Simple harmonic motion problem

1. Oct 23, 2015

NooDota

I translated it, so the notation may not be so great.

You have a vertical spring with a spring constant of 16 N.m^-1, we attach to its lower end a body with a mass of 0,4 KG and form a system which oscillates on a straight line with length of 8cm, we assume that at the start of the time, the body is in its equilibrium point and moving in the negative direction (the axis is directed downwards).

1. Find the period for oscillations and the displacement function.

2. Find the velocity and momentum function, calculate them at t = 1/2s. Calculate the kinetic energy at the same time.

2. Relevant equations

3. The attempt at a solution

I didn't have a problem with the first question, I got T=1s and the function for displacement: x=0.04 cos(2*π*t+π/2) What I'm having problem with is the kinetic energy,

if I calculate it by:

1/2 * m * v^2 I get 125*10^-4

Because v at t = 1/2 is 25*10^-2.

If I calculate it by:

1/2 * k * Xmax^2 I get 128*10^-4

Because at t=1/2, the mechanical energy is all kinetic energy.

What am I doing wrong? Which answer is correct, why is there a small 3 unit difference?

Last edited by a moderator: Oct 23, 2015
2. Oct 23, 2015

haruspex

Not sure how you are calculating the KE via the spring's PE, but I suspect you are forgetting a difference in gravitational PE between two situations.

3. Oct 23, 2015

JBA

This small error may only be due to number of decimal places of accuracy for either V or Xmax in the calculations.

4. Oct 23, 2015

NooDota

Nevermind, I forgot that I used a π approximation when calculating V, hence the small difference. Sorry.

I guess I should stick to 1/2*k*Xmax^2 because it's more accurate.

Also, haruspex, as I said in the original post, the entire energy is only kinetic since the body is at the equilibrium point, that's why I used 1/2*k*Xmax^2

Last edited: Oct 23, 2015
5. Oct 23, 2015

haruspex

1/2kxmax2 is the elastic potential energy when the body is at its lowest point, yes? In getting from there to the equilibrium point work is done against gravity. That will be about equal to your discrepancy.

6. Oct 23, 2015

JBA

For Example: For a .4 Kg Mass / 9.8 = .0408163265 as limited by my calculator's 10 digit limit on accuracy; and using that value rather than .04 for the mass in your first KE calculation results in KE = 127.551 X 10^-4 rather than your 125 x 10^-4 answer.

7. Oct 23, 2015

haruspex

Why would you divide the mass by g?

8. Oct 23, 2015

NooDota

Still no idea what you're talking about.

The discrepancy is because I used 4π=12.5

Also no idea why g is related here?

9. Oct 23, 2015

haruspex

Ok, I see the confusion. Your xmax is displacement from the equilibrium point, not the extension of the spring. As I said in post #2, I was not sure how you were using a formula based on kx2 to get the KE. It looked like you were using spring PE, but it seems you weren't.

[edit: I still don't know on what basis you used that equation. It isn't the extra spring PE corresponding to the extra extension either.]

Of, course, the period is not exactly 1s, it's $\frac{\pi}{\sqrt{10}}$. Correspondingly, it is not so much that you approximated 4pi as 12.5, but rather that you effectively approximated 4sqrt(10) as 12.5.

Last edited: Oct 23, 2015
10. Oct 24, 2015

NooDota

E = E_k + E_p

At x = 0, E_p = 0

Thus E = E_k

1/2 * k * Xmax^2 = E_k

I don't really see what's wrong in this.

11. Oct 24, 2015

haruspex

The complete energy expression would include a gravitational PE term. That will have a different value at max KE from its value at max elastic PE. On the other hand, the value, 4cm, you substituted for xmax to find max Ek is not the maximum extension. It is the difference between the maximum extension and the equilibrium extension. As it happens, these two errors(?) cancel.
If z is the actual extension and z0 the equilibrium extension then the delta in elastic PE is 1/2 k (z2-z02). If x=z-z0 then this becomes 1/2 k (x2+2xz0). But kz0=mg, so it becomes 1/2 kx2+mgx. The mgx cancels the gravitational PE term.

Now, maybe you understood all this, and it's how you arrived at the equation, but that's not evident so far.

12. Oct 24, 2015

NooDota

There is no gravitational E_p.

13. Oct 24, 2015

haruspex

It's a vertical spring with a mass bobbing up and down. How can there not be gravitational PE?

14. Oct 24, 2015

NooDota

It just ends up being the same numbers as before? It doesn't make a difference if you add it or not?