A merry-go-round makes one complete revolution in 9.1 s. An 18.8 kg child sits on the
horizontal floor of the merry-go-round 4.7 m
from the center.
Find the child’s acceleration. The acceleration of gravity is 9.8 m/s
2
2 pi r = 2 (3.14)(4.7) = 29.5 meters
v = 29.5 / 9.1 = 3.25 m/s
Ac = v^2/r = 3.25^2/4.7 = 2.24 m/s^2
= .23 g
To find the child's acceleration, we need to consider two forces acting on the child: the centripetal force and the gravitational force.
1. Centripetal Force:
The centripetal force is the force that keeps an object moving in a circle. It is given by the equation:
Fₙ = m * (v² / r)
where Fₙ is the net centripetal force required, m is the mass of the child, v is the velocity of the child, and r is the radius of the circular motion.
To find the velocity of the child, we need to calculate the circumference of the circle he is moving in, which is 2πr. Since he takes 9.1 seconds to complete one revolution, the velocity can be calculated as:
v = 2πr / t
where t is the time taken for one revolution.
Substituting the given values, we have:
v = 2π * 4.7 / 9.1
2. Gravitational Force:
The gravitational force acting on the child can be calculated using:
Fg = m * g
where Fg is the gravitational force, m is the mass of the child, and g is the acceleration due to gravity.
Substituting the given values, we have:
Fg = 18.8 * 9.8
Now, let's combine these two forces and solve for the child's acceleration:
m * (v² / r) + m * g = m * a
Substituting the values we calculated earlier:
18.8 * (v² / 4.7) + (18.8 * 9.8) = 18.8 * a
This equation allows us to solve for the child's acceleration. Simplify the equation and solve for 'a'.
The resulting value will be the child's acceleration.