#309 The dimension h is to be determined

The dimension h is to be determined - Mechanical Engineering

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Free Chegg Question

The dimension h is to be determined so that a worker can comfortably slide boxes weighing up to 100 lb up and down a frictionless incline. If the worker can apply a 501b horizontal force to the box, what is the largest value h should have?

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Free Chegg Answer

1. 1. 1. Step 1 of 4

         Given data:-

         Weight of the boxes, 

         Horizontal force applied by the worker, 

         Let,  be the angle made by the inclined plane with the horizontal.

         The normal reaction by the inclined slope on the boxes acts normal to the inclined surface. The weight of the boxes acts vertically downwards.

      2. Step 2 of 4

         The Free Body Diagram of the boxes is

         

      3. Step 3 of 4

         From the diagram, it is clear that

         

         So, 

         For the largest value of h, the forces on the boxes should just be in equilibrium.

         From the free body diagram, balancing the vertical forces, we get

         

          …… (1)

         Similarly, balancing the horizontal forces, we have

         

          …… (2)

      4. Step 4 of 4

         From equations (1) and (2), we can write

         

         Substituting in the values of the variables, we get

         

         Hence, the optimum value of the height is 

         For any value more than 1.5 ft, the weight component of the boxes down the inclined plane will be higher than the force applied along the same direction. Hence, 1.5 ft is the maximum value that h can take.