A basketball player is balancing a spinning basketball on the tip of his finger. The angular velocity of the ball slows down from 18.0 to 14.7 rad/s, during the slow down the angular displacement is 85.4 rad. Determine the time it takes for the ball to slow down.
To determine the time it takes for the ball to slow down, we can use the equation:
Δω = (ωf - ωi) / t
Where Δω is the change in angular velocity, ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time. We are given Δω = 14.7 rad/s - 18.0 rad/s = -3.3 rad/s (since the velocity is slowing down).
Rearranging the equation, we have:
Δω = -3.3 rad/s = (14.7 rad/s - 18.0 rad/s) / t
Now, let's solve for t:
-3.3 rad/s = (-3.3 rad/s) / t
To isolate t, we can divide both sides of the equation by -3.3 rad/s:
t = (-3.3 rad/s) / (-3.3 rad/s)
The negative signs in the numerator and denominator cancel out, and we get:
t = 1 second
Therefore, it takes 1 second for the basketball player to slow down the spinning basketball.
To determine the time it takes for the ball to slow down, we can use the formula:
angular acceleration (α) = (final angular velocity (ωf) - initial angular velocity (ωi)) / time (t)
We can rearrange this formula to solve for time:
time (t) = (final angular velocity (ωf) - initial angular velocity (ωi)) / angular acceleration (α)
Given:
Initial angular velocity (ωi) = 18.0 rad/s
Final angular velocity (ωf) = 14.7 rad/s
Angular displacement (θ) = 85.4 rad
To find the angular acceleration (α), we can use the formula:
angular acceleration (α) = (final angular velocity (ωf)^2 - initial angular velocity (ωi)^2) / 2* angular displacement (θ)
Substituting the given values:
angular acceleration (α) = (14.7^2 - 18.0^2) / (2 * 85.4)
Now we can calculate the angular acceleration (α):
α = (216.09 - 324) / 170.8
α ≈ -0.4233 rad/s² (negative sign indicates deceleration)
Now we can calculate the time (t):
t = (14.7 - 18.0) / -0.4233
t ≈ 8.5 seconds
Therefore, it takes approximately 8.5 seconds for the ball to slow down.