Physics Online

Hooke's Law

Introduction

The purpose of this lab is to test if springs individually and collectively obey Hooke's law. Hooke’s law states that a stretched or compressed spring exerts a force proportional to the distance it is stretched or compressed:

Fs = -k x

Fs = spring force

k = spring constant

x = displacement from the equilibrium position

If a mass is attached to a spring and it oscillates vertically, then Newton’s second law for the hanging mass is the following:

ΣF = ma

-mg + -kx = ma

This is a “differential equation” which requires techniques beyond the math required for this course. We will use the solution from the textbook which predicts simple harmonic motion. Simple harmonic motion can be described by a cosine function for position as a function of time. The formula for the time to complete an oscillation, the period, is as follows:

[1]

Further analysis that takes into account the mass of the spring estimates the period as follows:

[2]

m = hanging mass

ms = mass of spring

Equipment You Procure

  • digital camera
  • tape measure

Equipment from Kits

  • 2+ pendulum bobs
  • 2 springs
  • scale
  • C clamp
  • stopwatch
  • rubber band

Note: Do not use the formula k = mg/x to calculate the spring constant during any portion of this experiment.

Experimental Procedures

Hooke’s law for individual springs

1)Attach the C clamp to a door jam or other object.

2)Select a spring and attach it to the C clamp.

3)Attachthe smaller pendulum bob to the bottom of the spring andmeasure the distance from the top of the spring to the bottom of the spring. Measure the mass and calculate the weightof the smaller pendulum bob(W = mg). Be sure to use SI units (kg and N) here.

4)Remove the smaller pendulum bob and attach the larger pendulum bob. Measure the distance from the top of the spring to the bottom of the spring. Measure the mass and calculate the weight of the larger pendulum bob.

5)Attach both bobs to the spring. Measure the distance from the top of the spring to the bottom of the spring. Calculate the combined weight of the pendulum bobs.

6)If you have a third bob, repeat with various combinations of bobs.

7)Plot the results using Excel with the force in newtons on the vertical axis and the distancein meters on the horizontal axis (yes, this is a deviation from the convention of having the independent variable on the x-axis). Add a linear trendline and display both the equation and R2. Review and obey the general lab instructions regarding graph formats.

8)The slope of the equation given by Excel is the estimate of the spring constant kin N/m.You do not need to calculate the error in the spring constant. An R2 value close to 1 indicates a good fit to a straight line and is evidence in support of Hooke’s law.

9)Select the other spring and repeat steps3through 8.

Hooke’s law for springs in series

10)Attach the two springs used previously in series. Theorypredicts that the total spring constant will be:

ktotal = k1k2/(k1+k2)

11)Repeat steps3through 8 for the combined system. How does the predicted spring constant compare to the value derived by the graph in Excel? Calculate an encountered error = 100% * |theory – experiment|/theory. You do not need to obtain an estimated error for any of the k values.

Period of oscillation

12)Attach a pendulum bob to an individual spring. Record which spring and bob you use. Measure the length of the spring. Pull the pendulum bob down a cm or so from its equilibrium location. Measure the length of the spring. Calculate the amplitude by taking the difference between the equilibrium lengthand the length while pulling.

13)Release the bob and measure the time it takes to complete 10 oscillations. If the spring compresses completely or does not remain vertical, then throw out your results. Calculate the experimental period (measured time divided by the number of oscillations). Perform error propagation.

14)Calculate the theoreticalperiodusing equation [1] or [2].You do not need to calculate error in the theoretical period since you do not have an estimated error in the spring constant.

15)Repeat steps12through 14several times. You can vary the mass, spring, or amplitude. Which variables affect the period?

Hooke’s law for rubber bands

15) Repeat steps 3 through 8 with a rubber band.