# Simple Harmonic Motion: Block-spring system.

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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; 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 is given by , where is the displacement from equilibrium and 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.

Consider a block of mass attached to a spring with force constant , as shown in the figure . 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 . If the block is pulled to the right a distance and then released, 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.

A) After the block is released from , it will

remain at rest.

move to the left until it reaches equilibrium and stop there.

move to the left until it reaches and stop there.

move to the left until it reaches and then begin to move to the right.

B) The time it takes the block to complete one cycle is called the period. It is usually denoted and is measured in seconds.

The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ).

If the period is doubled, the frequency is

unchanged.

doubled.

halved.

C) An oscillating object takes 0.10 to complete one cycle; i.e., its period is 10 . What is its frequency ?

Express your answer in hertz.

=

D) If the frequency is 40 , what is the period ?

Express your answer in seconds.

=

The following questions refer to the figure which 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 from the equilibrium position?

R only

Q only

both R and Q

F) Suppose that the period is . Which of the following points on the t axis are separated by the time interval ?

K and L

K and M

K and P

L and N

M and P

G) Now assume that the x coordinate of point R is 0.12 , and the t coordinate of point K is 0.0050 .

What is the period ?

Express your answer in seconds.

=

H) How much time 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 does the object cover during one period of oscillation?

Express your answer in meters.

=

J) What distance does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

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#### Solution Summary

The block spring system on a horizontal surface is given. Related eight questions are discussed and answered.