In this lab students will explore the motion of a mass attached to a spring hanging vertically from a fixed support.
A spring has an equilibrium length L. When a spring is compressed, then a force with magnitude proportional to the decrease in its length from its equilibrium length is pushing each end away from the other. When a spring is stretched, then a force with magnitude proportional to the increase in its length from its equilibrium length is pulling each end towards the other.
If one end of a spring is fixed to a support and an object is attached to the other end, then the force exerted by the spring on the object has a magnitude proportional to the displacement DL of the free end from its equilibrium position. The direction of the force is always opposite to the direction of the displacement. If the x-axis of a coordinate system is chosen parallel to the spring and the equilibrium position of the free end of the spring is at x=0, then the displacement DL is equal to x, and the force is F=-kx. The proportional constant k is called the spring constant. It is a measure of the spring’s stiffness.
If an object attached to the free end of the spring is displaced along the x-axis (in the positive x-direction) and then released, the force exerted by the spring on the object will accelerate it towards the equilibrium position (in the negative x-direction). When it reaches the equilibrium position it will have its maximum speed. As it passes through the equilibrium position, the force the spring exerts on the object changes direction. The object now decelerates, until its speed is zero and it displacement is maximum in the negative x-direction. The acceleration is in the positive x-direction and the object again accelerates towards the equilibrium position. The object oscillates about the equilibrium position.
 | The position of the object as a function of time t is
x=xmaxcos(2pft+f). xmax is the amplitude of the oscillations and f is the frequency. f is the phase of the oscillations. The period T of the oscillations is T=1/f. |
| The velocity of the object as a function of time t is v=-vmaxsin(2pft+f). vmax is the maximum speed of the object as it passes the equilibrium position. |
| The acceleration of the object as a function of time t is a=-amaxcos(2pft+f). amax is the magnitude of the acceleration of the object at maximum displacement. The force acting on the object is F=ma. |
Students will measure the period of the oscillations of different masses attached to the spring. For a mass on a spring we expect
,
.
To find the spring constant k students will plot T2 versus m. The slope of the best fitting straight line equals 4p2/k.
Students will analyze 4 video clips, spring_x.avi, x=1-4. To play a video clip or to step through it frame-by-frame click the "Begin" button. The "Video Analysis" web page will open. You can toggle between the current page and the "Video Analysis" page by pressing Alt-Tab.
| Analyze each of the clips to find the period of oscillation. Track the point where the mass attaches to the spring. | ||||||||||||||||||||
| The weight of the object attached to the spring is given. Find its mass. | ||||||||||||||||||||
| Construct a spreadsheet as shown below.
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| Construct a plot of T2 versus m. | ||||||||||||||||||||
| Use the spreadsheet's regression function to find the slope of the best fitting straight line to this plot and the uncertainty in the slope. | ||||||||||||||||||||
| Find the spring constant k and the uncertainty in this value, Dk. Use Dk/k=D(slope)/(slope). | ||||||||||||||||||||
| If you have completed exercise 6, compare your value for k found in this lab with the value found in exercise 6. Both experiment use the same spring. |
Open Microsoft Word and prepare a report using the template shown below.
| In a few sentences summarize the experiment. |
| Show your spreadsheet containing your plot of T2 versus mass and your value for the slope of the best fitting straight line. |
| What value did you find for k and what is the uncertainty in k? |
| If you have completed exercise 6, did you find the same value for k in exercise 6? |
Save your Word document (your name_lab9.doc) and attach it to an e-mail message to mbreinig@utk.edu.