one end of a chain(end A) is hinged at O and other end(end B) is also raised up to O. calculate hinge reaction when the system is released and end B descends hieght L/4. chain mass M length L

To determine the hinge reaction when one end of a chain is hinged at point O and the other end is raised to the same point, we need to analyze the forces acting on the chain as it descends. When the system is released and end B descends a height of L/4, we can apply principles of mechanics to find the hinge reaction at point O.

Understanding the System

We have a uniform chain of mass M and length L, with one end (end A) fixed at point O. When end B is raised to point O and then released, it will fall due to gravity. As it descends, the chain will experience a change in potential energy, which will be converted into kinetic energy as it moves downward.

Forces Acting on the Chain

When the chain is in motion, the following forces are acting on it:

• Weight of the Chain: The total weight (W) of the chain is given by W = Mg, where g is the acceleration due to gravity.
• Hinge Reaction at O: This is the force exerted by the hinge to support the chain.

Calculating the Change in Potential Energy

When end B descends a height of L/4, the change in potential energy (ΔPE) of the chain can be calculated. The center of mass of the chain will also descend, and since the chain is uniform, the center of mass is located at L/2 from end A. The change in height of the center of mass when B descends L/4 is:

• Initial height of center of mass = L/2
• Final height of center of mass = L/2 - L/4 = L/4

The change in height (h) of the center of mass is:

h = (L/2) - (L/4) = L/4

The change in potential energy is then:

ΔPE = Mg * h = Mg * (L/4) = (MgL)/4

Applying Energy Conservation

As the chain descends, the potential energy lost will be converted into kinetic energy (KE) of the chain. When the chain has descended L/4, we can express the kinetic energy as:

KE = (1/2)Mv²

By conservation of energy, we have:

ΔPE = KE

Thus, we can set up the equation:

(MgL)/4 = (1/2)Mv²

Solving for v, we get:

v² = (gL)/2

v = √((gL)/2)

Finding the Hinge Reaction

To find the hinge reaction (R) at point O, we need to consider the forces acting on the chain at the moment just before it reaches the height L/4. The chain is in dynamic equilibrium, so we can apply Newton's second law in the vertical direction:

R - Mg = Ma

Where a is the acceleration of the center of mass. The acceleration can be derived from the change in velocity:

a = (dv/dt) = (d/dt)(√((gL)/2))

However, for simplicity, we can consider the instantaneous acceleration due to gravity acting on the chain:

a = g/2 (since the chain is falling under gravity and the center of mass is at L/4).

Substituting this back into our equation gives:

R - Mg = M(g/2)

R = Mg + M(g/2) = (3Mg)/2

Final Result

The hinge reaction at point O when end B descends a height of L/4 is:

R = (3Mg)/2

This indicates that the hinge must exert a force greater than the weight of the chain to support the dynamic conditions as the chain falls. This analysis illustrates the interplay between gravitational forces and the motion of the chain, providing insight into the mechanics involved in this scenario.