A playground marry-go-round has radius 2.40 m and moment of inertia 2100 kg m2 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?

Short Answer

Expert verified

(a) The angular speed is, ω=0.309rads.

(b) The work done on the merry-go-round by the child isW=100J.

(c) The average power supplied by the child is, P=6.67W.

Step by step solution

01

To state given data

Moment of inertia of the merry-go-round (I) = 2100 kg m2.

Radius of the marry-go-round(R) = 2.40 m.

Force exerted by the child (F) = 18.0 N.

Time interval (t) =15.0 s.

02

Mention the concepts

The torque is given by,

τ=FR

= ……. (1)

Here, F is the force, R is the radius, αis angular acceleration and I is the moment of inertia.

The angular acceleration is:

α=FRI ……. (2)

Here, F is the force, R is the radius, and I is the moment of inertia.

Consider the formula for the angular speed:

ω=αΔt …… (3)

Here, αis angular acceleration and Δtis the time interval.

03

(a) Find the angular speed

The initial time is ti=0s and final time is tf=15s.

Therefore, the time interval is Δt=15s.

Then using equation (2) in (3), angular speed is given by,

role="math" localid="1667982999190" ω=αΔtω=FRΔtIω=(18.0)(2.40)(15.0)2100ω=0.309rads

Hence, the angular speed is,.

04

(b)To find the work done on the merry-go-round

The work done on the merry-go-round by the child is equal to the rotational kinetic energy.

Therefore,

W=Krot=122W=12(2100)(0.309)2W=100J

Hence, the work done on the merry-go-round by the child isW=100J.

05

(c)To find the average power supplied by the child

The average power supplied by the child is given by,

P=WΔt.

Substituting values, we get,

localid="1667983259762" P=10015.0P=6.67W

Hence, the average power supplied by the child is, P=6.67W.

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