OSCILLATION OF A SPRING-MASS SYSTEM Exp. #23

(Alward/Harlow web File: "spring.doc" 3-30-04) 1.25 hours

Name: ______Partners: ______Section No. ______

Equipment Needed: 2 meter pole Large blue table clamp Utility clamp 45-cm rod Pasco spring (seven cm) 50 gram weight hanger (don't use the angled-bottom Beck hangers) 2-20 g mass 1-10 g mass 1-50 g mass 1-100 g mass Motion sensor Meter stick Masking tape ruler

WARNING: do not over-stretch the spring. The maximum-allowable hanging mass is 150 g, and the maximum stretch is 60 centimeters.

I. PURPOSE AND THEORY

The period of oscillation of a spring-mass system is the amount of time required for one "round-trip", i.e, the time required for one complete cycle. A mass m oscillating at the end of a spring has a period of oscillation T given by

T = 2p(m/k)1/2 (1)

where k is the spring constant. The mass m is measured in kilograms, the spring constant k is measured in Newtons/meter, and the period T is in seconds.

In this investigation the student will observe the period of oscillation for several different masses and compare the observed periods with the values predicted by Equation 1.

I.  SETUP

  1. Turn on the Pasco Interface box, then turn on the computer. Plug the motion sensor cables into the interface box.
  2. Attach a two-meter pole to the side of the table.
  1. Attach a small clamp and crossbar at the top of the pole, about 130 cm above the table.
  1. Use masking tape to attach the Pasco spring to the crossbar, about 15 cm from the pole.
  1. Place the motion sensor on the table below the spring, facing upward.
  1. Wrap a piece of masking tape around the pole 45 cm above the table.
  1. If the computer screen does not already display the graph of position versus time, turn on the signal interface, then turn on the computer. Open the file "Spring.sws".

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III. MEASURING THE SPRING CONSTANT

The spring constant k of a spring tells how much force is required to increase its length by a given amount. Thus,

k = DF/Dx (2)

where DF = added force and Dx = increase in length. The spring constant is measured in N/m.

1. Hang a 50-gram weight hanger from the spring. Of course, this weight will stretch the spring, and if we were to measure the amount by which the spring is stretched by this weight, we could compute k. Since the starting length of the spring is quite small, and is therefore difficult to measure with high accuracy, we will begin our measurements by starting out with a spring which is already stretched.

2. Place a meter-stick vertically along side the stretched spring with the zero mark resting on the table top and record the initial position x0 of the bottom of the weight hanger.

3. Stretch the spring by hanging the masses shown in Table I. These masses include the mass of the hanger. Record the new positions of the bottom of the hanger.

4. Calculate the ratios k = DF/Dx = Dmg/Dx. Note: each Dm is referenced to the hanger mass, i.e., Dm = m - m0, and each Dx is referenced to the initial position x0: Dx = x0 - x.

TABLE I. MEASURING THE SPRING CONSTANT K

Total Mass
g kg / Dm
kg / DF= Dmg N / Position x
cm m / Dx = x0 - x
m / k = DF/Dx
N/m /
m0 / 50 / .050 / 0 / ------/ x0 / ------/ ------/
m1 / 70 / .070 / .020 / x1
m2 / 100 / .100 / .050 / x2
m3 / 150 / .150 / .100 / x3
Average k: /

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III. MEASURING THE PERIOD OF OSCILLATION

For each of the masses in Table II on the next page, obtain a position versus time graph and measure the period of oscillation. The procedure to be followed is described below:

1. Attach the given mass (the value in the table includes the hanger’s mass). Tape the end of the spring to the hanger to prevent the hanger from flying off the spring.

2. Pull the mass down (about 4 cm for the lighter mass, up to about 8 cm for the heavier masses) and release it. If the spring coils crash together upon compression, reduce the initial stretch.

Make sure that the mass never passes below the masking tape mark on the pole: at distances closer than about 40 cm from the motion sensor, the data becomes invalid.

3. Click on RECORD to record data. A graph similar to the one shown below should appear.

4. Use the cross-hairs cursor to capture three time intervals between four consecutive peaks. See the figure above. Average these three times to obtain the average period for that mass.

5. Calculate the predicted value of the period using Equation 1. Use the work space below to show all work.

6. Calculate the percentage difference by dividing the absolute difference by the expected value, and multiply by 100.

Show all work here: /

TABLE II. MEASURING THE PERIOD

Mass m
g kg / m1/2
kg1/2 / Measured Period
T1 T2 T3 Ave
(seconds) / Expected Period
T = 2p(m/k)1/2
s / Percentage
Difference
% /
50 / .050 /
70 / .070
100 / .100
120 / .120
150 / .150

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8. Plot the average measured T versus m1/2. A straight line is expected since T=[2p(1/k)1/2]m1/2 , which is an equation linear in the variable m1/2 , with slope equal to [2p(1/k)1/2 ] . Draw the "best" straight line by eye, using a ruler; this line should pass among the data points, with the data points distributed equally above and below the line.

Measure the slope of the line by dividing a suitably chosen "rise" by the corresponding "run". The points chosen for the rise and run should be as far apart as possible on the line to minimize the importance of measurement errors. Indicate the chosen rise and run data points on your graph.

Report the rise, the run, the slope = rise/run, and the expected slope [2p(1/k)1/2 ] in Table III. Calculate and report the percentage difference between measured and expected slopes by dividing the absolute difference by the expected, and multiply by 100.

TABLE III: SLOPE

Rise: s /
Run: kg1/2
Slope: Rise/Run = s/kg1/2
Expected Slope: 2p(1/k)1/2 = s/kg1/2
Percentage Difference: %

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