Steps to Solve

1. Identify the forces acting on the mass The forces acting on the mass ( m ) in the pendulum motion are the tension ( T ) in the string and the gravitational force ( mg ). The component of the gravitational force acting along the direction of the string is ( mg \cos \theta ).

2. Write down the net force equation The net force acting on the mass can be expressed using Newton’s second law. The radial component of the forces gives: $$ T - mg \cos \theta = \frac{mv^2}{L} $$ This equation relates tension, gravitational force, and the centripetal force needed to keep the mass moving in a circular path.

3. Set up the equation for motion The tangential acceleration ( a_t ) can be defined as: $$ a_t = g \sin \theta $$ This will help in understanding the component of acceleration along the direction of motion.

4. Integrate to find velocities Using the relationship between acceleration and velocity, you can set up the integration: $$ - \int_{v_0}^{v_1} v , dv = -g R \int_0^{\theta} \sin \theta , d\theta $$ This establishes the connection between initial and final velocities with the angle swept during the motion.

5. Solve for change in velocity The integration yields: $$ \frac{v_0^2 - v_1^2}{2} = gR $$ Where ( R ) is the radius and gives an expression for the relationship between initial and final velocities in terms of gravitational acceleration.

6. Substitute initial and final velocities to determine tension From the net force equation, you can substitute ( v_1 ) to find tension ( T ): $$ T = \frac{mv_1^2}{L} + mg \cos \theta $$