(a) a 22.0-kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. what centripetal force is exerted if he is 1.25 m from its center? (b) what centripetal force is exerted if the merry-go-round rotates at 3.00 rev/min and he is 8.00 m from its center? (c) compare each force with his weight.

a)  F₁ = 483 N, b) F₂ = 176 N  and c)  F₁ / W = 2.2 ,    F₂ / W = 0.82

Explanation:

To find force we use Newton's second law, where acceleration is centripetal

    F = m a

The centripetal acceleration is

    a = v² / r

The relationship between linear and angular velocity is

    v = w r

Substituting Newton's law

    F = m (w r)² / r

    F = m w² r

Let's calculate for each case

a) Let's reduce the units to the SI system

      w = 40.0 rev / min (2pi rad / rev) (1min / 60s)

      w = 4,189 rad/s

      F₁ = 22.0 4.189² 1.25

      F₁ = 482.6 N

      F₁ = 483 N

b)   w = 3.00 rev / min = 0.314 rad / s

     F₂ = 22.0 0.314 2 8.00

     F₂ = 176 N

c) the child's weight is

    W = m g

    W = 22.0 9.8

    W = 215.6 N

To make this comparison we can divide the two quantities

    F₁ / W = 483 / 215.6

    F₁ / W = 2.2

   F₂ / W = 176 / 215.6

   F₂ / W = 0.82

Answer is in the photo. I couldn't attach it here, but uploaded it to a file hosting. link below! Good Luck!

cutt.ly/wzUiVyK