Uniform Wire with Attached Masses

When a uniform wire of mass M and length L is hinged at the center, it can freely rotate in a vertical plane about a horizontal axis. In this scenario, two masses of 2m and M are rigidly attached to the ends of the wire.

Initial State and Release

Initially, the wire is at rest and in a horizontal position. When the system is released, it starts to rotate due to the gravitational forces acting on the attached masses.

Analysis of the System

To determine the initial angular acceleration of the bar, we need to consider the torques acting on the system.

1. Torque due to the attached mass M:
- The mass M is attached at one end of the wire, creating a torque about the hinge.
- The torque can be calculated using the formula: τ = r x F, where r is the perpendicular distance from the axis of rotation to the line of action of the force.
- In this case, the torque due to mass M is zero since the perpendicular distance is zero (the mass is attached at the center of the wire).

2. Torque due to the attached mass 2m:
- The mass 2m is attached at the other end of the wire, creating a torque about the hinge.
- The torque can be calculated using the formula: τ = r x F, where r is the perpendicular distance from the axis of rotation to the line of action of the force.
- In this case, the torque due to mass 2m is non-zero as the perpendicular distance is L/2.

Calculating the Torque

The torque due to mass 2m can be calculated as follows:

τ = r x F
= (L/2) x (2mg)
= mgL

Calculating the Angular Acceleration

The net torque acting on the system is equal to the moment of inertia (I) multiplied by the angular acceleration (α). Since the wire is uniform, its moment of inertia can be calculated as I = (1/3)ML^2.

τ = Iα
mgL = (1/3)ML^2 α

Simplifying the equation, we get:

α = (3g)/2L

Therefore, the initial angular acceleration of the bar is (3g)/2L.

Conclusion

In conclusion, the initial angular acceleration of the uniform wire with attached masses is (3g)/2L. This result is obtained by analyzing the torques acting on the system due to the attached masses. The torque due to the mass M is zero, while the torque due to the mass 2m is non-zero and causes the wire to rotate when released.