A 50.0-kg child stands at the rim of a merry-go-round of radius 1.65 m, rotating with an angular speed of 3.00 rad/s.
(a) What is the child's centripetal acceleration?
m/s2
(b) What is the minimum force between her feet and the floor of the carousel that is required to keep her in the circular path?
N
(c) What minimum coefficient of static friction is required?
To find the answers to these questions, we can use the following steps:
Step 1: Find the child's centripetal acceleration.
We can use the formula:
centripetal acceleration (a) = radius (r) × angular speed (ω)^2
Given:
radius (r) = 1.65 m
angular speed (ω) = 3.00 rad/s
Substituting these values into the formula gives:
centripetal acceleration = 1.65 m × (3.00 rad/s)^2 = 14.85 m/s^2
So, the child's centripetal acceleration is 14.85 m/s^2.
Step 2: Find the minimum force between her feet and the floor of the carousel that is required to keep her in the circular path.
We can use the formula:
force (F) = mass (m) × centripetal acceleration (a)
Given:
mass (m) = 50.0 kg (child's mass)
centripetal acceleration (a) = 14.85 m/s^2 (from the previous calculation)
Substituting these values into the formula gives:
force (F) = 50.0 kg × 14.85 m/s^2 = 742.5 N
So, the minimum force between her feet and the floor of the carousel that is required to keep her in the circular path is 742.5 N.
Step 3: Find the minimum coefficient of static friction required.
We can use the formula:
coefficient of static friction (μ) = force of static friction (F_static) / normal force (N)
Given:
force of static friction (F_static) = force (F) = 742.5 N (from the previous calculation)
normal force (N) = weight of the child = mass (m) × gravity (g)
In order to find the minimum coefficient of static friction, we need to know the value of gravity (g). Assuming we are on Earth, the standard value of gravity is approximately 9.8 m/s^2.
Substituting the values into the formula gives:
normal force (N) = 50.0 kg × 9.8 m/s^2 = 490 N
Finally, substituting these values into the formula for the coefficient of static friction:
coefficient of static friction (μ) = 742.5 N / 490 N = 1.515
So, the minimum coefficient of static friction required is 1.515.