In a DVD player, the centripetal acceleration is measured to be 200 m/s2, at a point that is 0.02 m from the center of the disc.

Question

In a DVD player, the centripetal acceleration is measured to be 200 m/s², at a point that is 0.02 m from the center of the disc.

1. What is the rotating speed of the DVD?
2. What is the centripetal acceleration at a point that is 0.06 m from the center of the disc?
3. Is the rotating speed of the DVD different at different radial locations?

Answer 1

Hello Gift,

So given in the problem, we have centripetal acceleration = 200 m/s^2 and radius (r) = 0.02m.

1. The rotating speed is also another name for the tangential speed, which we can solve from the centripetal acceleration formula.

Centripetal acceleration = (v^2)/r

So 200 m/s^2 = (v^2)/0.02m

4 = v^2

v = 2 m/s

1. Now before we can simply plug into our formula to find the new centripetal acceleration, we need to calculate the new rotating speed. Tangential velocity/Rotating speed changes if the radius also changes.
2. You can think about this in a few ways. Centripetal force is the force that pushes things into the center of the rotating disk. So if I put a penny on the edge of this rotating disk then centripetal force would hold the penny in place and not let it fly off the disk.
3. So when you place the penny farther and farther away from the center of the disk, then the centripetal force decreases in value. The force decreases because you are dividing by a larger radius in the formula. This makes sense because the penny is more likely to fly off the disk the farther away from the center it is placed. Similarly, the rotating speed actually increases the farther away from the center the penny is placed. This can be represented by the following formula
4. rotating speed = (angular velocity) * (radius)

With a new radius of 0.06m, we need to calculate the new rotating speed. But first, the angular velocity can be calculated using the old radius. Angular velocity is the value that is not affected by changing the radius.

2 m/s = (0.02m) * (angular velocity)

angular velocity = 100 radians/s

(100 radians/s) *(0.06m) = 6 m/s

Now we can plug into our centripetal acceleration formula,

((6 m/s)^2) / (0.06m) = 600 m/s^2

1. As we mentioned before the angular velocity does not change due to a change in the radius, but the tangential velocity/rotating speed is affected by a change in the radius.

I hope this helps!