A 22 kg child is riding a playground merry-go-round that makes 40 revolutions in 60 seconds. Determine the magnitude of force (in Newtons) necessary to make the child stay on the merry-go-round if she is 1.25m from its center.
To solve this problem, we need to use the formula for centripetal force:
F = (m * w^2 * r)
Where:
F is the centripetal force (in Newtons)
m is the mass of the child (in kg)
w is the angular velocity (in rad/s)
r is the distance of the child from the center of the merry-go-round (in meters)
First, let's determine the angular velocity (w) of the merry-go-round using the given information that it makes 40 revolutions in 60 seconds.
To calculate the angular velocity, we use the formula:
w = (2 * pi * n) / t
Where:
w is the angular velocity (in rad/s)
pi is a mathematical constant approximately equal to 3.14159
n is the number of revolutions
t is the time taken (in seconds)
Plugging in the values, we get:
w = (2 * 3.14159 * 40) / 60
Simplifying further, we get:
w = 4.18879 rad/s
Now, we can calculate the centripetal force (F) using the formula:
F = (m * w^2 * r)
Plugging in the given values, we get:
F = (22 kg * (4.18879 rad/s)^2 * 1.25 m)
Simplifying further, we get:
F = 115.6613 N
Therefore, the magnitude of force necessary to make the child stay on the merry-go-round is approximately 115.6613 Newtons.