In a playground, there is a small merry-go-round of radius 1.20 m and mass 190 kg. Its radius of gyration (see Problem 85 of Chapter 10) is 91.0 cm. A child of mass 44.0 kg runs at a speed of 4.50 m/s along a path that is tangent to the rim of the initially stationary merry-go-round and then jumps on. Neglect friction between the bearings and the shaft of the merry-go-round.
b) Calculate the magnitude of the angular momentum of the child, while running, about the axis of rotation of the merry-go-round.
c) Calculate the angular speed of the merry-go-round and child after the child has jumped on.
3 rad/s
To calculate the magnitude of the angular momentum of the child while running, we can use the formula:
Angular Momentum = Moment of Inertia * Angular Velocity
The moment of inertia of the child can be calculated using the formula:
Moment of Inertia = Mass * Radius^2
Plugging in the given mass of the child (44.0 kg) and the radius of the merry-go-round (1.20 m), we get:
Moment of Inertia = 44.0 kg * (1.20 m)^2
Now, the angular velocity of the child is the rate at which the child is running around the merry-go-round. Since the child is running tangentially to the rim, their linear speed is equal to the tangential speed of the merry-go-round. Therefore, the angular velocity of the child is equal to the tangential speed divided by the radius of the merry-go-round.
Angular Velocity = Linear Speed / Radius
Plugging in the given linear speed of the child (4.50 m/s) and the radius of the merry-go-round (1.20 m), we get:
Angular Velocity = 4.50 m/s / 1.20 m
Now we have all the values we need to calculate the magnitude of the angular momentum of the child while running:
Angular Momentum = Moment of Inertia * Angular Velocity
Substituting the values we calculated earlier, we get:
Angular Momentum = (44.0 kg * (1.20 m)^2) * (4.50 m/s / 1.20 m)
Simplifying the expression:
Angular Momentum = 44.0 kg * (1.20 m) * 4.50 m/s
Calculating the result:
Angular Momentum ≈ 239.76 kg·m²/s
Therefore, the magnitude of the angular momentum of the child, while running, about the axis of rotation of the merry-go-round is approximately 239.76 kg·m²/s.
To calculate the angular speed of the merry-go-round and child after the child has jumped on, we can use the law of conservation of angular momentum. The total angular momentum before the child jumps on is equal to the total angular momentum after the child jumps on.
The total angular momentum before the child jumps on is given by the angular momentum of the merry-go-round alone, which is:
Angular Momentum of the merry-go-round before = Moment of Inertia * Angular Velocity
The total angular momentum after the child jumps on is given by the angular momentum of the merry-go-round and the child combined, which is:
Angular Momentum of the merry-go-round and child after = (Moment of Inertia of the merry-go-round + Moment of Inertia of the child) * Angular Velocity
We can set these two quantities equal to each other and solve for the angular velocity of the merry-go-round and child after the child has jumped on:
Moment of Inertia of the merry-go-round * Angular Velocity (before) = (Moment of Inertia of the merry-go-round + Moment of Inertia of the child) * Angular Velocity (after)
Substituting the values given in the problem:
Moment of Inertia of the merry-go-round * Angular Velocity (before) = (Moment of Inertia of the merry-go-round + Moment of Inertia of the child) * Angular Velocity (after)
Now, the moment of inertia of the merry-go-round can be calculated using the formula given in the problem:
Moment of Inertia of the merry-go-round = Mass * Radius^2
Plugging in the given mass of the merry-go-round (190 kg) and the radius of gyration (91.0 cm), we get:
Moment of Inertia of the merry-go-round = 190 kg * (91.0 cm)^2
The moment of inertia of the child was calculated earlier:
Moment of Inertia of the child = Mass * Radius^2
Plugging in the given mass of the child (44.0 kg) and the radius of the merry-go-round (1.20 m), we get:
Moment of Inertia of the child = 44.0 kg * (1.20 m)^2
Now we can substitute these values into the equation and solve for the angular velocity (after):
(190 kg * (91.0 cm)^2) * Angular Velocity (before) = ((190 kg * (91.0 cm)^2) + (44.0 kg * (1.20 m)^2)) * Angular Velocity (after)
Note that we need to convert the radius of gyration from centimeters to meters, and the angular velocity (before) is equal to zero since the merry-go-round is initially stationary.
Simplifying the equation:
(190 kg * (0.91 m)^2) * 0 = ((190 kg * (0.91 m)^2) + (44.0 kg * (1.20 m)^2)) * Angular Velocity (after)
Now solving for Angular Velocity (after):
Angular Velocity (after) = [(190 kg * (0.91 m)^2) * 0] / [(190 kg * (0.91 m)^2) + (44.0 kg * (1.20 m)^2)]
Simplifying further:
Angular Velocity (after) = 0 / [(190 kg * (0.91 m)^2) + (44.0 kg * (1.20 m)^2)]
Since the numerator is zero, the angular velocity (after) will be zero as well.