To find: 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 ?
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 to Problem 45AYU
a. It is damped motion with a bob of mass 20 kg and a damping factor of .
b. Initial displacement is meters leftward
c. Graph is plotted
d. Displacement of the bob at the second oscillation is meters leftwards.
e. Hence increases displacement tends to become 0 or the bob comes to rest.
Explanation of Solution
Given:
The distance (in meters) of the bob of a pendulum of mass (in kilograms) from its rest position at time (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
Formula used:
The displacement of an oscillating object from its at–rest position at time is given by ,
where is the damping factor or damping coefficient and is the mass of the oscillating object. Here is the displacement at , and is the period under simple harmonic motion (no damping).
Calculation:
a.
,
From the given equation
the damping factor is and
mass of the bob is
It is damped motion with a bob of mass 20 kg and a damping factor of
b. To find the initial displacement let us substitute in the given equation.
Initial displacement is meters leftward
c. The graph of
d. From the graph below we see that the displacement of the bob at the second oscillation is . that is meters leftwards.
e. Displacement of the bob as increases tends to 0. Hence increases displacement tends to become 0 or the bob comes to rest.
Chapter 8 Solutions
Precalculus
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