A ball is rotating in a horizontal circle at the
end of a string of length 3.8 m at an angular
velocity of 8.5 rad/s. The string is gradually
shortened to 3.3 m without any force being
exerted in the direction of the ball’s motion.
Find the new angular velocity of the ball.
Answer in units of rad/s
To find the new angular velocity of the ball, we can use the principle of conservation of angular momentum. The angular momentum of an object remains constant as long as no external torque is applied.
The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In this case, the moment of inertia remains constant since the ball's mass and shape do not change. Therefore, we can equate the initial angular momentum to the final angular momentum.
Initial angular momentum = Final angular momentum
I_initial * ω_initial = I_final * ω_final
The moment of inertia I is given by the formula I = m * r^2, where m is the mass of the ball and r is the radius of the circular path.
Since the mass of the ball is not given and we are only interested in the ratio of the initial and final angular velocities, we can cancel out the mass from the equation:
(r^2 * ω_initial) = (r^2 * ω_final)
Simplifying further, we get:
ω_initial = ω_final
Therefore, the new angular velocity of the ball after shortening the string is the same as the initial angular velocity, which is 8.5 rad/s.