A steel washer is suspended inside an empty shipping crate from a light string a

Question

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 39 degrees above the horizontal. The crate has mass 244 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 75 degrees with the top of the crate. What is the coefficient of kinetic friction between the ramp and the crate?

Explanation / Answer

Have a look at this image : http://i48.tinypic.com/b9gcnp.png In the last image a = actual acceleration of the crate g = gravity You can see that a / sin(a) = g / sin(f) so a = g sin(a) / sin(f) a = 9.8 sin(24) / sin(105) a = 4.127 m/s² Given the angle of the incline, the acceleration without friction should have been: a' = 9.8 sin(39) = 6.167 m/s² So the deceleration by friction is: f = 6.167 - 4.127 = 2.036 m/s² The component of gravitation that is perpendicular to the incline is: g' = 9.8 cos(39) = 7.617 m/s² And finally, the coefficient of kinetic friction is: µ = f / g' µ = (2.036 m/s²) / (7.617 m/s²) µ = 0.267 < - - - - - - - - - - - - - - - - - - - - answer - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Edit: The above is the 'normal way' to solve a question like this. That is... if you don't know the shortcut formula shown by RickB Below, I'll derive that shortcut. Let ?1 = angle of the incline ?2 = angle of the the string a, ß, f the angles as shown in the image. a = ?1 + ?2 - 90 ß = 90 - ?1 f = 180 - ?2 (follow the steps on the image to find these) a = g sin(a) / sin(f) a = g sin(?1 + ?2 - 90) / sin(180 - ?2) a = -g cos(?1 + ?2) / sin(?2) a' = g sin(?1) f = a' - a f = (g sin(?1)) - (-g cos(?1 + ?2)/sin(?2)) f = g (sin(?1) + cos(?1 + ?2)/sin(?2)) g' = g cos(?1) µ = f / g' µ = g (sin(?1) + cos(?1 + ?2)/sin(?2)) / g cos(?1) µ = (sin(?1) + cos(?1 + ?2)/sin(?2)) / cos(?1) µ = tan(?1) + cos(?1 + ?2) / cos(?1)sin(?2) by cos(?1 + ?2) = cos(?1)cos(?2) - sin(?1)sin(?2) you can say: µ = tan(?1) + (cos(?1)cos(?2) - sin(?1)sin(?2)) / cos(?1)sin(?2) µ = tan(?1) + (cos(?1)cos(?2) / cos(?1)sin(?2)) - (sin(?1)sin(?2) / cos(?1)sin(?2)) µ = tan(?1) + cos(?2)/sin(?2) - sin(?1)/cos(?1) µ = tan(?1) + cotg(?2) - tan(?1) µ = cotg(?2) µ = 1/tan(?2)

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