PHYS 2425 – Engineering Physics I

Simple Harmonic Motion

 

Leader: _____________________________  Recorder: ___________________________

Skeptic: _____________________________ Encourager: _________________________

 

Materials

Helical Spring

Ring Stand with meter stick

Clamp(s)

Newton Weight Set

Pasco mass set

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 Newtons’ second law equation for the spring as a 2nd order differential equation.

 

 

Q5.  Show that x(t) = A cos(ωt + δ) is a solution provided that .

 

 

 

 

 

Figure 1 Geometry of Hooke's law

 

Procedure

Part 1  Hooke’s Law

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 Pasco mass set.  Arrange the spring so that it is about 75 cm above the motion detector with the bottom of the mass hanger directly above the motion detector.  Place 100 g on the mass hanger and tape it to the hanger so that the mass doesn’t fall off onto the sonic motion detector.  Gently start the mass oscillating.  Press the collect button and you should obtain simultaneous graphs of distance, velocity, and acceleration.

 

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 ~ ______.