What is the displacement of the bob at the start of the second oscillation?

Question

The distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.

(a) Describe the motion of the object. Be sure to give the mass and damping factor.

(b) What is the initial displacement of the bob? That is, what is the displacement at t = 0?

(c) Graph the motion using a graphing utility.

(d) What is the displacement of the bob at the start of the second oscillation?

(e) What happens to the displacement of the bob as time increases without bound?

Answer 1

Step 1:

(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 40 kg.

Solution:

Damping factor is .

Mass of the object is 40 kg.

Answer 2

Step 1:

(b)

Find Initial displacement of the bob.

Substitute in the displacement equation.

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

Initial displacement of the bob is .

Solution:

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

Answer 3

Step 1:

(c)

and gives the path of motion.

 Graph the function .

Graph the and .

Note: The blue color in the graph indicates the motion of the function.

Solution:

Graph the motion of bob is

Answer 4

Step 1:

(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.

Solution:

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

Answer 5

Step 1:

(e)

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

Displacement of the bob decreases and gets closer to 0 as time increases without bound.

Solution:

Displacement of the bob decreases and gets closer to 0 as time increases without bound.