a disk of radius 25 cm spinning at a rate of 30 rpm slows to a stop over 3 seconds. what is the angular acceleration?
b. how many radians did the disk turn while stopping?
c. how many revolutions?
To find the angular acceleration, you need to use the formula:
Angular Acceleration (α) = Change in Angular Velocity (Δω) / Time (t)
Given that the disk is spinning at a rate of 30 rpm and slows to a stop over 3 seconds, we can calculate the change in angular velocity:
Change in Angular Velocity (Δω) = Final Angular Velocity (ωf) - Initial Angular Velocity (ωi)
Since the disk slows to a stop, the final angular velocity is 0, and the initial angular velocity can be calculated as follows:
Initial Angular Velocity (ωi) = Initial Speed / Radius
The initial speed can be calculated using the formula:
Initial Speed = Initial Angular Velocity (ωi) * Radius
Now, let's substitute the given values into the equations:
Initial Speed = 30 rpm * (2π rad/min) * 25 cm = 30 * 2π * 25 cm/min
Initial Angular Velocity (ωi) = (30 * 2π * 25 cm/min) / 25 cm = 30 * 2π cm/min
Change in Angular Velocity (Δω) = 0 - (30 * 2π cm/min)
Finally, we can find the angular acceleration:
Angular Acceleration (α) = (0 - (30 * 2π cm/min)) / 3 sec
For part b, to find the number of radians the disk turns while stopping, we can use the formula:
Angular Displacement (θ) = Initial Angular Velocity (ωi) * Time (t) + 0.5 * Angular Acceleration (α) * Time (t)^2
Substituting the given values:
Angular Displacement (θ) = (30 * 2π cm/min) * 3 sec + 0.5 * ((0 - (30 * 2π cm/min)) / 3 sec) * (3 sec)^2
Finally, to convert the angular displacement from radians to revolutions, we use the conversion factor:
1 revolution = 2π radians
Thus, for part c, you can convert the angular displacement in radians to revolutions by dividing it by 2π radians.