A clothes dryer is spinning at 75 rpm. A student opens its door and it comes to rest after rotating through four revolutions. Assume constant deceleration. What was the magnitude of the dryer's angular deceleration?
To find the magnitude of the dryer's angular deceleration, we can use the following equation:
ω^2 = ω0^2 + 2αθ
Where:
- ω is the final angular velocity (0 rpm since it comes to rest)
- ω0 is the initial angular velocity (75 rpm)
- α is the angular deceleration (what we want to determine)
- θ is the angle through which the dryer rotates (4 revolutions or 4*2π radians)
First, we convert the initial and final angular velocities from rpm to radians per second (rad/s):
ω0 = 75 rpm * (2π rad/1 min) * (1 min/60 s) = 75*2π/60 rad/s ≈ 7.853 rad/s
ω = 0 rpm * (2π rad/1 min) * (1 min/60 s) = 0 rad/s
Now, we substitute these values into the equation:
(0 rad/s)^2 = (7.853 rad/s)^2 + 2α * (4*2π rad)
0 = 7.853^2 + 16πα
0 = 61.525 + 16πα
Next, we rearrange the equation to solve for α:
16πα = -61.525
α = -61.525 / 16π
Using a calculator, we can evaluate the expression:
α ≈ -0.978 rad/s^2
Therefore, the magnitude of the dryer's angular deceleration is approximately 0.978 rad/s^2.