Physics

Problem

![](https://storage.googleapis.com/diraq-contents/question-bank/question-images/f3e26ff12748/2025_11_20_4fb16085cfb86337503bg-19.jpg) 2009M3 A block of mass

M / 2

rests on a frictionless horizontal table, as shown above. It is connected to one end of a string that passes over a massless pulley and has another block of mass

M / 2

hanging from its other end. The apparatus is released from rest. a. Derive an expression for the speed

v_{\mathrm{h}}

of the hanging block as a function of the distance

d

it descends. Now the block and pulley system is replaced by a uniform rope of length

L

and mass

M

, with one end of the rope hanging slightly over the edge of the frictionless table. The rope is released from rest, and at some time later there is a length

y

of rope hanging over the edge, as shown below. Express your answers to parts (b), (c), and (d) in terms of

y, L, M

, and fundamental constants. ![](https://storage.googleapis.com/diraq-contents/question-bank/question-images/8f8e115b48b0/2025_11_20_4fb16085cfb86337503bg-19.jpg) b. Determine an expression for the force of gravity on the hanging part of the rope as a function of

y

. c. Derive an expression for the work done by gravity on the rope as a function of

y

, assuming

y

is initially zero. d. Derive an expression for the speed

v_{\mathrm{r}}

of the rope as a function of

y

. e. The hanging block and the right end of the rope are each allowed to fall a distance

L

(the length of the rope). The string is long enough that the sliding block does not hit the pulley. Indicate whether

v_{\mathrm{h}}

from part (a) or

v_{\mathrm{r}}

from part (d) is greater after the block and the end of the rope have traveled this distance. __

v_{\mathrm{h}}

is greater.

\_\_\_\_

v_{\mathrm{r}}

is greater.

\_\_\_\_

The speeds are equal. Justify your answer.

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