# Spring System Graphing

Excel Spreadsheet Problem

A mass M is connected to the end of a spring with mass m and force constant k. The mass of the spring is 7.40 g. When the mass/spring system is set into simple harmonic motion, the period of its motion is

T = 2*PI*SQRT((M + (m/3))/K)

Part 1: A two-part experiment is conducted with use of various masses suspended vertically from the spring. First, the masses are hung from the spring and allowed to come to rest. The static extensions are measured for six masses, M. The following is a list of the extensions for each mass:

Mass, M (g) Extension (cm)

20.0 17.0

40.0 29.3

50.0 35.3

60.0 41.3

70.0 47.1

80.0 49.3

Construct a graph of Weight (Mg) versus extension (x). Add a linear trendline to the data. From the slope of your graph, determine a value for k for this spring. (Re-write your equation in terms of weight and extension. Separately, write the physics equation you would use to answer this question by hand. Comparing the two equations, you should see that the slope and k are related.)

Part 2: The mass/spring system is now set into simple harmonic motion. The experiment is repeated with the same various masses and the periods are measured using a stopwatch. For ease of measuring, the total time is measured for 10 oscillations. The following is a list of the periods for 10 oscillations obtained for each mass:

Mass, M (g) Period (s) -- for 10 oscillations

20.0 7.03

40.0 9.62

50.0 10.67

60.0 11.67

70.0 12.52

80.0 13.41

a) If you notice, the measured periods are slightly different than the predicted values from Part 1.b. What could be the cause of the differences (list at least three plausible causes)?

b) Construct a graph of T2 versus mass, M, and add a linear trendline to the data. From the slope of the line, determine a value for k (compare the trendline equation to the equation given at the beginning of the problem squared). Compare this value to the value of k obtained in part 1. What is the percentage difference between the two values?

%Difference = ((K"-K')/K')*100%

c) Using the trendline equation and k value obtained in Part 2.b, determine a value for m. Compare this value with the given value of 7.40 g.

%Difference = ((m"given" - m"measured")/m"given")*100%

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#### Solution Summary

This solution is provided in 254 words in an attached .doc and .xls file. It uses Hooke's Law to determine several equations, including force and mass, and the Excel spreadsheet includes graphing information.

Simple Harmonic Motion: Block-spring system.

(See attached file for full problem description)

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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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