Physics 1115

Theory

Hooke's Law states that the force F exerted by a spring is proportional and in the opposite direction to the displacement x of the end of the spring.   The constant of proportionality k is called the spring constant.

F_spring = -kx

When the force is in the opposite direction from displacement, this is an example of a restoring force.  When you have a restoring force, you get oscillatory motion called harmonic motion: if you attach a mass to the end of the spring, pull it away from equilibrium and let it go, it will oscillate about its equilibrium position (equally in both directions).  It is called simple harmonic motion (SHM) if the force is directly proportional to the displacement (as it is in this case). 

Notice that a body undergoing harmonic motion is not influenced by a constant force, so if you apply Newton's Law you find the acceleration is not constant.  This means that the kinematic equations that we have been using all along for constant acceleration do not hold, and we will have to develop a new set of kinematic relations.  We will explore the equations describing SHM in a later chapter.

Equilibrium/Non-Equilibrium

In equilibrium, a mass on a spring will hang in a position where all the other forces balance the spring force. What is the net force on the mass at this position? How would one calculate the equilibrium position with a mass versus the equilibrium position with no mass (the "unstretched" position)?

If you pull the mass away from its equilibrium position, it will oscillate back and forth about that position.  The oscillation cannot be described from our constant acceleration equations of motion since the acceleration varies.  We can characterize the oscillation with a few variables like amplitude (A), frequency (f) and period (T).

The amplitude of oscillation A is the maximum displacement from the equilibrium position.  This means that when a mass on a spring oscillates, it goes from a position of A away from equilibrium, stops, turns around, goes through equilibrium to -A.  It continues this type of oscillation if there are no dissipative forces.

The period of oscillation T is the time it takes for one complete oscillation.  It is the time elapsed between any two instants at which it completely reproduces its motion.

The frequency f is the number of oscillations per unit time, and is the inverse of period: f = 1/T.

Apparatus

Spring 
Weight hanger
Mass set
Meter stick
Stopwatch

Determine a procedure to verify the linear relation between the spring force F and x, and to measure the spring constant. Be sure to report k with 90% confidence.

The amplitude A of the motion is the distance you pull the mass from equilibrium.  Using a fixed value of mass, measure the periods T of oscillation for different amplitudes of motion and try to determine the relationship between A and T for constant mass. Try to measure the period as precisely as you can. You are not supplied with an equation for this relationship. A normal way to try to determine a relationship for two variables is to try plotting them in various ways.  Hint: try to plot T on the y-axis and A on the x-axis. Does it appear to be linear? Pay attention to the y-axis scale. What is the resulting slope with 90% confidence? 

Note: Keep the amplitude of oscillation small enough that the oscillation stays reasonably uniform.  For measuring A and T, if you want to use the motion sensor, keep at default sampling rate of 20 and check that displacement is properly zeroed.

Write-Up

1. If this is a formal lab (as indicated on the lab syllabus), you have been instructed as to whether this is an individual or group write-up, or an oral presentation. If it is a write-up, each person must submit the lab electronically as a word-processed document in Moodle before the next lab meeting. If it is a group write-up, you should all be submitting the same document. For written formal labs, remember to check the "write-up hints" page to be sure everything is included and check your write-up against the grading rubric.
2. If this is an informal lab, work on the results together in your groups, and be sure to have your complete informal lab in your lab notebook and checked by the instructor before you leave.
3. Remember to read the next lab and do the pre-lab before you come to lab next week! You may work on the pre-lab with others, but each person must submit her or his own work.

Department of Physics