A 214-kg crate rests on a surface that is inclined above the horizontal at an angle of 19.4°. A horizontal force (magnitude = 541 N and parallel to the ground, not the incline) is required to start the crate moving down the incline. What is the coefficient of static friction between the crate and the incline?
forces down slope = Fd
Fd = m g sin 19.4 + 541 cos 19.4
forces normal to slope
= m g cos 19.4 - 541 sin 19.4
so force up slope =
Ff = mu ( m g cos 19.4 - 541 sin 19.4)
no acceleration so Ff=Fd
mu (m g cos 19.4 - 541 sin 19.4)=m g sin 19.4 + 541 cos 19.4
To determine the coefficient of static friction between the crate and the incline, we can use the following steps:
1. Resolve the weight of the crate into its components. The weight of the crate is given by W = mg, where m is the mass of the crate (214 kg) and g is the acceleration due to gravity (9.8 m/s^2). So, the weight is W = (214 kg)(9.8 m/s^2) = 2097.2 N. The vertical component of the weight can be calculated as W_vertical = W * cos(19.4°) and the horizontal component as W_horizontal = W * sin(19.4°).
2. Use the horizontal component of the weight to determine the maximum static friction force. Since the crate is not moving, the force of static friction balances out the horizontal component of the weight. So, the maximum static friction force, Fs_max, is equal to the horizontal component of the weight: Fs_max = W_horizontal.
3. Calculate the force required to start the crate moving down the incline. The force required to overcome static friction and initiate motion is given as F = 541 N.
4. Compare the force required to start the crate moving (F) with the maximum static friction force (Fs_max). If the applied force F is smaller than or equal to the maximum static friction force Fs_max, then the crate will not move. The coefficient of static friction is then given by the ratio of the static friction force to the normal force: μ_static = Fs_max / W_vertical.
Using these steps, we can now calculate the coefficient of static friction:
1. W_vertical = W * cos(19.4°) = 2097.2 N * cos(19.4°) ≈ 1960.5 N
2. Fs_max = W_horizontal = W * sin(19.4°) = 2097.2 N * sin(19.4°) ≈ 684 N
3. F = 541 N
4. Since F (541 N) is less than Fs_max (684 N), the crate will not move and the coefficient of static friction is given by μ_static = Fs_max / W_vertical = 684 N / 1960.5 N ≈ 0.349.
Therefore, the coefficient of static friction between the crate and the incline is approximately 0.349.
To find the coefficient of static friction between the crate and the incline, we need to use the concept of forces and their relationships on inclined surfaces.
First, let's break down the forces acting on the crate on the inclined plane:
1. The gravitational force acting straight down is given by: Fg = mg, where m is the mass of the crate (214 kg) and g is the acceleration due to gravity (9.8 m/s²).
2. There is a normal force acting perpendicular to the incline exerted by the surface. This force cancels out the vertical component of the crate's weight and is given by: Fn = mg * cos(θ), where θ is the angle of the incline (19.4°).
3. The force of static friction (Fs) opposes the applied horizontal force, preventing the crate from sliding down the incline until the force of static friction is overcome.
To start the crate moving down the incline, we need to find the maximum static friction force (Fs_max) that can be exerted. This maximum static friction force can be calculated using the equation: Fs_max = μs * Fn, where μs is the coefficient of static friction.
Since the crate is on the verge of moving, the horizontal force applied (F_applied) is equal to the maximum static friction force: F_applied = Fs_max.
Substituting the equations mentioned above, we get:
μs * Fn = F_applied
Now, let's plug in the known values:
Fn = mg * cos(θ)
μs * (mg * cos(θ)) = F_applied
Rearranging the equation to solve for μs:
μs = F_applied / (mg * cos(θ))
Plugging in the given values:
F_applied = 541 N
m = 214 kg
g = 9.8 m/s²
θ = 19.4°
μs = 541 / (214 * 9.8 * cos(19.4°))
Calculating the value using a calculator, we find that the coefficient of static friction (μs) between the crate and the incline is approximately 0.312.