(a)If a crate slides down a 10 degree incline at uniform speed, what is the coefficient of friction between the crate and the incline?(b)What is the coeffiecient of friction using a 20 degree incline?(c)What is the coefficient of friction using an n degree incline?
To find the coefficient of friction between the crate and the incline, we can use the concept of forces and equations of motion. The following explanation will help you find the coefficient of friction using any degree of incline.
(a) When the crate slides down a 10-degree incline at uniform speed, we can assume that the force of friction opposing the crate's motion is equal to the component of the weight of the crate acting parallel to the incline. This force can be calculated using trigonometry.
1. Start by drawing a diagram of the problem. Label the forces acting on the crate, including the force of gravity (mg) acting vertically downwards and the normal force (N) acting perpendicular to the incline.
- The weight of the crate (mg) can be resolved into two components: one acting perpendicular to the incline (mg cosθ) and the other acting parallel to the incline (mg sinθ).
- The force of friction (f) acts in the opposite direction of the crate's motion, parallel to the incline.
2. Now, write down the forces acting on the crate in terms of their components:
- Force of friction (f) = coefficient of friction (μ) * normal force (N).
- Normal force (N) = mg cosθ.
- Force of gravity parallel to incline (mg sinθ) = force of friction (f).
3. Since the crate is moving at a constant speed, the acceleration is zero. Thus, the net force on the crate along the incline is zero.
- Net force = force of gravity parallel to incline (mg sinθ) - force of friction (f) = 0.
- Substitute the expressions for the forces to get: mg sinθ - μmg cosθ = 0.
4. Simplify the equation by canceling out the common factor of mg:
- sinθ - μcosθ = 0.
5. Now, solve the equation for the coefficient of friction (μ):
- μ = tanθ.
Therefore, the coefficient of friction between the crate and the incline is equal to the tangent of the incline angle.
(b) Similarly, for an incline of 20 degrees, the coefficient of friction can be found by following the same steps:
- μ = tan20°.
(c) In general, for an n-degree incline, the coefficient of friction can be found as:
- μ = tan(n°).
By substituting the value of the angle (θ) into the equation, you can find the coefficient of friction for any given incline angle.