assignment
harmonic
1
Mass on a Spring
Walker, Chapter 13
Simulations:
https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html
http://physics.bu.edu/~duffy/HTML5/mass_on_spring_graphs.html
https://phet.colorado.edu/sims/html/hookes-law/latest/hookes-law_en.html
2
Vibrations, Oscillations,�
and Periodic motion
• Some everyday examples: pendulum, object on a
spring, swing, bridge, vibrations of guitar string
• Goal: describe the motion, forces, and energy of
the system
• Last week: pendulum à what did we learn?
• This week: mass on a spring à examples?
3
Object (Mass) on a Spring
Horizontal
• x is the displacement from equilibrium
Vertical
• gravity also comes into play
Can change:
• mass (m)
• spring constant (k): stiffness of spring
• amplitude (A): max that you stretch it
Equilibrium position
(spring unstretched)
Table is
frictionless
x = 0
at
equilibrium
position
+x -x
4
Elastic Potential Energy
• Everyday example: launching a ball across the
room with a compressed spring (ping pong ball
gun). Energy in compressed spring converts to
kinetic energy of ball.
• Other examples: archery, kangaroo, bungee cord
game at fairs, compression of femur
• Conservation of energy: Efinal = Einitial
o Total energy remains constant, energy is transformed
• Elastic potential energy:
U = (1/2)kx2
o Energy stored in spring when stretched or compressed
o Energy is zero when x = 0
o More potential energy when displacement is bigger
o Energy is a scalar, and always positive here
5
Energy of Object on a Spring
Given what you know about energy, where do you
think each of the following will be highest? Why?
• Gravitational potential energy highest:
When object is at greatest height
U = mgy
• Kinetic energy highest when:
When object is at greatest speed
K = ½ mv2
• Elastic potential energy highest when:
When object is at greatest displacement
U = ½ kx2
• Total energy:
Stays the same
Einital = Efinal
6
Period of Mass on Spring
We will determine how various factors affect the
motion of the mass on a spring.
Factors we might test:
• Mass?
• Spring constant?
• Amplitude?
• Anything else?
Equilibrium position
(spring unstretched)
Table is
frictionless
x = 0
at
equilibrium
position
+x -x
7
Characteristics of Motion
• Amplitude (A): max displacement relative to
equilibrium. Object reaches +/- A during each
cycle. (Units are meters.)
• Period (T): time to complete a cycle (seconds)
+A à -A à +A
(in simulation it goes from middle to ends back to middle)
• Frequency (f): number of cycles per second
(units are 1/s = s-1 = Hz)
• Relationship between frequency and period:
f = 1/T
8
Period of a Mass on a Spring
A mass on a spring has a period:
T = 2π √(m/k)
o Bigger mass à _____ T à oscillation takes _____
o Stiff spring (____ k) à ______ T à _____ oscillation
o Does / does not depend on amplitude
What you observe in the simulations should agree
with this formula
bigger longer
big shorter rapid
does not
9
Spring Force: Hooke’s Law
• Force is proportional to displacement
• Force is in the
of displacement:
F = – kx
F = _______
x = ____________
k = ______________ (stiff spring = large k)
“-” means _______________
• When is spring force biggest? Smallest?
• Spring force is a restoring force: object is pushed or pulled
towards equilibrium. What’s the direction at various points
in its motion?
force
displacement
spring constant
opposite direction
PHYS 242 Lab Week 11 Lab San Francisco State University
Copyright 2020 San Francisco State University
Simple Harmonic Motion
In this lab we will investigate the properties of a mass on a spring, using a simulation from the
PhET team.
The simulation is available at the following link:
https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html
The simulation can be run in a browser. If you have issues with the simulation, try using
another browser. If you are unable to run the simulation, your TA will provide you with remote
assistance. When you run the simulation, choose the “Lab” option.
NOTE: For this activity, you do not need to write a full lab report. Instead, answer the questions
that are included in this lab. Please type up your answers, if possible. For the sketches, you can
take pictures and include the images in your write up. Submit your answers to these questions
on iLearn as a PDF. If you need help with this process, please ask your TA for assistance.
Part I: Masses and Springs
This lab will investigate the motion of a mass hanging vertically on a spring, as shown in
Figure 1.
Question: Draw a free body diagram for the hanging mass.
Figure 1: A 100 g mass hanging vertically on a spring, as shown in the PhET
simulation. The spring constant is set to the default value.
PHYS 242 Lab Week 11 Lab San Francisco State University
Copyright 2020 San Francisco State University
Part II: Familiarize yourself with the simulation
Again, make sure that you choose the “Lab” option for the simulation. At the very bottom of
the screen you will see the other options for the simulation, including a home button, “Intro,”
“Vectors,” “Energy,” and “Lab.” If you accidentally navigate to another area, you can return to
the Energy option by clicking the “Lab” button.
The simulation provides a spring with a certain default spring constant, as shown in Figure 1. By
moving the 100 g mass up to the spring, we can attach it to the spring. As you showed in Part I,
if the system is not in equilibrium the mass will begin to oscillate up and down. We can pause
or slow down the simulation with the buttons in the lower right corner of the screen. We can
also drag the ruler and the timer (initially in the box on the right-hand side of the screen) over
to the system to measure the displacement of the mass at different times.
