(a)

The displacement of the bob of the pendulum is .

The equation represents the damped Motion of the object.

The displacement by an oscillating object from rest position at time  is

.

Compare the displacement equation with the above equation.

Here, damping factor is 

Mass of the object, .

Mass of the object is .

(b)

Find Initial displacement of the bob.

Substitute  in the displacement equation.

The negative sign indicates the displacement to the left of the resting position.

Initial displacement of the bob is .

(c)

Motions are

and gives the path of motion.

Graph :

(1) Draw the coordinate plane.

(2) Graph the function .

Graph the and on the same plane.

(d)

Find displacement at the start of second oscillation.

Time period of the function is .

Here .

The second oscillation starts at .

Substitute in the displacement equation.

The negative sign indicates the displacement to the left of the rest position.

Displacement at the start of second oscillation is from the left of the rest position.

(e)

Displacement of the bob as time increases without bound(  tends to infinity).

Displacement of the bob decreases and gets closer to as time increases without bound.

(a) 

Damping factor is . 

Mass of the object is .

(b)

Initial displacement of the bob is from the left of the rest position.

(c)

Graph :

(d)

Displacement at the start of second oscillation is  from the left of the rest position.

(e)

Displacement of the bob decreases and gets closer to as time increases without bound.