Final answer:
The centripetal force exerted on a 27-kg child riding a merry-go-round rotating at 45.0 rev/min and located 1.26 m from the center is approximately 792.52 newtons.
Step-by-step explanation:
A 27-kg child is riding a playground merry-go-round that is rotating at 45.0 rev/min. To calculate the centripetal force that is exerted on the child, we first need to convert the rotational speed from revolutions per minute to meters per second. Then, we use the formula for centripetal force:
F = m × v^2 / r
where
- m is the mass of the child (27 kg),
- v is the linear velocity (which we need to find), and
- r is the radius of the merry-go-round (1.26 m).
First, convert 45.0 rev/min to radians per second:
45 rev/min × (2π radians/1 rev) × (1 min/60 sec) = 4.712 radians/sec
Next, calculate the linear velocity:
v = ω × r
where ω is the angular velocity (4.712 radians/sec) and r is the radius (1.26 m).
v = 4.712 × 1.26 = 5.937 m/s
Finally, plug these values into the centripetal force formula:
F = 27 kg × (5.937 m/s)^2 / 1.26 m = 792.52 N (approx)
Therefore, the centripetal force exerted on the child is approximately 792.52 newtons.