Rotational Kinematics: Slowing Down Turntable

[hb20007]

Homework Statement

A turntable is a uniform disc of mass m and radius R. The turntable is initially spinning clockwise when looked down on from above at a constant frequency f_0. The motor is turned off at t=0 and the turntable slows to a stop in time t with constant angular deceleration.

a. What is the magnitude of the angular acceleration of the turntable? Express your answer in terms of f_0 and t

b. What is the magnitude of the total angle Δθ in radians that the turntable spins while slowing down? Express your answer in terms of f_0 and t

Homework Equations

We have to find them!

The Attempt at a Solution

I know that the initial angular velocity is = 2 * pi * f_0

I'm not very good at maths but I think the solution to part a is derived from differentiating the equation for angular velocity. I did that and got my answer as angular acceleration = - 2 * pi * (f_0)^2 which is wrong. I'm not sure where I went wrong though. As for part b, I tried integrating the equation of angular velocity but I'm getting a wrong answer as well?

 

[gneill]

You can get away without using any calculus if you make use of the usual kinematic formulas. All the formulas for linear motion have their counterparts for angular motion. They are even of the same form, but use angular variables instead of linear ones.

Take for example the formula for velocity from rest when there is a constant acceleration: v = at. Its angular motion analog is $ω = \alpha t$. All the other basic kinematic formulas can be similarly "translated".

Imagine that the same problem was given to you in linear form: "An object of mass m is moving to the right on a level surface with initial velocity v_o. At time t_o its propulsion is switched off and It slows to a stop after time t with constant acceleration. What's the magnitude of the acceleration? How far will the object travel while it is slowing down?" How would you go about solving that problem? What formulas would you use?

 

[hb20007]

That's a really useful simplification, thanks.
However, what would be the linear motion counterpart for frequency?

 

[gneill]

That's a really useful simplification, thanks.
However, what would be the linear motion counterpart for frequency?

Frequency is a sort of "lap counter" or "event counter" form of angular frequency; $ω = 2 \pi f$. When dealing with angular motion we usually convert f's to ω's as soon as possible It's ω that has velocity as its analog.

 

[hb20007]

Thanks once again. The question actually wasn't difficult at all but I struggled with it because I expected it to be so. Once you showed me how to eliminate the concepts of angular acceleration and frequency I could think clearly and I just got the answer

 

[gneill]

Thanks once again. The question actually wasn't difficult at all but I struggled with it because I expected it to be so. Once you showed me how to eliminate the concepts of angular acceleration and frequency I could think clearly and I just got the answer

Excellent news It can often be helpful to look at "new" concepts from the point of view of ones you've already mastered.