Learning Goal: To learn the basic terminology and relationships among the main c
ID: 1793951 • Letter: L
Question
Learning Goal:
To learn the basic terminology and relationships among the main characteristics of simple harmonic motion.
Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth.
The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows:
There must be a position of 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 force F is given by F =kx , where x is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.
The resistive forces in the system must be reasonably small.
In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them.
Figure (1)
Consider a block of mass m attached to a spring with force constant 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 at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.
Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.
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A: After the block is released from x=A, it will
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The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.
The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).
B: If the period is doubled, the frequency is
C: An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?
Express your answer in hertz.
D: If the frequency is 40 Hz, what is the period T ?
Express your answer in seconds.
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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 the x coordinate of the oscillating object, and the horizontal axis represents time.
E: Which points on the x axis are located a distance A from the equilibrium position?
F: Suppose that the period is T. Which of the following points on the t axis are separated by the time interval T?
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Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.
G: What is the period T ?
Express your answer in seconds.
H: How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?
Express your answer in seconds.
I: What distance d does the object cover during one period of oscillation?
Express your answer in meters.
J: What distance d does the object cover between the moments labeled K and N on the graph?
Express your answer in meters.
Intro:Learning Goal:
To learn the basic terminology and relationships among the main characteristics of simple harmonic motion.
Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth.
The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows:
There must be a position of 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 force F is given by F =kx , where x is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.
The resistive forces in the system must be reasonably small.
In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them.
Figure (1)
-A 0Explanation / Answer
A) move to the left until it reaches x = -A and then begin to move to the right.
As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and block will slow down, temporarily coming to rest at x = -A.
B) If the period is doubled, the frequency is Halved.
C) f = 1/T = 1/0.10 = 10 Hz
D) T = 1/f = 1/40 = 0.025 s
E) both R and Q
F) K and P are separated by phase interval of 2 , or the block gets back to the same point.
G) In moving from the point t = 0 to the point K, what fraction of a full wavelength is covered? Call that fraction a. Then you can set aT = 0.005. Dividing by the fraction a will give the period T
T = 0.02 s
H) t = 0.01 s
The block travels only half the period.
I) d = 0.48 m 4 times of the amplitude.
J) d = 0.36 m 3 times of the amplitude.
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