PHYS 2425 – Engineering Physics I
Simple Harmonic Motion
Leader: _____________________________ Recorder:
___________________________
Skeptic: _____________________________ Encourager:
_________________________
Materials
Helical Spring
Ring Stand with meter stick
Clamp(s)
LabPro
Laptop
Motion Detector
Masking tape
Introduction
In this lab, we will study motion that
repeats itself periodically. If the
repeating - or periodic - motion can be described by a simple sine function,
then it is referred to as simple harmonic motion. Such motion is
ubiquitous in nature and technology. Examples include time-keeping devices, the
planets orbiting the sun, atoms in a solid and so on. Harmonic motion occurs when the net force on
an object has certain properties. The
properties the force has to have include that when an
object is displaced from its equilibrium position, the force on the object is
directed towards the equilibrium position.
Such a force is called a restoring force since the force is
restoring the object to its equilibrium position. For the motion to be simple harmonic the
magnitude of the force must be proportional to the distance the object is
displaced from the equilibrium position.
The force is typically written as F = -kx (1)
where x is the displacement from equilibrium.
k is a proportionality constant.
Q1. What is the significance of the – sign in
equation (1)
Q2. You displace an object from equilibrium and a
linear restoring force accelerates the object towards the equilibrium
point. What is the force on the object
at the instance it returns to the equilibrium point?
Q3. Does the object stop at the equilibrium
point? Explain.
A simple physical system showing an excellent linear restoring is a spring with a weight suspended from it as shown in figure 1. We will suspend different masses from the spring and measure the change in length of the spring. The relationship between the suspended mass and the length of the spring is called Hooke’s Law.
Q4.
A mass is attached to a spring and sits on a frictionless horizontal
surface. The mass is displaced from the
equilibrium point. i) Sketch the situation. ii) Draw a free body diagram and iii)
write the
Q5. Show that x(t) = A cos(ωt + δ) is a solution provided that .
Figure 1 Geometry of Hooke's law
Procedure
1. Set-up
Use clamps to suspend the provided springs in front of the meter stick mounted to a ring stand. Hang the 50 g mass hanger from the bottom of the spring. You want to arrange your experiment such that the position of the bottom of the mass hanger with respect to the meter stick can be easily determined. Your set up should appear as in figure 2.
Figure 2 Experimental set up for the determination of Hooke's law.
2. Data Acquisition
We will now apply a force to stretch the spring by hanging a known weight from the spring. Note the position of the bottom of the mass hanger and then place a .50 N weight on the mass hanger. Record the stretch of the spring from the original position of the mass hanger in a data table in the space below. Repeat your measurements by adjusting the weights to increase the force on the spring to 1.0 N and then 1.50 N, 2.00 N, and 2.50 N, respectively. For each weight, record the stretch of the spring in your data table.
Q6. Use Excel, LoggerPro (disconnect the LabPro), or GraphAnalysis to construct a properly labeled graph of Force vs. stretch.
Q7. What type of relationship is shown by your data?
Q8. Find and record the slope of the best-fit line through your data. Include units.
Q9. Interpret the slope by completing the following: It takes a force of _____ to change the length of the spring by ______.
The constant you have determined is called the spring constant and is
usually denoted by the letter k. Hooke’s Law is then given by F = -kx. The spring constant measures how stiff the
spring is - the stiffer the spring, the greater the spring constant.
Q10. For this spring the value of
the spring constant is _____________ (Include units)
Part 2 The spring and Mass System
Procedure
1. Set up
We will make simultaneous measurements of
the position, velocity, and acceleration of the mass. We will make the measurements using a sonic
motion detector. Make sure that the LabPro is plugged into the computer and turned on. If it is not already open, start LoggerPro. Click on
the open folder icon, open the “Probes & Sensors” folder, open the “Motion
Detector” folder and open the file called “Motion Detector”. Plug the motion
detector into the DIG/SONIC 1. Replace
the 50 g mass hanger with the 5 g hanger in the
2. Determine the phase relationships between x,
v and a
Start the oscillator and press the collect
button. Once the data is collected, use
the mouse to highlight a peak of the distance versus time graph. The same location in time will be highlighted
on the v vs. t and a vs. t graphs.
Q11. Describe the phase relationships between
these three sine waves? Print a copy of
this graph.
3. Determine the factors that affect the Period of Oscillation of Spring Mass System
Q12. Sketch the spring and mass system, and list at least three factors which might affect the period of oscillation. Discuss your list with the instructor.
Q13. Describe how you can determine the period of oscillation from the graph of position vs. time obtained from the computer.
Q14. Design an experiment to test whether two of the factors you identified in question Q12 affect the period of oscillation of the spring (provided you can conduct the experiment). Describe your procedure in the space below. Be sure to indicate what variables you will change, what variables you will keep the same and what you will measure. Complete a data table for each and then discuss which factors do affect the period of oscillation. You may want to attach a separate sheet. (Hint: an important variable you should consider is the mass and you should look at small masses as well as large.)
Q15. Construct a graph of the period of the spring vs. the mass (including the hanger). Does this graph show a linear relationship?
Q16. Construct a graph of T2 vs. m. Does this graph show a better linear relationship than the T vs. m graph?
Q17. Complete the following relationship: T ~ ______.