Two playground swings start out together. After ten completeoscillations the swi

Question

Two playground swings start out together. After ten completeoscillations the swings are out of step by a half-cycle. Find thepercentage difference in the length of the swings.

I tried out the following procedures:

One makes 10 oscillations in the time and the other makes 10.5(could be 9.5, but I'm gonna go with 10.5)
The distance each swing goes is pi x (radius squared) eachoscillation.

So, 10.5 = (pi x radius squared)
Radius = square root of (10.5 / pi)

This is the length of the first swing. Do this for the other swingwith 10 instead of 10.5 and then take one length away from theother.

But keep getting the wrong answer. Please help me and give mespecific explanation.

Explanation / Answer

Assuming that the swings are at smallamplitudes, they roughly behave as simple pendulums undergoingsimple harmonic motion.


The period of such a swing is given by:
T = 2*Pi*sqrt(L/g)

g is constant, Pi and 2 are obviously constant. The relation isbetween T and L.

Swing 1 is the slow swing, it swings exactly 10 cycles in the timeinterval:
10*T1 = 10.5*T2

10*2*Pi*sqrt(L1/g) = 10.5*2*Pi*sqrt(L2/g)

Cancel 2, cancel Pi, cancel g:
10*sqrt(L1) = 10.5*sqrt(L2)

Square both sides:
10^2*L1 = 10.5^2 * L2

Solve for L1/L2:
L1/L2 = (10.5/10)^2

Result:
L1/L2 = 1.1025

Conclusion, the percentage difference is 10.25% in the chain lengthof the swings.

Swing 1 is 10.25% longer than swing 2.

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