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How to Calculate the Torque Needed to Accelerate a Spinning Disc

You can use physics to calculate the amount of torque needed to accelerate (or decelerate) the speed of a spinning disc. Without the ability to change the speed of a disc, it would be impossible for you to watch a movie on your DVD player.

Here’s an interesting fact about DVD players: They actually change the angular speed of the DVD to keep the section of the DVD under the laser head moving at constant linear speed.

Say that a DVD has a mass of 30 grams and a diameter of 12 centimeters. It starts at 700 revolutions per second when you first hit play and winds down to about 200 revolutions per second at the end of the DVD 50 minutes later. What’s the average torque needed to create this acceleration? You start with the torque equation:

image0.png

A DVD is a disk shape rotating around its center, which means that its moment of inertia is

image1.png

The diameter of the DVD is 12 centimeters, so the radius is 6.0 centimeters. Putting in the numbers gives you the moment of inertia:

image2.png

How about the angular acceleration,

image3.png

Here’s the angular equivalent of the equation for linear acceleration:

image4.png

But because the angular velocity always stays along the same axis, you can consider just the components of the angular velocity and angular acceleration along this axis. They are then related by

image5.png

First, you need to express angular velocity in radians per second, not revolutions per second. You know that the initial angular velocity is 700 revolutions per second, so in terms of radians per second, you get

image6.png

Similarly, you can get the final angular velocity this way:

image7.png

Now you can plug the angular velocities and time into the angular acceleration formula:

image8.png

The angular acceleration is negative because the angular velocity of the disk decreased. The negative acceleration then leads to a reduction in this angular velocity.

You’ve found the moment of inertia and the angular acceleration, so now you can plug those values into the torque equation:

image9.png

To get an impression of how easy or difficult this torque may be to achieve, you may ask how much force is this when applied to the outer edge — that is, at a 6-centimeter radius. Torque is force times the radius, so

image10.png

Slowing down the DVD doesn’t take much force.

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