Final answer:
To find the merry-go-round's final angular velocity and the son's tangential speed, we calculate the moment of inertia, use the angular impulse formula, and assess the effects of the father's three consecutive pushes from rest while taking into account the mass distribution without friction.
Step-by-step explanation:
In this physics problem, we need to calculate the final angular velocity of a merry-go-round and the tangential speed of the son sitting on it. The merry-go-round has a mass of 160 kg, a diameter of 2.40 m, and is being treated dynamically like a disc. The son has a mass of 35 kg and is located 0.80 m from the center. A tangential force of 45.0 N is applied by the father for 1.50 seconds, three times, and we are given that the merry-go-round starts from rest.
The moment of inertia (I) for a solid disc is ½MR², where M is the mass and R is the radius (half the diameter). Here, R is 1.20 m, so I for the merry-go-round is ½(160 kg)(1.20 m)². We shall also consider the moment of inertia for the son as a point mass, which is m*r², where m is the son's mass and r is the distance from the axis, which in this case is 0.80 m.
To calculate the final angular velocity, we need to use the equation for angular impulse, which tells us that torque (τ) times time (t) equals change in angular momentum (I*δω). Since the father applies the force three times, we need to calculate the angular impulse for each push and sum them up. The torque for each push is the force applied (45.0 N) times the radius (1.20 m). We multiply this torque by the time of each push (1.50 s) to find the angular impulse. It should be noted that we are ignoring any friction or air resistance in this calculation.
The son's tangential speed will be his distance from the center (0.80 m) multiplied by the final angular velocity of the merry-go-round. The tangential speed reflects how fast the son is moving along the circular path.
After all calculations, we'll have the merry-go-round's final angular velocity in radians per second (rad/s) and the son's tangential speed in meters per second (m/s).