## To solve these problems, we can use the formula for angular acceleration and the conversion factor between rpm and radians per second.

The formula for angular acceleration is:

angular acceleration = change in angular velocity / time

Let's start with the first question:

(a) The turntable starts with an initial angular velocity of 33.3 rpm and slows down to a stop in 60 seconds. We need to find the angular acceleration.

First, we convert the initial angular velocity from rpm to radians per second. There are 2π radians in one revolution, and 60 seconds in one minute:

Initial angular velocity in radians per second = (33.3 rpm) * (2π radians / 1 revolution) * (1 revolution / 60 seconds)

= (33.3 * 2π) / 60 radians per second

Now we can calculate the angular acceleration using the formula:

angular acceleration = (Change in angular velocity) / time = (0 - Initial angular velocity) / time = (-Initial angular velocity) / time

So the angular acceleration is equal to - (Initial angular velocity) / time.

Plugging in the values, we have:

angular acceleration = - ((33.3 * 2π) / 60) / 60 rad/s²

(b) For the second question, we are given an angular acceleration of 50 rad/s² and we need to find the time required to attain a final angular velocity of 1800 rpm.

First, we convert the final angular velocity from rpm to radians per second, using the same conversion factor:

Final angular velocity in radians per second = (1800 rpm) * (2π radians / 1 revolution) * (1 revolution / 60 seconds)

= (1800 * 2π) / 60 radians per second

Now we can rearrange the formula for angular acceleration:

angular acceleration = (Change in angular velocity) / time

Rearranging the formula to solve for time, we have:

time = (Change in angular velocity) / angular acceleration

Plugging in the values, we have:

time = ((1800 * 2π) / 60) / 50 seconds

Simplifying the expression, we can calculate the time required to attain the final angular velocity.