Aplayground merry-go-round has radius 2.40 m and moment of inertia 2100 kgm2 about a vertical axle through its center, and it turns with negligible friction. (a) a child applies an 18.0-n force tangentially to the edge of the merry-go-round for 15.0 s. if the merry-go-round is initially at rest, what is its angular speed after this 15.0-s interval? (b) how much work did the child do on the merry-go-round? (c) what is the average power supplied by the child?

a) 0.31 rad/sec b) 101 J c) 6.73 W

Explanation:

a) If the merry-go-round starts from rest, we can apply the definition of angular acceleration, to get the value of the angular speed, as follows:

ωf = γ *t (1)

We don´t know the value of the angular acceleration, but we know that there was an external force applied tangentially to the edge of the body, which produced a torque.

Newton´s 2nd Law has also a rotational equivalent, so we can say the following:

τ = I * γ (2)

Applying the definition of torque, we can write the following equation:

τ = F*r (as both are perpendicular) = 18.0 N * 2.4 m = 43.2 N.m

Replacing in (2) and solving for the angular acceleration γ:

γ = 0.02 rad/sec2

Now, replacing this value in (1) we can get the value of the angular speed, as follows:

ωf = 0.31 rad/sec.

b) Applying the work-energy theorem, we know that the change in the kinetic energy, is equal to the work done by the child (neglecting any friction).  

As the merry-go-round only rotates, the kinetic energy is only rotational.

As it started from rest, the change is just equal to the final value, as follows:

∆K = ½ I * ωf2 = ½ * 2100 kg.m2 * (0.31)2 rad/sec2 = 101 J

W= 101 J

c) By definition, the average power is the work done in a given time, so we can write the following equation:

P = W/t = 101 J / 15.0 sec = 6.73 W

the working equation for the angular size is the product of the physical size and 360 degrees divided by the product of two pi and distance from earth. the minimum size would be 0.559 degrees while the maximum size would be 0.49 degrees. 

the area between the graph and the t axis