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^circ above the horizontal. The crate has mass 244{ m kg} . You are sitting inside the crate (with a flashlight); your mass is 67{ m 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^circ 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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