PERIOD OF THE LEG
The period of the leg can be approximated by treating the leg as a physical pendulum, with a period of 2pi*sqrt(I/mgh), where I is the moment of inertia, m is the mass, and h is the distance from the pivot point to the center of mass.
The leg can be considered to be a right cylinder of constant density. For a man, the leg constitutes 16% of his total mass and 48% of his total height.
Find the period of the leg of a man who is 1.82 m in height with a mass of 72 kg. The moment of inertia of a cylinder rotating about a perpendicular axis at one end is ml^2/3
To find the period of the leg of a man, we need to determine the moment of inertia (I), the mass (m), and the distance from the pivot point to the center of mass (h) of the leg.
First, let's calculate the mass and height of the leg. We know that the leg constitutes 16% of the man's total mass and 48% of his total height.
Mass of the leg = Total mass * 16% = 72 kg * 16% = 11.52 kg
Height of the leg = Total height * 48% = 1.82 m * 48% = 0.8736 m
Given that the leg is considered a right cylinder of constant density, the moment of inertia (I) can be calculated using the formula I = ml^2/3, where l is the length of the cylinder (which is equal to the height in this case).
Moment of inertia of the leg (I) = (mass of the leg * height^2)/3 = (11.52 kg * (0.8736 m)^2)/3
Now, substitute the values of I, m, and h (I = (11.52 kg * (0.8736 m)^2)/3, m = 11.52 kg, h = 0.8736 m) into the formula for the period of a physical pendulum:
Period(T) = 2π * sqrt(I / (m * g * h))
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Calculate the period using the given values and the formula:
Period(T) = 2π * sqrt((11.52 kg * (0.8736 m)^2)/3 / (11.52 kg * 9.8 m/s^2 * 0.8736 m))
Simplify the expression inside the square root:
Period(T) = 2π * sqrt(0.8736 m^2 / (3 * 9.8 m/s^2))
Period(T) = 2π * sqrt(0.0296 s^2)
Calculate the square root:
Period(T) = 2π * 0.172 s
Finally, multiply by 2π to get the final result:
Period(T) ≈ 1.08 s
Therefore, the period of the leg of a man who is 1.82 m in height with a mass of 72 kg is approximately 1.08 seconds.