# Differentiation of damped motion function - Need help ly!

1. Sep 4, 2009

### Shaybay92

Differentiation of damped motion function - Need help urgently!

1. The problem statement, all variables and given/known data

Basically my task was to come up with a function to model the swing of a pendulum. The model I came up with was:

0.16e$$^{-0.25t}$$cos(($$\stackrel{2\pi}{1.22}$$)t-0.8) + 0.814

The next part of my task asks me to find the point where the pendulum is first stationary. I attempted to differentiate this function, and set it equal to 0 to find t. When I graph it in my graphics calculator, I get approximately 0.18seconds, but when I do it by hand I get 0.165seconds. Help would be appreciated.

The attempt at a solution
When I differentiated using the product rule I got:

-0.04e$$^{-0.25t}$$ [cos((2$$\pi$$/1.22) t-0.8)-4(2$$\pi$$/1.22)sin((2$$\pi$$/1.22) t-0.8) ]

If you could please go through the word document attached, ths contains the working out of the derivative and solving when set to 0.

Help is greatly appreciated, thankyou!!

#### Attached Files:

• ###### Stationary point working.doc
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Last edited: Sep 4, 2009
2. Sep 4, 2009

### Staff: Mentor

Re: Differentiation of damped motion function - Need help urgently!

How can a pendulum have an offset in its displacement? Shouldn't it oscillate about 0 displacement? The initial phase in the cos() argument should be able to take care of an initial condition of an offset release...

3. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

We made a model off experimental data. A motion detector measured the distance of the pendulum from it, and it did not exactly begin measuring when the pendulum was released either.

What do you mean the cos() argument?

4. Sep 4, 2009

### Staff: Mentor

Re: Differentiation of damped motion function - Need help urgently!

The argument to the cos() function in your equation has a time component and an offset constant phase component.

Did the pendulum oscillate about 0 displacement, or about 0.814m displacement? Zero is the bottom of the pendulum swing, right?

5. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

The motion detector sat on a table 0.814m away from the centre of oscillation. The pendulum moved backwards and forwards, towards the motion detector.

6. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

Do you understand what I mean? I'm not sure why I'm getting 0.15 off the actual value.

7. Sep 4, 2009

### Redbelly98

Staff Emeritus
Re: Differentiation of damped motion function - Need help urgently!

0.165 is approximately 0.18. How accurate, in your estimation, is the value from the graph?

That looks right, and is zero at 0.165 s just as you calculated.

8. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

The minimum was calculated using my graphics calculator. I graphed the function and pressed 'MIN' where it solved for the first minimum point at t=0.1805seconds. I realise it is close, however should it not be almost exact, as I kept all the values in the calculator without any rounding?

9. Sep 4, 2009

### Redbelly98

Staff Emeritus
Re: Differentiation of damped motion function - Need help urgently!

Wait, I just spotted an error. It should be a + in front of the sin term.

That changes things. Neither 0.165 or 0.18 are correct :grumpy:

10. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

It's negative because i factorized out -0.04? It was originally 0.016, so I had to make it -4.

11. Sep 4, 2009

### Redbelly98

Staff Emeritus
Re: Differentiation of damped motion function - Need help urgently!

What is the derivative of cos?

12. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

-sin?

13. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

Oh! I think I see the problem ...lol

14. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

15. Sep 4, 2009

### Redbelly98

Staff Emeritus
Re: Differentiation of damped motion function - Need help urgently!

I don't see anything squared here.

16. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

Are you looking at the attached document?

0 =[cos(2π/1.22 t-0.8)+(8π/1.22)sin(2π/1.22 t-0.8) ]
That's after changing it back to a positive sin.. Then I let y = 2pi/1.22 t - 0.8, changed cos into a sin function using pythagorean identity, and then let w = sin y

-(8π/1.22)w= √(1-w^2 )
Then I squared both sides
[-(8π/1.22)w]= [√(1-w^2 )]^2

Which got rid of the +/- problem anyway

17. Sep 4, 2009

### Redbelly98

Staff Emeritus
Re: Differentiation of damped motion function - Need help urgently!

Just saw your attachment. There's a much easier way to solve this.

If

0 = cosy + A siny

then

cosy = ____ ? (solve above equation for cosy)​

and

tany = siny / cosy = ____ ?​

or

y = arctan(____?)​

18. Sep 4, 2009

### Shaybay92

Re: Differentiation of damped motion function - Need help urgently!

0 = cos y + (8pi/1.22)siny
sin y = -cosy/(8pi/1.22)
siny/cosy = -0.0485
tan y = -0.0485
y = -0.0485

Recall value of y:
(2pi/1.22)t - 0.8 = -0.0485
2pi/1.22t = 0.7514
t = 0.1459

19. Sep 4, 2009

### Redbelly98

Staff Emeritus
Re: Differentiation of damped motion function - Need help urgently!