Introducing SHM to high school students

[Mayan Fung]

I need to introduce Simple Harmonic Motion to a group of high school students studying physics. They don't know anything about differential equation except the method of separation of variables. Also, they have limited knowledge on complex numbers like e^iωt. However, I don't want to just give the solution and let they check that the solution fits the equation. Is there any method that can clearly show the logic?

I have an idea here.
$$ \frac {d^2x}{dt^2} = -ω^2 x $$
$$ v\frac {dv}{dx} = -ω^2 x \text{ } (v = \frac{dx}{dt}) $$
then I can get sth relating dx/dt and x, thus finding x(t)
If I did the math correctly, it should be
$$ x(t) = Asin(ωt+φ) $$

I would think this is a logical way that the students would be convinced with the result.
However, it still looks a bit clumsy and I have never seen anyone do this before. Is there any simpler way to solve this equation logically? Thanks!

 

[cnh1995]

I need to introduce Simple Harmonic Motion to a group of high school students studying physics. They don't know anything about differential equation except the method of separation of variables. Also, they have limited knowledge on complex numbers like e^iωt. However, I don't want to just give the solution and let they check that the solution fits the equation. Is there any method that can clearly show the logic?

I have an idea here.
$$ \frac {d^2x}{dt^2} = -ω^2 x $$
$$ v\frac {dv}{dx} = -ω^2 x \text{ } (v = \frac{dx}{dt}) $$
then I can get sth relating dx/dt and x, thus finding x(t)
If I did the math correctly, it should be
$$ x(t) = Asin(ωt+φ) $$

I would think this is a logical way that the students would be convinced with the result.
However, it still looks a bit clumsy and I have never seen anyone do this before. Is there any simpler way to solve this equation logically? Thanks!

You can show an animation of SHM as a projection of Uniform Circular Motion. I think high school students are familiar with UCM.
There are plenty of images and videos available on google and youtube. This is one of them.

 

[Mayan Fung]

Thanks for your materials!

 

[lychette]

You can approach it from a 'forces' point of view.
You have studied uniformly accelerated motion and are familiar with Newtons laws and the use of F = mA
If you are familiar with circular motion you have been dealing with the case where F is at right angles to the direction of motion and applied F = ma to these situations.
F is called CENTRIPETAL force
Simple Harmonic Motion (harmonic motion!) involves dealing with situations where the force acts on a particle that is DISPLACED from an equilibrium position. The force is called a restoring force and the
SIMPLEST to analyse is when F is proportional to DISPLACEMENT (this is why it is called SIMPLE harmonic mtion.
The fundamental equation is F= -kx where x is the displacement, the - sign indicates the direction of the force is opposite to the direction of displacement.
You can also see that the constant k ( =F/x) is STIFFNESS or spring constant and SHM is closely linked with oscillations of springs etc

As mentioned above there is a very very strong link between circular motion and SHM. When you get into it you will see that all of the equations of SHM can be found in the analysis of circular motion . ( this is where all the sine and cosines are hiding !)

 

[Mayan Fung]

Thanks for all your suggestion. However, it seems that they are qualitative arguments. I am thinking whether it can be presented in rigorous mathematical form.

 

[robphy]

If they know some simple derivatives, one can ask if they know of a function whose second derivative is equal to "minus a constant" times itself.
Their guesses can be checked by substitution.

You could start with an expression for the total energy (kinetic plus spring-potential energies) being a constant in time.
Solve for v, then write as dx/dt. Separating variables, you end up having to compute an integral:
http://www.wolframalpha.com/input/?i=integral(1/sqrt(1-x^2),x)
in the form
http://www.wolframalpha.com/input/?i=integral(1/sqrt(A^2-x^2),x).

While these are fine mathematical "solutions", they lack the physical mechanism suggested by @lychette .
If one introduces deviations from ideal conditions, the closed-form solutions may fail.
However, one can still use a discretized solution suggested by the force law.
One can tell a story like this:
Use a set of graphs of position, velocity, acceleration, and force... with their time axes lined up.
Declare that the force applied by the spring is minus the displacement of the spring from equilibrium.
Start at time t=0 with an initial position and initial velocity. Determine the spring force, and then
(since the spring force is equal to the net force) determine the acceleration.
The acceleration tells the velocity how to change, and the velocity tells the position how to change.
[The way to perform the update steps may have some subtleties.]
With a new position at the next interval of time, repeat... eventually producing a sequence of positions.
You can start this manually, then have a computer carry out the numerical details.
Check out, for example, https://phys221.wordpress.com/2015/03/04/glowscript-tutorial-4-mass-on-a-spring/ .

 

[Mayan Fung]

I appreciate your help! I believe that showing a numerical solution is a goo idea!

