A point on a rim of a 0.15-m-radius rotating disk has a centripetal acceleration of 5.0 m/s^2. What is the angular speed of a point 0.075 m from the center of the disk?

To find the angular speed of a point on the disk, we need to relate the centripetal acceleration, radius, and angular speed using the formula:

a = rω^2

where:
a is the centripetal acceleration,
r is the radius,
ω (omega) is the angular speed.

We are given the centripetal acceleration (a = 5.0 m/s^2) and the radius (r = 0.15 m). We need to calculate the angular speed (ω) for a point that is 0.075 m from the disk's center.

Plugging in the given values into the formula, we have:

5.0 m/s^2 = (0.075 m)ω^2

To isolate ω, we can rearrange the equation:

ω^2 = a/r

ω^2 = 5.0 m/s^2 / 0.075 m

ω^2 = 66.67 s^-2

Taking the square root of both sides, we can find ω:

ω = √(66.67 s^-2)

Using a calculator, we find:

ω ≈ 8.164 s^-1

Therefore, the angular speed of a point 0.075 m from the center of the disk is approximately 8.164 s^-1.