At Home Laboratory 4: Oscillations and
Waves
Set yourself the following objectives for this week’s exercises:
● Understand the connection between waves and oscillations
● Explore oscillations as a sinusoidal function
● Experimentally measure the amplitude and angular frequency of an oscillator
● Determine what physical parameters affect the angular frequency of your oscillato
The following shows the value of all the questions in this lab:
Laboratory 4 Grading Scheme Totals
Ex XXXXXXXXXX)
Points XXXXXXXXXX
Ex XXXXXXXXXX)
Points XXXXXXXXXX
Ex XXXXXXXXXX)
Points XXXXXXXXXX
Ex XXXXXXXXXX)
Points 5 5
Total number of points in this lab (each lab is worth 5% of your final grade) 25
Physicists love springs. From modelling the bonds between atoms in a molecule, to the pulsing
of Neutron stars, nearly every
anch of physics uses concepts related to springs to describe
the way certain systems store and release energy.
One of the consequences of a system being “spring-like” is that it may display oscillatory
ehaviour. In the following exercises you will be exploring how springs store and transfer energy
https:
en.wikipedia.org/wiki/Molecular_vi
ation#Newtonian_mechanics
https:
en.wikipedia.org/wiki/Neutron-star_oscillation#Oscillation_excitation
in the form of oscillations, and how oscillations can be related to waves. By understanding how
springs work, you will be developing a physics toolkit that will allow you to approach many
complicated phenomena.
Warm-Up: From waves to oscillations
In this lab you will be looking at the oscillations of an object connected to a spring. Oscillations
are a type of repeated motion around a central or average position. In this lab, your oscillations
will be up and down (stretching and un-stretching your spring) but many of the principles we
discuss are transferable to other oscillators, like a pendulum or even the way an electronic
device keeps track of time or does computations.
Since oscillators display repeated behavior, we can try to use a sinusoidal function to describe
its motion. You may be familiar with sine functions in the context of waves, and in fact,
oscillations share a lot of similarities with waves. The purpose of this Warm-Up is to get you
thinking about the connection between waves and oscillations.
Oscillations and waves
The equation of a travelling wave is the following:
?(?, ?) = ? ???(?? − ??),
where y is the vertical displacement of a point, x is horizontal displacement, t is time, A is the
amplitude of the wave, k is the wave number, and ? is the angular frequency. Below is a
sequence of snapshot graphs at different times, with x = 0 labeled with a red dot:
a) How does y at x = 0 change as t increases? (Note that time is increasing; T4 > T3 > T2 >
T1.)
) Imagine plotting y at x = 0 as a function of time (a history plot of y at x = 0). What do you
think this plot would look like? (Hint: the answer is a sine curve). NOTE: Remember here
the difference between a snapshot graph (y vs x) and a history graph (y vs t). Rewriting
the equation ?(?, ?) = ? ???(?? − ??), for x = 0, is equivalent to giving
the equation for the history graph for a point on the travelling wave positioned at x = 0. If
you think about this, you will realize that each point on the travelling wave is oscillating,
thus the equation you have written down is the equation for a “sinusoidal oscillator”,
?(?) = ? ???(−??), also known as simple harmonic motion.
c) How is this new equation related to an object hanging from a spring? Does this equation
describe the object’s vertical oscillations? (Hint: the answer is ‘yes’)
You may find it useful to refer back to the Warm Up in the following
exercises.
Exercise 1: Developing your hypothesis
In the Warm-Up exercise, you started to explore the connection between oscillations and
waves. In the following exercises, you will be looking at how a real-life oscillator compares to the
equation of an oscillator that you developed in the Warm-Up, ie:
?(?) = ? ???(−??).
In this lab, you will use a weight (a household item) hanging from your spring (included in the
lab kit) as your oscillator. When the weight is lifted and released, you will see that it oscillates (or
obs) up and down as the force of gravity and spring force compete against each other.
In Exercise 2, you will take measurements of your weight-spring system and see how it relates
to the above equation. Specifically, you will compare the motion of your weight to a sine-
function, and determine the amplitude A of your oscillations.
Finally in Exercise 3, you will perform an experiment to determine what about your system
determines its angular frequency ?, and how does the value of ? influence the motion of your
oscillator.
