A moving company uses the pulley system in figure 1 (which you may have to just

Question

A moving company uses the pulley system in figure 1 (which you may have to just imagine, provided I give you enough info) to lift heavy crates up a ramp. The ramp is coated with rollers that make the crate's motion essentially frictionless. A worker piles cinder blocks onto the plate until the plate moves down, pulling the crate up the ramp. Each cinder block has mass 10kg. The plate has a mass 5 kg. The rope is nearly massless, and the pulley is essentially frictionless. The ramp makes a 30 degree angle with the ground. The crate has mass 100kg. Let W1 denote the combined weight of the plate and the cinder blocks piled on the plate. Let T denote the tension in the rope. And let W2 denote the crate's weight.
a) what is the smallest number of cinder blocks that need to be placed on the plate in order to lift the crate up the ramp?
b) the ramp exerts a normal force on the crate, directed perpendicular to the ramp's surface. This normal force has magnitude W2cos30 degrees. Explain.

Explanation / Answer

If I have the picture correctly in my head, then what helps us most is orientating the axis so that the y-axis is normal to the surface of the incline and the x-axis runs down the slope of the ramp. Drawing a free body diagram with the fores on the crate (weight, normal force, and tension by the rope), we see that a triangle is formed from the components of the crates weight that is congruent to the triangle formed by the ramp. Looking at these triangles, one can then see, that the vertical component of the crates weight must equal the normal force applied on the crate, as the crate is not moving in the y-direction as defined earlier. Thus, the normal force equals W2*cos(30), for part b. For part a., one should recognize that the two tensions in the rope will cancel as they are equal and opposite. Therefore, using our free body diagram, we should see that weight_plate+weight_blocks=W2*Sin(30). So, canceling the g's, we have: (5+10*x)=100*sin(30), where x is the number of blocks Solving, we find that x=4.5, so at minimum, we will need 5 blocks.

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