A ball hanging from a twisted string is released from rest, and spins around in a circle (reversing direction periodically). The angular velocity of the ball is given by: 𝜔(𝑡)=(16 𝑟𝑎𝑑/𝑠) sin[(𝜋/5 𝑟𝑎𝑑/𝑠)𝑡]
Find the number of revolutions the ball makes before it comes to rest again.
To find the number of revolutions the ball makes before it comes to rest again, we need to determine the time it takes for the ball to complete one full revolution.
The angular velocity of the ball is given by 𝜔(𝑡) = (16 rad/s) sin[(𝜋/5 rad/s)𝑡]
We know that for one full revolution, the angle traversed is 2𝜋 radians. So, we need to determine the time it takes for 𝜔(𝑡) to reach the value of 2𝜋 rad.
Setting 𝜔(𝑡) = 2𝜋, we can solve for 𝑡:
(16 rad/s) sin[(𝜋/5 rad/s)𝑡] = 2𝜋
To solve this equation, we need to isolate the sin term:
sin[(𝜋/5 rad/s)𝑡] = (2𝜋)/(16 rad/s)
sin[(𝜋/5 rad/s)𝑡] = 𝜋/8 rad
Now, we can solve for 𝑡 by taking the inverse sine of both sides:
(𝜋/5 rad/s)𝑡 = sin^(-1)(𝜋/8 rad)
𝑡 = [sin^(-1)(𝜋/8 rad)] / (𝜋/5 rad/s)
Simplifying the equation, we get:
𝑡 = 5 / 8 seconds
Since 𝑡 represents the time it takes for one full revolution, the number of revolutions the ball makes before it comes to rest again is given by:
Number of revolutions = Total time / Time for one revolution
Number of revolutions = 5 / 8 seconds / (5 / 8 seconds per revolution)
Number of revolutions = 1 revolution
Therefore, the ball makes one full revolution before it comes to rest again.