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? U_g, system = mgd 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? Now, find a new reference height (measured again from the bottom of the stairs) such that the highest two objects have the exact same potential energy.

Explanation / Answer

let us consider the value of m1 , m2 , m3 values as given below

m1 = 5.20m

m2 =1.46m

m3 =m

A) Ep = g*(m1*d + m2*2d + m3*3d) = gd(5.20m + 2x1.46m + 3m)=11.1mgd

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B) h = Ep / (g(m1 + m2 + m3)) = d(5.20m + 2x1.46m + 3m) / (5.20m + 1.46m + m)=1.44d

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C) In order for m2 and m3 to have the same potential energy, m3 < m2. For reference heights near the bottom of the stairs, m3 must be significantly larger than m2.
Let's say that for the equipotential condition, m2 is at height h from the reference point. It follows that m3 is at height (h + d). It further follows that
m2 * h = m3 * (h + d) = m3 * h + m3 * d
(m2 - m3) * h = m3 * d
h = m3 * d / (m2 - m3)

=mxd/(1.46m-m)

=2.17d

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