Good Vibes: Introduction to Oscillations
Β
The conditions that lead to simple harmonic motion are as follows:
There must be a position ofΒ stable equilibrium.
There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring forceΒ FβΒ Β is given by β, where is the displacement from equilibrium and kΒ is a constant that depends on the properties of the oscillating system.
The resistive forces in the system must be reasonably small.
Consider a block of massΒ mΒ attached to a spring with force constantΒ k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located atΒ x=0. If the block is pulled to the right a distanceΒ AΒ and then released,Β AΒ will be theΒ amplitudeΒ of the resulting oscillations.
Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.
Part A
After the block is released fromΒ x=A, it will
(a) remain at rest
(b) move to the left until it reaches equilibrium and stop there.Β
(c) move to the left until it reaches x= βA and stop there.Β
(d)Β move to the left until it reachesΒ x=βA and then begin to move to the right.
Β
Β
Β
The time it takes the block to complete one cycle is called theΒ period. Usually, the period is denotedΒ TΒ and is measured in seconds.
TheΒ frequency, denotedΒ f, is the number of cycles that are completed per unit of time:Β f=1/T. In SI units,Β fΒ is measured in inverse seconds, or hertz (Hz).
Part B
If the period is doubled, the frequency is
(a) unchangedΒ
(b) doubledΒ
(c) halved
Β
Part C
An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequencyΒ f?
Express your answer in hertz.
Β
Part D
If the frequency is 40 Hz, what is the periodΒ T?
Express your answer in seconds.
Β
The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.
Note that the vertical axis represents theΒ x-coordinate of the oscillating object, and the horizontal axis represents time.
Β
Part E
Which points on theΒ x-axis are located at a distance AΒ from the equilibrium position?
(a) R only
(b) Q only
(c) both R and Q
Β
Part F
Suppose that the period isΒ T. Which of the following points on theΒ t-axis are separated by the time intervalΒ T?
(a) K and L
(b) K and MΒ
(c) K and P
(d) L and NΒ
(e) M and PΒ
Β
Β
Now assume for the remaining Parts G - J, that theΒ xΒ coordinate of point R is 0.12 m and theΒ tΒ coordinate of point K is 0.0050 s.
Part G
What is the periodΒ T?
Express your answer in seconds.
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Part H
How much timeΒ tΒ does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?
Express your answer in seconds.
Β
Part I
What distanceΒ dΒ does the object cover during one period of oscillation?
Express your answer in meters.
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Part J
What distanceΒ dΒ does the object cover between the moments labeled K and N on the graph?
Express your answer in meters.