Making up a question for Hooke's Law based on Calcullus

[bankedroads]

Homework Statement

I'm not sure if this should be in this thread of the calculus one as it involves a bit of both. Basically I have to to make a question up that applies Hooke's Law. The only other condition is that k (the spring constant) must be equal to 20N/m. It should involve integration or differentiation.

Homework Equations

F = -ks
F = ma
a = -Rw²sinwt

The Attempt at a Solution

My current idea is to use to use the simple harmonic equations associated with an oscillating spring. I plan on giving out the acceleration and some conditions so that the person solving the problem has to integrate it twice to get s = blah (where s equals displacement). I will also give out the force so that then they can calculate k, which will end up being 20. Does anyone think this will work or if it's too easy? (the question should be suited for an advanced maths class but not overly difficult)

The main issue I'm having is if it's applying Hooke's law. Does anyone else have any other ideas about how to come up with a challenging question using this very simple law (perhaps using period or that equation for a mass attached to the end of the string with the square root sign)? A teacher was telling me about using acceleration in the F = ma equation and using that to differentiate but i was a ltitle confused by it.

Also, what does w mean? I understand it is angular velocity but I'm not quite sure what that means in relation to an oscillating spring.

Thanks for your time!

 

[heth]

Also, what does w mean? I understand it is angular velocity but I'm not quite sure what that means in relation to an oscillating spring.

Think of it as angular frequency rather than angular velocity. In circular motion, the angular frequency is the number of radians per second, omega = 2 * pi * f

In the context of SHM, you still have omega = 2 * pi * f

Omega is a useful tool in your equations as it stops you writing 2*pi all over the place.

 

[Hootenanny]

Sounds okay to me. Of course you will have to specify some initial conditions in order to allow them to determine the value of the arbitrary constants.

As you correctly say ω is the magnitude of the angular velocity (called the angular frequency), which is defined as 2∏f, where f is the frequency of oscillation.

 

[bankedroads]

Thanks guys!

So can I just assign a random value for w? or would I need to give some reasoning for it? so say acceleration = -100sin10t, does that mean w = 10? Should i try to include the period but get the person to work out w for themselves? Is it of a high enough standard do you think? I'm just really worried it doesn't apply Hooke's law...

Also, one of my teachers was telling me that if:
s = 3sin2t and you differentiated till you got
a = -12sin2t, you could make an equation
F = ma, F = -ks
thus, ma = -ks (assume mass is 10kg)
10*-12sin2t = -k x 3sin2t
10 * 4 = k
therefore k = 40

does that work/make sense? Is it a better way of using Hooke's law than the previous method (is it harder?)? Can you make the F = ma force equal the F = -ks force? Doesn't the F = ma force mean that it is the force required for it to travel in the direction of displacement but the F = -ks force is the restoring force. And so since they're in opposite directions, for them to equal each other, shouldn't one of them be multiplied by negative?

sorry for askig so many questions!

 

[Hootenanny]

Also, one of my teachers was telling me that if:
s = 3sin2t and you differentiated till you got
a = -12sin2t, you could make an equation
F = ma, F = -ks
thus, ma = -ks (assume mass is 10kg)
10*-12sin2t = -k x 3sin2t
10 * 4 = k
therefore k = 40

does that work/make sense?

This method looks okay to me.

Is it a better way of using Hooke's law than the previous method (is it harder?)?

Hooke's law is Hooke's law, there is no better way of using it, one simply uses it. Difficulty is virtually impossible to quantify, particularly from an applied mathematics/physics point of view. Often, the difficulty lies not in the actual solution, but in the wording of a question. There are infinite possibilities with a question such as this, the wording would also depend if you would like to focus more on the mathematics (i.e. solving ODE's) or the physics (i.e. interpreting) of the problem.

What age group are the students at which this question is to be pitched? Advanced Mathematics can mean a variety of courses.

Can you make the F = ma force equal the F = -ks force?

Yes

Doesn't the F = ma force mean that it is the force required for it to travel in the direction of displacement but the F = -ks force is the restoring force.
And so since they're in opposite directions, for them to equal each other, shouldn't one of them be multiplied by negative?

No. Newton's second law simply states that the vector sum of the forces is proportional to the acceleration, there is no requirement that the force be in the same direction as motion, if there was then one wouldn't be able to apply Newton's second law to any problems that have a restoring force (such as this).

sorry for askig so many questions!

No need to apologise, that's what we're all here for