A pendulum consisting of a small sphere of mass m, suspended by an inextensible and massless string of length l, is made to swing in a vertical plane. If the breaking strength of the string is 2 mg then the maximum angular amplitude of the disc placement from the vertical can be

The problem is asking for the maximum angular amplitude of the pendulum's displacement from the vertical before the string breaks.

Step 1: Understand the forces acting on the pendulum When the pendulum is at its maximum displacement, the forces acting on it are gravity (mg) and tension (T). The tension is at its maximum when the pendulum is at its maximum displacement.

Step 2: Apply Newton's second law According to Newton's second law, the net force acting on an object is equal to its mass times its acceleration. In this case, the net force acting on the pendulum is the difference between the tension and the component of gravity acting perpendicular to the tension, which is mgcos(θ), where θ is the angle of displacement. This net force is equal to the centripetal force required to keep the pendulum moving in a circle, which is ml*(d^2θ/dt^2).

Setting these equal gives T - mgcos(θ) = ml*(d^2θ/dt^2).

Step 3: Solve for the maximum tension The problem states that the maximum tension the string can withstand is 2mg. At the maximum displacement, the tension is at its maximum, so we can set T = 2mg.

Step 4: Solve for the maximum angle Substituting T = 2mg into the equation from step 2 gives 2mg - mgcos(θ) = ml*(d^2θ/dt^2). Since the pendulum is at its maximum displacement, it is momentarily at rest, so (d^2θ/dt^2) = 0. This simplifies the equation to 2mg - mg*cos(θ) = 0. Solving for cos(θ) gives cos(θ) = 2/3.

Therefore, the maximum angular amplitude of the pendulum's displacement from the vertical before the string breaks is θ = cos^-1(2/3).