A 2.00 kg rim with a radius of 20.0 cm is setting on an 8.00 kg wedge of angle 30 degrees. The kinetic coefficient of kinetic friction between the wedge and the floor is 0.20
A) With F = 0.00 N, find the minimum coefficient of static friction between the rim and the wedge required for rolling without slipping.
B) Find the minimum force required to keep the rim from rolling downward.
C) Find the maximum force required to get the rim rolling upwards at a constant speed.
To solve this problem, we need to analyze the forces acting on the rim and the wedge. Let's go step by step to find the answers to each question.
A) To find the minimum coefficient of static friction required for rolling without slipping, we need to consider the forces acting on the rim.
The forces acting on the rim are:
1. Weight (mg) acting downward at the center of mass of the rim.
2. Normal force (N) exerted by the wedge on the rim.
3. Force of static friction (f_s) acting between the rim and the wedge, opposing the tendency of the rim to slip.
For rolling without slipping, the static friction force must provide the torque required to prevent slippage.
The torque due to the weight can be calculated as follows:
τ_weight = r * mg * sin(θ)
where r is the radius of the rim and θ is the angle of the wedge.
The torque due to the static friction can be calculated as:
τ_friction = r * f_s
For rolling without slipping, these two torques must be equal:
τ_weight = τ_friction
Substituting the expressions for torque:
r * mg * sin(θ) = r * f_s
The mass of the rim does not cancel out since it appears in the torque due to weight but not in the torque due to friction.
Now, solving for f_s:
f_s = mg * sin(θ) / 2.00 kg
Therefore, the minimum coefficient of static friction (μ_s) required for rolling without slipping is:
μ_s = f_s / N = (mg * sin(θ) / 2.00 kg) / (mg * cos(θ))
The mass cancels out, leaving:
μ_s = sin(θ) / 2.00 kg * cos(θ)
B) To find the minimum force required to prevent the rim from rolling downward, we need to consider the forces acting on the rim.
The forces acting on the rim are the same as mentioned before, i.e., weight (mg), normal force (N), and static friction force (f_s).
To prevent the rim from rolling downward, the static friction force must balance the component of the weight that wants to make the rim roll.
The component of weight trying to make the rim roll can be calculated as:
F_roll = mg * sin(θ)
Therefore, the minimum force required to keep the rim from rolling downward is the static friction force:
F_min = f_s = mg * sin(θ) * μ_k = 8.00 kg * 9.8 m/s^2 * sin(30°) * 0.20
C) To find the maximum force required to get the rim rolling upwards at a constant speed, we need to consider the forces acting on the rim.
The forces acting on the rim are still weight (mg), normal force (N), and the kinetic friction force (f_k).
To get the rim rolling upwards at a constant speed, the kinetic friction force must balance the component of the weight that wants to make the rim roll back downward.
The component of weight trying to make the rim roll back downward can be calculated as:
F_roll = mg * sin(θ)
Therefore, the maximum force required to get the rim rolling upwards at a constant speed is equal to the component of weight that wants to make the rim roll back downward:
F_max = mg * sin(θ) = 8.00 kg * 9.8 m/s^2 * sin(30°)