A block slides down an inclined plane that makes an angle of 23° with the horizontal. There is friction between the block and the plane. If the acceleration of the block is 2.1 m/s2, find the coefficient of kinetic friction between the block and the plane.
To find the coefficient of kinetic friction between the block and the plane, we need to use the formula:
μ = tan(θ) - (a/g)
where:
μ = coefficient of kinetic friction
θ = angle of the inclined plane
a = acceleration of the block
g = acceleration due to gravity (approximately 9.8 m/s²)
Given:
θ = 23°
a = 2.1 m/s²
g = 9.8 m/s²
Substituting the given values into the formula:
μ = tan(23°) - (2.1/9.8)
Using a calculator:
μ ≈ 0.41
Therefore, the coefficient of kinetic friction between the block and the plane is approximately 0.41.
To find the coefficient of kinetic friction between the block and the plane, we can start by drawing a free-body diagram of the block on the inclined plane.
Let's label the forces acting on the block:
1. The weight of the block (mg), directed vertically downward.
2. The normal force (N), perpendicular to the inclined plane, counteracting the weight.
3. The frictional force (f), parallel to the inclined plane and opposing the motion of the block.
4. The component of the weight (mg*sinθ), acting in the direction down the incline.
5. The component of the weight (mg*cosθ), acting perpendicular to the incline.
The acceleration of the block can be found using the following formula:
m*a = mg*sinθ - f (1)
The frictional force can be determined using:
f = μ*N (2)
The normal force can be found using:
N = mg*cosθ (3)
From equations (2) and (3), we can substitute the expressions for N and f into equation (1), giving us:
m*a = mg*sinθ - μ*N
m*a = mg*sinθ - μ*(mg*cosθ)
Now, we can rearrange the equation to solve for the coefficient of kinetic friction (μ):
μ = (m*a + mg*sinθ) / (mg*cosθ)
Substituting the given values into the equation, we have:
μ = (m*a + mg*sinθ) / (mg*cosθ)
= (m*2.1 + m*9.8*sin(23)) / (m*9.8*cos(23))
= (2.1 + 9.8*sin(23)) / (9.8*cos(23))
Using a calculator to compute the values, we can find the coefficient of kinetic friction between the block and the plane.