Exercises
(1) Two blocks (each of width , depth , height and weight density ) initially stacked on the ground are lifted to side by side positions on a shelf which is above the ground as in scene 1. Compute the work performed to lift the blocks to their final positions on the shelf. (For all of these exercises assume the weight of the hoist and all frictional forces are negligible.)
(2) Is the distance the hoist moves each block along the rail parallel to the ground before releasing it on the shelf relevant to your computation in exercise (1)? Explain.
(3) Now suppose the width, depth, height, and weight density of the (initially) lower and upper blocks are , , , , and , , , , respectively, and the height of the shelf above the ground is . Construct an expression in terms of these parameters for the work performed in lifting the blocks to side by side positions on the shelf. (See scene 2.)
(4) Are the dimensions of the (initially) lower block interchangeable in your formula from the previous problem? How about the dimensions of the upper block? Explain.
(5) Revise your formula from problem (3) for the situation depicted in scene 3 where the initially lower block is stacked on top of the other block on the shelf.
(6) Now revise your formula for the situation in scene 4 where the final stacking order on the shelf matches the initial stacking order on the ground. Simplify your answer algebraically as much as possible, and provide an interpretation for your simplified formula.
(7) Do your formulas from problems (5) and (6) always yield the same values (regardless of the values of the parameters)? Do they ever yield the same values? Explain.
(8) A stack of four short cylinders with weight density , height and radii , , , and are lifted to side by side positions on a shelf of height as in scene 5. Compute the work performed to lift the blocks to their final positions.
(9) Suppose a stack of cylinders, , of common height and common weight density are lifted from the ground to side by side positions on a shelf of height . Given that cylinder has radius , for , write an expression in sigma notation for the work performed in lifting the stack. (Assume the cylinders are stacked in order of increasing index with at the bottom and at the top.)
(10) Now suppose that the function gives the radii of the cylinders as a function of the initial height of (the bottom of) each cylinder, and that the total height of the stack of cylinders is . Rewrite your sigma expression in terms of the function and stack height .
(11) Write the definite integral that your last sigma expression tends to as goes to (assuming that is defined throughout the interval and is continuous there). Can you think of any physical quantity that could be modelled by this integral?
