\frac{2\pi\times 80.0}{60}=8.38\ rad/s$

Now, using equation of motion for rotational motion, we have:

$\omega=\omega_0+\alpha t\\\\\alpha\to angular\ acceleration$

Plug in the given values and solve for α. This gives,

$8.38=0+\alpha \times 3.50\\\\\alpha=\frac{8.38}{3.50}=2.39\ rad/s^2$

Therefore, the angular acceleration is 2.39 rad/s².

Now, again using rotational equation of motion relating angular displacement, we have:

$\omega^2=\omega_0^2+2\alpha\theta\\\\\theta=\frac{\omega^2-\omega_0^2}{2\alpha }$

Plug in the given values and solve for 'θ'. This gives,

$\theta=\frac{(8.38)^2-0}{2\times 2.39}\\\\\theta=\frac{70.2244}{4.78}=14.69\ rad$

Convert radians to degrees using the conversion factor. This gives,

π radians = 180°

So, 1 radian =( 180 ÷ π ) degrees

Therefore, $14.69\ rad=14.69\times (\frac{180}{\pi})=841.68^\circ$

So, the number of degrees it rotates is 841.68 degrees.