The answer to each of the following questions can be written as a numeric coeffi

Question

The answer to each of the following questions can be written as a numeric coefficient times some combination of the variables m, g, and d (representing mass, acceleration due to gravity, and the height of one step, respectively). The appropriate combination of variables is indicated. Enter only the numeric coefficient. (Example: If the answer is 1.23mgd, just enter 1.23) Three different objects, all with different masses, are initially resting at the bottom of a set of steps, each with a uniform height d. In this position, the total gravitational potential energy of the three object system is said to be zero. If the objects are then relocated as shown, what is the new total potential energy of the system?

This potential energy was calculated relative to the bottom of the stairs. If you were to redefine the reference height such that the total potential energy of the system becomes zero, how high above the bottom of the stairs would the new reference height be?

Explanation / Answer

Total PE = mgd + (2.21m)*g*(d + d) + (3.1m)*g*(d + d + d) = 14.72*mgd

Suppose reference height be h.

So, Total PE = 0 = mg*(d - h) + (2.21m)*g*(2d - h) + (3.1m)*g*(3d - h)

(1 + 2.21*2 + 3.1*3)d = h(1 + 2.21 + 3.1)

h = 2.33 d

Suppose reference height be H from bottom .

in the third case assume height to be H*d

then, 2.21(H-2) = 3.10(3-H)

solving, H = 2.58

Hence answer is : 2.675d

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