Consider a rope with length l , mass per unit length ? , experiencing a gravitat

Question

Consider a rope with length l, mass per unit length ?, experiencing a gravitational acceleration g and hanging vertically as shown. (Figure 1) Let y refer to the height of a point P above the bottom of the rope.

Consider a rope with length l, mass per unit length ?, experiencing a gravitational acceleration g and hanging vertically as shown. (Figure 1) Let y refer to the height of a point P above the bottom of the rope. The force exerted on the rope by the ceiling is in the direction. Find F, the magnitude of the force exerted on the rope by the ceiling. What is the tension TP at point P in the rope?

Explanation / Answer

Let:
dy be the length of an infinitesimal portion of the rope at height y from the lower end of the rope,
T1 be the tension at a point immediately below,
T2 be the tension at a point immediately above,
g be the acceleration due to gravity.

The downward forces on the portion are T1 and its own weight lambda * g * dy
= T1 + lambda * g * dy

The upward force is T2.

Equating the two for equilibrium of the portion:
T2 = T1 + lambda * g * dy.
T2 - T1 = lambda * g * dy ...(1)

T2 - T1 is the increase in tension as you move up the rope a distance dy.

Therefore (1) can be written:
dT = lambda * g * dy.

Integrating:
int ( 0 to y) [ dT ] = lambda * g * int(0 to y) [ dy ]
T_P = lambda * y * g.

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