The quantities in the right-hand box allow us to change the gravity and the damping for the
system. Make sure that the gravity is set to Earth’s gravity, and the damping is set to “None.”
On the left side, we can expand the “Energy Graph” box to see how the kinetic, potential, and
total energy of the system vary with time.
Part III: Simple Harmonic Motion
You will investigate the motion of the mass. Use the following initial parameters for your
simulation:
Damping = “None”
Gravity = 9.8 m/s2 (Earth)
Mass = 100 g
Spring Constant = Default value
Mass Equilibrium box = Checked
The dashed black line will now show the equilibrium position of the system. Set up your ruler
so that 50 cm falls on the dashed black line. Pause the simulation, and set your timer to play.
Move the mass so that the center is at about 90 cm. Your setup should now look similar to that
in Figure 2.
When you hit play, the timer will begin to count down. You can reset the timer by hitting the
bottom left arrow button on the timer.
Change the animation speed to “Slow.”
PHYS 242 Lab Week 11 Lab San Francisco State University
Copyright 2020 San Francisco State University
Question: Record the displacement of the mass at the times that are given in Table 1.
(This is simplest to do by running the animation at “slow” speeds, pausing it
when the clock hits one of the values in Table 1, and recording the
displacement at that time.) Find y’ by subtracting 50 cm from y.
Make a plot of the adjusted y-displacement (y’ in Table 1) as a function of
time, t. What shape does this plot have?
Figure 2: The initial setup for your experiment in Part III.
PHYS 242 Lab Week 11 Lab San Francisco State University
Copyright 2020 San Francisco State University
Table 1: Displacement versus time. Note that y’ = 0 cm when y = 50 cm.
t (s) y (cm) y’ (cm) = y – 50 cm
0
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
PHYS 242 Lab Week 11 Lab San Francisco State University
Copyright 2020 San Francisco State University
The movement of the mass can be described as “simple harmonic motion,” where the mass
obeys the equation:
!′ = %&'((*+ + -)
where A is the amplitude, ω is the angular frequency (in rads/s), and φ is a phase constant
(don’t worry about this too much for now). The angular frequency is also related to the
frequency and period of the system.
Question: What is the amplitude of the system?
Question: The period of the oscillation is the time it takes to complete one full cycle.
What is the period of this system? What are the frequency and the angular
frequency?
The angular frequency of a mass on a spring is directly related to the mass and the spring
constant, according to the equation:
* = /01
Question: What is the spring constant for this system?
Part IV: Velocity, Acceleration, and Energy
Recall that the maximum velocity and acceleration of a single harmonic oscillator can be found
by taking the first and second derivatives, respectively, of the displacement equation and
setting them equal to zero. If we do that, we find:
2!”# = *%
3!”# = *$%
Question: Where in the mass’s oscillation is the velocity at a maximum? Where is the
acceleration at a maximum? Show these points on your plot of y’ versus t and
explain.
PHYS 242 Lab Week 11 Lab San Francisco State University
Copyright 2020 San Francisco State University
Looking at the graph on the left, you can see how the kinetic and potential energy vary as the
mass oscillates. Note that PEelas is the potential energy of the spring, whereas PEgrav is the
potential energy due to gravity.
Question: Where is the kinetic energy at a maximum? Where is the potential energy at
a maximum? Show these points on your plot of y’ versus t and explain.
Part V: Mystery Masses
You will now determine the masses for the blue and red weights at the bottom using the same
spring constant.
Question: Given what we have learned from the 100 g mass, how would you determine
the masses of the other two blocks?
Question: What are the masses of the other two blocks?
Lab: Simple Harmonic Motion Post-Lab Knowledge Check PHYS 242
Copyright 2020 San Francisco State University
Simple Harmonic Motion
Suppose we have a mass of 100 g hanging from a spring
with a spring constant of k, as in Figure 1. The spring makes
one complete oscillation in 0.125 sec, covering 30 cm from
top to bottom.
1. Find the following properties of the system.
a. What is the period of the system? Explain.
b. What is the frequency of the system? Explain.
c. What is the angular frequency of the system? Explain.
d. What is the amplitude of the system? Explain.
Figure 1: A 100 g mass hanging vertically on a spring, as
shown in the PhET simulation.
Lab: Simple Harmonic Motion Post-Lab Knowledge Check PHYS 242
Copyright 2020 San Francisco State University
2. Draw the vertical motion of the system over time and come up with an equation to
describe this motion. On this plot, indicate where the velocity and acceleration are at a
maximum.
3. How would the motion of the spring change if the mass were doubled to 200 g?
Simple Harmonic Motion: Summary
Answer the following questions and submit your responses as a PDF.
1. Write down one major conclusion you can draw from this week’s laboratory.
Please explain.
2. Describe the experimental evidence that supports your conclusion. Please
explain.
3. Give one example of applications/situations for the finding(s) you described
above in your everyday life outside of physics lab.
4.What
did
you
like
and
dislike
about
week’s
lab?