A steel washer is suspended inside an empty shipping crate f

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 37 above the horizontal. The crate has mass 198 kg . You are sitting inside the crate (with a flashlight); your mass is 67 kg . 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 69 with the top of the crate.

What is the coefficient of kinetic friction between the ramp and the crate?

Solution

Incase the box was stationary the sting would hang vertically making a 53 degree angle (90-37.

But since the angle is 68 deggrees apparent \'down\' is 15deg from the vertical. (68-53)

Since the vertical acceleration is 9.8 (g) the horizontal acceleration must be 9.8 x sin15 (from the Triangle made by the string and vertical) = 2.54m/s^2. Since the horizontal acceleration is 2.54, the acceleration down the slope must be 2.54/cos37 (from the triangle made by the slope and horizontal) = 3.18m/s^2

Total force that accelerates the 230kg box down the slope at 3.18m/s^2 is F=m x a = 230 x 3.18 = 730.5. N

And the weight of the box pushing down the slope is 230 x 9.8 x sin37 = 1356.5 N

retarding force due to friction is 1356.5-730.5 = 626.0 N

Compare that to the normal force from the weight of the box which is 230 x 9.8 x cos37 = 1800.1, and the coef of friction is 626.0/1800.1 = 0.348

0.348 is the answer