A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is:
To determine the initial angular acceleration of the rod when the string is cut, we need to consider the torque acting on the rod due to gravity and the moment of inertia of the rod.
Step 1: Calculate the moment of inertia
The rod is hinged at point P, and we need the moment of inertia about point P. For a rod of mass \( M \) and length \( L \) hinged at one end, the moment of inertia \( I \) is given by:
I = \frac{1}{3}ML^2
Step 2: Calculate the torque due to gravity
The force of gravity acts at the center of mass of the rod, which is at a distance \( \frac{L}{2} \) from point P. The torque \( \tau \) about point P is given by:
\tau = Mg \cdot \frac{L}{2}
Step 3: Use the relation between torque and angular acceleration
The torque is related to the angular acceleration \( \alpha \) by the equation:
\tau = I \cdot \alpha
Substituting the expressions for torque and moment of inertia, we have:
