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20 Nov 2019

I am attempting to answer question 10 from this sheet:

In which of the above situations will the period of oscillations T be the greatest? Explain the reason for your answer, (assume that all the surfaces are frictionless, the springs are massless and the weights are identical in all cases) Consider the two simple pendulums with identical masses but with different lengths shown above. Suppose they are released from rest in position A (same phase) at the same time as shown. So you understand that they will not oscillate in the same way (i.e. will not oscillate in phase) since they will be having different periods of oscillations. But at a later time we will see both pendulums reaching the position A simultaneously. When will it take place? (calculate the time taken from the start)

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Related textbook solutions

Physics: Principles with Applications

7th Edition, 2013

ISBN: 9780321625922

Related questions

A mass m rests on a frictionless horizontal table and isconnected to rigid supports via 2 identical springs each of relaxedlength L0 and spring constantk. Each spring is stretched to a length L considerable greater thanL0. Horizontal displacementsof m from its equilibrium position are labeled x and y.

(Diagram insert didn't work, all it really tells you is that xis on the axis of the springs and y is perpendicular to thesprings)

a) Write down the differential equation of motion (i.e.,Newton's Law) governing small oscillations in the xdirection.

b) Same as a but in the y direction (assume that y <<L).

c) In terms of L and L0, calculate the ratio of the periods ofthe oscillations along x and y.

d) If at t=0 the mass m is released from the point x = y = A0with zero velocity, what are its x and y coordinates at any latertime t?

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So, I started with just ma = -2kx for a, leading me to thedifferential equation x''(t) + (2k/m) x = 0.

Then I did basically the same thing with y.

However, when I get to c, there's my problem - I don't knowhow I could write any of this in L and L0.

Also, b doesn't really make sense to me because if y is smallwhen compared to L, it seems like there shouldn't be anyoscillation in the y direction.

The answer the back of the book shows for c and d are:

c) Tx/Ty = (1 - L0 / L)1/2

d) x(t) = A0cos( (2k/m)1/2 t)

y(t) = A0cos( (2k(L-L0)/mL)1/2 t)

I have no idea how they got to these answers. Can somebodyhelp me? It has something to do with springs already beingstretched, but it seems they are making an assumption somewherethat I don't understand.

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