A solid sphere of uniform density starts from rest and rolls without slipping down an inclined plane with angle θ = 30o. The sphere has mass M = 8 kg and radius R = 0.19 m . The coefficient of static friction between the sphere and the plane is μ = 0.64. What is the magnitude of the frictional force on the sphere?
I got 21.168N and the computer told me I was wrong because I was trying to find force of the max static instead of the actual static force. Please help!
To find the magnitude of the frictional force on the sphere, you need to consider the components of the forces acting on it. Here's how you can solve this problem step by step:
1. Identify the forces acting on the sphere:
- Gravitational force (mg): Acts vertically downwards with magnitude equal to the mass of the sphere (M) multiplied by the acceleration due to gravity (g ≈ 9.8 m/s^2). So, F_gravity = Mg.
- Normal force (N): Acts perpendicular to the inclined plane. For a solid sphere rolling without slipping, the normal force is equal to the gravitational force (N = Mg).
- Frictional force (F_friction): Acts parallel to the inclined plane, opposing the motion of the sphere. This is the force you want to find.
2. Determine the maximum static friction force:
The maximum static friction force (F_static_max) can be calculated using the coefficient of static friction (μ) and the normal force (N). The formula is F_static_max = μN. In this case, μ = 0.64 and N = Mg.
3. Determine if the sphere will start rolling:
To determine if the sphere will start rolling or remain in equilibrium, compare the maximum static friction force (F_static_max) with the component of gravitational force that acts parallel to the inclined plane (mg*sin(θ)). If F_static_max > mg*sin(θ), the sphere will not roll and the static friction force will be equal to the component of gravitational force parallel to the incline (F_friction = mg*sin(θ)).
4. Calculate the magnitude of the frictional force:
If F_static_max ≤ mg*sin(θ), then the sphere will start rolling and the frictional force will be kinetic instead of static. In this case, the magnitude of the frictional force is given by the kinetic friction coefficient (μ_kinetic) multiplied by the normal force (N): F_friction = μ_kinetic * N.
In this problem, we will focus on the case where the sphere is rolling without slipping, so the frictional force is static.
Given:
- θ (angle of the inclined plane) = 30°
- M (mass of the sphere) = 8 kg
- R (radius of the sphere) = 0.19 m
- μ (coefficient of static friction) = 0.64
Now let's apply the steps:
Step 1: Identify the forces acting on the sphere:
- F_gravity = Mg
- N = Mg
- F_friction (the force you want to find)
Step 2: Determine the maximum static friction force:
- F_static_max = μN
= μ(Mg)
Plugging in the given values:
- F_static_max = 0.64 * (8 kg * 9.8 m/s^2)
- F_static_max ≈ 50.176 N
Step 3: Determine if the sphere will start rolling:
- mg*sin(θ) = (8 kg * 9.8 m/s^2) * sin(30°)
- mg*sin(θ) ≈ 39.2 N
Since F_static_max (50.176 N) is greater than mg*sin(θ) (39.2 N), the sphere will not start rolling. Therefore, we use the static friction force.
Step 4: Calculate the magnitude of the frictional force:
- F_friction = mg*sin(θ)
= (8 kg * 9.8 m/s^2) * sin(30°)
= 39.2 N
So, the magnitude of the frictional force acting on the sphere is approximately 39.2 N.
Please note that the value you obtained (21.168 N) is incorrect because you calculated the maximum static friction force instead of the actual static friction force. In this scenario, since the sphere does not start rolling, the actual static friction force is equal to mg*sin(θ), as explained in Step 4.