a crate weighing 3500 newton down an inclined plane at constant speed. The plane is 7.5 meter long. Its height is 3.0 meter. What is the coefficient of sliding friction between the crate and the inclined plane?
To find the coefficient of sliding friction between the crate and the inclined plane, we can use the following steps:
Step 1: Determine the force due to gravity acting on the crate.
The force due to gravity is equal to the weight of the crate, which is given as 3500 Newtons.
Step 2: Calculate the normal force acting on the crate.
The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. In this case, it is equal to the component of the weight acting perpendicular to the inclined plane. To find the normal force, we need to consider the angle of the inclined plane. Since the crate is moving at a constant speed, the force of friction must be equal and opposite to the force component along the inclined plane.
Given that the height of the inclined plane is 3.0 meters and the length is 7.5 meters, we can use trigonometry to find the angle of the inclined plane. The angle (θ) can be calculated as the inverse sine of the height divided by the length:
θ = sin^(-1)(vertical component / hypotenuse)
= sin^(-1)(3.0 / 7.5)
Step 3: Calculate the normal force.
The normal force (N) can be found by taking the component of gravity acting perpendicular to the inclined plane, which is determined by multiplying the weight of the crate by the cosine of the angle:
N = weight × cos(θ)
= 3500 N × cos(θ)
Step 4: Calculate the force of friction.
The force of friction between the crate and the inclined plane can be calculated using the equation:
force of friction = coefficient of sliding friction × normal force
Since the crate is moving at a constant speed, the force of friction is equal to the force of gravity acting along the inclined plane. Therefore:
force of friction = weight × sin(θ)
= 3500 N × sin(θ)
Step 5: Calculate the coefficient of sliding friction.
To find the coefficient of sliding friction (μ), we divide the force of friction by the normal force:
μ = (force of friction) / (normal force)
= (3500 N × sin(θ)) / (3500 N × cos(θ))
The normal force cancels out, leaving:
μ = sin(θ) / cos(θ)
Now you can substitute the value of θ in radians into the equation to find the coefficient of sliding friction.