 

[tygerdawg]

Borrow a metronome from the music department.

 

[Student100]

I need to introduce Simple Harmonic Motion to a group of high school students studying physics. They don't know anything about differential equation except the method of separation of variables. Also, they have limited knowledge on complex numbers like e^iωt. However, I don't want to just give the solution and let they check that the solution fits the equation. Is there any method that can clearly show the logic?

I have an idea here.
$$ \frac {d^2x}{dt^2} = -ω^2 x $$
$$ v\frac {dv}{dx} = -ω^2 x \text{ } (v = \frac{dx}{dt}) $$
then I can get sth relating dx/dt and x, thus finding x(t)
If I did the math correctly, it should be
$$ x(t) = Asin(ωt+φ) $$

I would think this is a logical way that the students would be convinced with the result.
However, it still looks a bit clumsy and I have never seen anyone do this before. Is there any simpler way to solve this equation logically? Thanks!

Why not just take a class period to teach auxiliary equations and Euler's formula. Both should be within reach assuming they've had some calculus. Then from the general solution you can easily show to get $x(t) = Asin(ωt+φ)$

Maybe a good compromise is to explain the technique to solve the differential equation, and expect students to be able to do it, while providing a handout with a deeper discussion of Euler's formula and aux equations for those curious beyond the technique. (I don't see why all of them shouldn't be able to factor to find roots, and then plug them into the dummy expression, you could even that on a exam or something)

More importantly, I think more time should invested teaching how to arrive at the DE, because that would provide far more insight into the physical significance.

With that approach you could also introduce free damped motion, forced motion would probably be pushing it though.

 

[Mayan Fung]

Why not just take a class period to teach auxiliary equations and Euler's formula. Both should be within reach assuming they've had some calculus. Then from the general solution you can easily show to get $x(t) = Asin(ωt+φ)$

Maybe a good compromise is to explain the technique to solve the differential equation, and expect students to be able to do it, while providing a handout with a deeper discussion of Euler's formula and aux equations for those curious beyond the technique. (I don't see why all of them shouldn't be able to factor to find roots, and then plug them into the dummy expression, you could even that on a exam or something)

More importantly, I think more time should invested teaching how to arrive at the DE, because that would provide far more insight into the physical significance.

With that approach you could also introduce free damped motion, forced motion would probably be pushing it though.

I am afraid that I may not have enough time to do so. I am not holding a regular class. Anyway, thanks for your help!

 

[Kajal Sengupta]

My approach is to bring in the Physics first and then the Mathematical equations. I tell them that in Kinematics we use equations of motion to find position, velocity acceleration etc. When we are dealing with S.H.M. we have to describe a motion which is repetitive in nature so we need an expression which repeats itself after regular interval of time. Sine and Cosine functions are the ones which satisfy this condition and hence the expressions. After this only I try to embark upon differentiating or solving as have been shown. Don't know whether it works.

 

[ZapperZ]

My approach is to bring in the Physics first and then the Mathematical equations. I tell them that in Kinematics we use equations of motion to find position, velocity acceleration etc. When we are dealing with S.H.M. we have to describe a motion which is repetitive in nature so we need an expression which repeats itself after regular interval of time. Sine and Cosine functions are the ones which satisfy this condition and hence the expressions. After this only I try to embark upon differentiating or solving as have been shown. Don't know whether it works.

While this may work if the students are not well-versed in linear algebra, it is not accurate. Sin and cos functions are not the only unique functions that "repeats itself after regular interval of time". There's an infinite number of these functions.

For example, I can write f(t) = |sin(t)|. This is also a periodic function that repeats itself at regular time interval. Yet, this is certainly not the solution to the SHM that we often deal with at the elementary level.

What you could have done is to simply stop at the point where you say that we want to find a solution that is periodic in time, and leave it at that and start with the mathematical derivation. You may even want to say that sinusoidal function is one such solution, rather than giving the impression that it is the only function that is periodic.

Zz.

 

[Dadface]

I would approach it from a qualitative and practical point of view.
. Group discussion on what vibrations, in are and why a knowledge of them is important.
.Introduce SHM. Explain why it is known as SIMPLE harmonic motion.
.Demonstrate systems that move (approximately) with SHM...simple pendulum,mass on spring etc
.Define terms such as frequency,time period amplitude
.Ask them to predict what factors affect time period of a pendulum and then get them to carry out an experiment to test their predictions.

That should be enough for the first session.
To get them ready for the second session ask them to revise the concepts of velocity,acceleration etc

In the second session discuss velocity acceleration etc have some worksheets prepared for example velocity time graphs from which they have to sketch acceleration time graphs. Get them used to concepts such as the gradient of a vt graph at any point (dv/dt) equals the acceleration at that point. Having done the introductory stuff you will be in a strong position to introduce the mathematical treatment of the subject.

