Two boys of the same mass, 45 kg, are on a playground merry-go-round that has a diameter of 4.0 meters. Both boys are at the very edge of the ride. One boy stands with the coefficient of friction between his shoes and eh ride being .45 whereas the other is on his knees with a coefficient of friction of .3. Who will fly off first? At what speed will each boy fly off?
To determine which boy will fly off the merry-go-round first and at what speed, we need to compare the maximum frictional force acting on each boy with the centripetal force required to keep them moving in a circle. Let's break down the steps to find the answers:
Step 1: Calculate the centripetal force required to keep each boy moving in a circle.
The centripetal force can be determined using the following equation:
F = m * (v^2 / r)
Where:
F is the centripetal force,
m is the mass of the boy,
v is the velocity of the boy,
r is the radius of the merry-go-round (which is half the diameter).
For both boys, the mass (m) is 45 kg, and the radius (r) can be calculated as half the diameter, which is 4.0 meters / 2 = 2.0 meters.
Step 2: Calculate the maximum frictional force acting on each boy.
The maximum frictional force can be calculated using the equation:
Ffriction = μ * N
Where:
Ffriction is the frictional force,
μ is the coefficient of friction,
N is the normal force acting on the boy.
The normal force (N) is equal to the weight of the boy, which can be calculated using:
N = m * g
Where:
g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
For both boys, the mass (m) is 45 kg, so we can calculate the normal force (N).
Step 3: Compare the maximum frictional force to the centripetal force.
For each boy, compare the maximum frictional force (Ffriction) with the centripetal force (F). The boy with the greater frictional force will not fly off until the centripetal force exceeds that maximum frictional force.
Step 4: Calculate the speed at which each boy will fly off.
Once we determine which boy will fly off first, we can calculate the speed at which they will fly off using the equation:
v = √(Ffriction * r / m)
Let's carry out these steps for both boys:
Boy standing with a coefficient of friction (μ) of 0.45:
Step 1: F = 45 kg * (v^2 / 2.0 m)
Step 2: Ffriction = 0.45 * (45 kg * 9.8 m/s^2)
Step 3: Compare Ffriction with F
Step 4: Calculate v = √(Ffriction * 2.0 m / 45 kg)
Boy on his knees with a coefficient of friction (μ) of 0.3:
Step 1: F = 45 kg * (v^2 / 2.0 m)
Step 2: Ffriction = 0.3 * (45 kg * 9.8 m/s^2)
Step 3: Compare Ffriction with F
Step 4: Calculate v = √(Ffriction * 2.0 m / 45 kg)
By following these steps, you can determine which boy will fly off first and at what speed.