Tether ball is a game children play in which a ball hangs from a rope attached to the top of a tall pole. The children hit the ball, causing it to swing around the pole. What is the total initial acceleration of a tether ball on a 2.0 m rope whose angular velocity changes from 13 rad/s to 7.0 rad/s in 15 s

a_total = 14.022 m/s²

Explanation:

The total acceleration of a uniform circular motion is given by the following formula:

        (1)

ac: centripetal acceleration

aT: tangential acceleration

Then, you first calculate the centripetal acceleration by using the following formula:    

r: radius of the circular trajectory = 2.0m

w: final angular velocity  of the ball = 7.0 rad/s

Next, you calculate the tangential acceleration. aT is calculate by using:

   (2)

α: angular acceleration

The angular acceleration is:

wo: initial angular velocity = 13 rad/s

t: time = 15 s

Then, you use the expression for the angular acceleration in the equation (1) and solve for aT:

Finally, you replace the values of aT and ac in the equation (1), in order to calculate the total acceleration:

The total acceleration of the ball is 14.022 m/s²