 

[Mayan Fung]

I would approach it from a qualitative and practical point of view.
. Group discussion on what vibrations, in are and why a knowledge of them is important.
.Introduce SHM. Explain why it is known as SIMPLE harmonic motion.
.Demonstrate systems that move (approximately) with SHM...simple pendulum,mass on spring etc
.Define terms such as frequency,time period amplitude
.Ask them to predict what factors affect time period of a pendulum and then get them to carry out an experiment to test their predictions.

That should be enough for the first session.
To get them ready for the second session ask them to revise the concepts of velocity,acceleration etc

In the second session discuss velocity acceleration etc have some worksheets prepared for example velocity time graphs from which they have to sketch acceleration time graphs. Get them used to concepts such as the gradient of a vt graph at any point (dv/dt) equals the acceleration at that point. Having done the introductory stuff you will be in a strong position to introduce the mathematical treatment of the subject.

That's a good approach to this topic! Thanks!

 

[Kajal Sengupta]

While this may work if the students are not well-versed in linear algebra, it is not accurate. Sin and cos functions are not the only unique functions that "repeats itself after regular interval of time". There's an infinite number of these functions.

For example, I can write f(t) = |sin(t)|. This is also a periodic function that repeats itself at regular time interval. Yet, this is certainly not the solution to the SHM that we often deal with at the elementary level.

What you could have done is to simply stop at the point where you say that we want to find a solution that is periodic in time, and leave it at that and start with the mathematical derivation. You may even want to say that sinusoidal function is one such solution, rather than giving the impression that it is the only function that is periodic.

Zz.

Thanks for your suggestions. It means a lot. By the way, may be the way I wrote about Sine and Cosine functions makes it seem that they are the only functions which repeats itself after regular interval of time but I really did not mean they are the unique functions doing so. It was wrong language , pardon me.

 

[vela]

I need to introduce Simple Harmonic Motion to a group of high school students studying physics. They don't know anything about differential equation except the method of separation of variables. Also, they have limited knowledge on complex numbers like e^iωt. However, I don't want to just give the solution and let they check that the solution fits the equation. Is there any method that can clearly show the logic?

I have an idea here.
$$ \frac {d^2x}{dt^2} = -ω^2 x $$
$$ v\frac {dv}{dx} = -ω^2 x \text{ } (v = \frac{dx}{dt}) $$

When you integrate the equation, you end up with
$$\frac 12 mv^2 = -\frac 12 kx^2 + C$$ where $\omega^2 = k/m$ for a mass and spring system. You could write this result down directly by starting with conservation of energy and avoid the initial calculus complications.

I would think this is a logical way that the students would be convinced with the result.

Do you think the students at this level need convincing? Your limited class time might be better spent on the physics rather than on the mathematics of deriving the solution.

 

[Mayan Fung]

um.. I am not sure about that. Maybe personally I like analytical methods more. I think I can do a better job with the animations and ideas mentioned here

 

[Dr. Courtney]

Last time I did it, I took an experimental approach.

I put a mass on a spring on a Vernier dual range force sensor and plotted the F vs t for all to see.

Didn't try to derive the why from first principles, just asserted the what and proved it all with a little calculus.

All the experimental bits of the theory are easily confirmed.

Lecture demo first, followed by DIY lab with different mass and spring constant.

Good theories do not need to be derived from first principles, but they do need to be experimentally proven.

This one is easy.

 

[mpresic]

Can they solve the integral of dx over square root of a squared - x squared.? I remember I could in my senior year just before the last semester with first semester calculus.

Anyway kinetic energy is 1/2 m (dx/dt) squared. Hookes law potential energy is 1/2 k x squared. The sum of these energies is total energy E.
Solve this conservation of energy equation for dx / dt. then you can get dt = dx over square root of ( E/m - k/m x squared). Factor out square root k/m and you get square root k/m dt = dx over square root E/k - x squared. Now take the integral of both sides.
You get: square root of (k / m) times t = arcsin ( x divided by the square root (E/k)). take sine of both sides any you get sine ( square root (k/m) times t ) = x divided by square root (E/k). now solve for x by multiplying both sides by square root E/k .

Lots of steps but they are straightforward. Also you do not have to "guess" a solution to the differential equation, you just have to hope the calculus teacher has done a good job in teaching integrals by trigonometric substitution or memorizing inverse sine integrals. I think using conservation of energy in this way beats dealing with forces. I remember wishing someone showed me this as I was (as a freshman undergraduate) uncomfortable about "guessing" the solution to a second order differential equation.