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skysupercool · Feb 1, 2011, 09:00 AM

2.Two pendulum of length 1.21m and 1m start vibrating. At some instant the two are in the mean position in same phase. After how many vibrations of the longer pendulum will they be in the same phase again?

Unknown008 · Feb 1, 2011, 10:24 AM

You need to know the formula:

$T = 2\pi \sqrt{\frac{l}{g}}$

g is the acceleration due to gravity.
T is the period.

Then find omega, from $\omega = \frac{2\pi}{T}$

Then, the general equation for an oscillation:

$x = \sin(\omega t)$

Put the different values of omega in the equations.

Now, what I think would be the easiest is to plot those two curves on the same graph, then see where they meet and at either both increasing or both decreasing. Then, you estimate the time where they meet (or if you have a graphic calculator, this should be a piece of cake) and use simultaneous equations to get a more accurate value of t.

Otherwise, the longer method would be to find all the points where they meet and check the gradients of each graph at each point.

skysupercool · Feb 1, 2011, 10:37 AM

Can you plug in the values for me and tell me the answer? Its not supposed to be very tough since it's a multiple choice question. Help appreciated

Unknown008 · Feb 1, 2011, 11:00 AM

Well, putting in the two equations instead of working through a lot of values, we get:

$\omega = \sqrt{\frac{l}{g}}$

$\omega_1 = \sqrt{\frac{1}{9.8}} = 0.31\ rad/s$

$\omega_2 = \sqrt{\frac{1.21}{9.8}} =0.35\ rad.s$

So,

$x_1 = \sin(0.31t)$

$x_2 = \sin(0.35t)$

NOTICE: I'm switching partitions on my computer to make use of the graphing package to get the points. I'll be right back.

EDIT: I did a mistake here, please see my post below.

Unknown008 · Feb 1, 2011, 11:14 AM

Okay, the time I got is 157 s approximately.

Unknown008 · Feb 1, 2011, 12:12 PM

Dash, I made a mistake while using the formulae.

The values of omega need to be reversed;

$\omega_1 = \sqrt{\frac{9.8}{1}} = 3.13$

$\omega_2 = \sqrt{\frac{9.8}{1.2}} = 2.86$

This time, we get this:

As you can see, they meet at time t about 24 s

You can now compute the number of oscillations by knowing the time that the longer pendulum takes.

This will average to 24/2.2 = 10.87 oscillations.

EDIT: What? I wasn't even finished with your question and getting a picture was so time consuming... :(