1. (12 pt) A pendulum consists of a small object of mass m fastened to the end of an inextensible cord of length L. Initially, the pendulum is drawn aside through an angle of θ = 60° with the vertical and held by a horizontal string as shown in the diagram above. This string is burned later so that the pendulum is released to swing to and fro.
a. In the space below draw a force diagram identifying all of the forces (not components) acting on the object while it is held at rest by the string.
b. Derive the formula of the force of tension in the cord before the string is burned. You may use only m, L, θ, and fundamental constants. Do not substitute any numbers. If you need to draw anything to assist your solution, use the space provided below.
C. Derive the formula of the force of tension in the string before the string is burned. You may use only m, L, θ, and fundamental constants
D. The string is burned. Draw a FBD for the pendulum at the moment when it passes through the bottom of the swing.
E. Draw the vectors of velocity and acceleration of the pendulum as it passes the lowest point swinging from left to right. If some vectors are equal to zero, make a clear statement.
F. The pendulum starts swinging from left to right and back. What quantities will define the period of the pendulum?
G. In a different imaginary experiment you bring the same pendulum on a newly discovered planet that has one-half the Earth’s mass and twice the Earth’s radius. Is the period of the pendulum in this imaginary environment greater than, less than or equal to your answer in part (F). Justify.
H. After several swings, the cord breaks when the ball passes through the bottom of the swing. Briefly describe the motion of the ball after the break.