Find a step or block or just lean against a wall or door frame and let your leg

Question

Find a step or block or just lean against a wall or door frame and let your leg swing naturally, without forcing motion. Time the period (probably by measuring the time for several cycles, remember a cycle means returning to the same point) and determine the frequency of oscillation. Measure or estimate the length of your leg, and using I = mL2/3 for a rod rotating around one end (your hip) and the center of mass being halfway down (or somewhere else if you want to try to be more accurate), estimate what frequency you’d expect. The mass will cancel out of Eq: frequency = 1/2pi SQRT (mgd/I), when you put in the moment of inertia. If you’re way off, speculate on possible reasons. Describe your processes briefly (how you measured/estimated various numbers)

Explanation / Answer

Total time = 20 s

Number of cycles = 10

So period = 2s and frequency measured =10/20 = 0.5 Hz

Now estimation :

mass of my one leg m = 15 kg

Length L = 1 m

moment of inertia i = ML^2/3 = 15*1^2/3 = 5 kgm^2

Center of mass distance d = 0.4L = 0.40 m because thighs are thicker.

Estimated frequency = 1/2pi * sqrt(mgd/i)

= 1/2pi * sqrt(15*9.8*0.40/5)

= 0.546 Hz . answer

Measured frequecy is slightly less than estimated frequency. The air friction or inaccuracy in estimation of center of mass may be the reason.

Mass of leg was estimated by the fact that each legs weigh about 20-22% of the body weight. Length was measured by measuring tape. center of mass location was assumed to be at 0.4L, because mass is more concentrated in thighs so 0.4L makes more sense than 0.5L.

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