A computer is reading data from a rotating CD-ROM. At a point that is 0.037 m from the center of the disc, the centripetal acceleration is 130 m/s2. What is the centripetal acceleration at a point that is 0.064 m from the center of the disc?
To solve this problem, we can use the formula for centripetal acceleration:
a = (v^2) / r
where:
a = centripetal acceleration
v = linear velocity
r = distance from the center of rotation
We're given the centripetal acceleration at a point 0.037 m from the center of the disc, which is 130 m/s^2. We need to find the centripetal acceleration at a point 0.064 m from the center of the disc.
At different points on a rotating CD-ROM, the linear velocity changes, but the centripetal acceleration remains constant, as long as the disc is rotating with a constant angular velocity.
So, to find the centripetal acceleration at 0.064 m from the center, we need to find the linear velocity at that point.
We can use the formula for linear velocity:
v = ωr
where:
v = linear velocity
ω = angular velocity
r = distance from the center of rotation
Since the angular velocity remains constant throughout the disc, we can find the linear velocity at 0.064 m from the center by using the linear velocity at 0.037 m from the center.
Given:
r1 = 0.037 m
a1 = 130 m/s^2
Using the formula for centripetal acceleration, we can solve for v1:
a1 = (v1^2) / r1
130 = (v1^2) / 0.037
Now, let's solve for v1:
v1^2 = 130 * 0.037
v1^2 = 4.81
v1 ≈ √4.81
v1 ≈ 2.19 m/s
Now, we have the linear velocity at 0.037 m from the center, v1 = 2.19 m/s. We can use this velocity to find the centripetal acceleration at 0.064 m from the center.
Given:
r2 = 0.064 m
v1 = 2.19 m/s
Using the formula for centripetal acceleration:
a2 = (v2^2) / r2
a2 = (v1^2) / r2
a2 = (2.19^2) / 0.064
Now, let's solve for a2:
a2 = (4.7961) / 0.064
a2 ≈ 74.94 m/s^2
Therefore, the centripetal acceleration at a point that is 0.064 m from the center of the disc is approximately 74.94 m/s^2.