A steel washer is suspended inside an empty shipping crate from a light string attached to the top of the crate. The crate slides down a long ramp that is inclined at an angle of 38∘ above the horizontal. The crate has mass 171kg . You are sitting inside the crate (with a flashlight); your mass is 62kg . As the crate is sliding down the ramp, you find the washer is at rest with respect to the crate when the string makes an angle of 56∘ with the top of the crate.
What is the coefficient of kinetic friction between the ramp and the crate?
To find the coefficient of kinetic friction between the ramp and the crate, we can start by analyzing the forces acting on the crate.
First, let's determine the gravitational force acting on the crate. The weight of the crate can be calculated using the formula:
Weight = mass * gravity
where mass is the mass of the crate and gravity is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Weight of the crate = 171 kg * 9.8 m/s^2 = 1675.8 N
Next, let's examine the forces acting on the crate as it slides down the ramp. There are two main forces: the gravitational force and the force of friction.
The force of friction can be calculated using the formula:
Frictional Force = coefficient of kinetic friction * normal force
where the normal force is the perpendicular force exerted by the ramp on the crate.
In this case, the normal force is equal to the component of the gravitational force perpendicular to the ramp. This can be calculated by:
Normal Force = Weight of the crate * cosine(angle of inclination)
Normal Force = 1675.8 N * cos(38∘)
Now, let's calculate the frictional force using the equation above. However, we need to note that the washer is at rest with respect to the crate, which means the net force on the washer is zero. This implies that the frictional force is equal to the tension in the string.
So we have:
Tension in the string = Frictional Force = coefficient of kinetic friction * Normal Force
Tension in the string = coefficient of kinetic friction * 1675.8 N * cos(38∘)
Finally, we know that the tension in the string can be represented by the vertical component of the tension force, which is given by:
Tension in the string = Weight of the washer * sin(angle of inclination - angle of the string with the top of the crate)
Weight of the washer = mass of the washer * gravity
Plugging in the given angle of the string and the mass of the washer, we can solve for the tension in the string.
Tension in the string = 62 kg * 9.8 m/s^2 * sin(38∘ - 56∘)
Now we can equate both expressions for the tension in the string and solve for the coefficient of kinetic friction.
coefficient of kinetic friction * 1675.8 N * cos(38∘) = 62 kg * 9.8 m/s^2 * sin(38∘ - 56∘)
After solving the equation, we can find the value of the coefficient of kinetic friction between the ramp and the crate.