But before you do any of this, let’s think about each variable in the equation above. First, let’s
consider the amplitude A:
(1.1) i) If your oscillator follows the function ?(?) = ? ???(−??), what is the maximum and
minimum value of y? ii) If the range of y values you measure in your experiment go from
0.22 m to -0.35 m, what is A, and where (what value of y) is the spring-mass-system in
equili
ium? (2 marks)
The other parameter in your equation is angular frequency, ?. ?? (or the product of angular
frequency and time) appears in the argument of your sine function. This means two things; a)
? affects the rate at which your oscillator moves through its maximum and minimum y values
(over time) and b) the product of ?and t must in radians, since the argument of a sine function
must be an angle. We note that radians are a unit that is a dimensionless quantity!
(1.2) i) If ?? is in units of radians, what units does ?have? Note that the answer to this
question will make it clear why ? is called an angular frequency. ii) If ? = ?. ?? rad/s,
what is your oscillator’s period T? In other words, how many seconds does it take for
your oscillator to go through a full cycle? (1 mark) Remember: One full period goes from
sin(x) -> sin(x + 2?).
In question 1.2, you determined the units of ?. Now you are going to create a hypothesis about
what aspects of your system determine ?. For example, does the stiffness of the spring matter?
How about the mass of the object you use? What other physical quantities do you think would
affect angular frequency (aka, the rate your oscillator moves through a full period)? Think about
what forces cause your object to accelerate upwards, and which cause it to accelerate
downwards. Also, keep in mind that in physics, often the simplest answer is the co
ect answer,
so try to keep your theory as simple as possible. Take a minute or two to think about this before
continuing.
From your previous labs, you saw that the magnitude of your spring force is proportional to the
spring constant k, which has units of N/m. Likewise, the force of gravity is proportional to the
mass of the object m. Since these forces are involved in the oscillation of the spring it would be
easonable to guess that k and m would be related to the angular frequency ?. Let’s use
dimensional analysis to try and understand how all of these parameters might be related.
(1.3) i) What are the base SI units (ie. kg, m, s, etc.) of the spring constant k? ii) Use
dimensional analysis to come up with a combination of k and m that gives you that same
dimensions as ? and is therefore, proportional to ?. Report your answer in the form: ? ∝
????. (2 marks) Hint: remember that the units of radians are dimensionless. Also remember
that you can multiply or divide k and m by each other any number of times to get your answer.
For example, what units do you get if you try k2/m2?
Note: the answer to 1.3 ii) and 1.3 iii) are the basis of your hypothesis, and are what you will be
testing in Exercise 4. This should look familiar to the proportionality exercise from Lab 1. You
may wish to review Lab 1, Exercise 1 for help with question 1.3 in this lab.
https:
en.wikipedia.org/wiki/Occam%27s_razo
https:
en.wikipedia.org/wiki/SI_base_unit
Exercise 2: Measuring oscillations (checking if
sinusoidal)
The experiment you will be performing in Exercise 2 has aspects that are similar to Lab 2.
However, instead of using your spring to drag an object along a surface, you will be hanging the
object vertically for an extended period of time. This means the object’s full weight will be
suspended from your spring, and you will need to be extra careful not to over-stretch your
spring.
Before you continue collect the following:
1. One object of known mass.
Tip #1: It is up to you to select an object. We recommend something around 500 g or
less. To know the mass, you may need to use your spring constant (see previous Lab 2)
to weigh your object!
Tip #2: Consider using a water bottle. You will need to use different masses in Exercise
4, and a water bottle will let you use different volumes of water (or sand, rocks, rice,
eans etc…) to achieve this
2. Spring (lab kit)
3. Ruler (lab kit)
4. Elastic band (lab kit)
5. Measuring tape (lab kit)
In Exercise 2 you will observe the way your object oscillates, and verify that it follows a sine
function. You will want to verify the distance markings behind your object are set up so that
a) y = 0 m marks where your object is when the spring and object are left hanging,
stretched and at rest (the equili
ium position)
) When your spring stretches, y decreases (so y < 0 m) and when it rebounds
upward, y increases (and