one can determine a persons center of mass by having them recline on a plank resting on two scales. A person 2m tall lies on such a plank and the scale near her head reads 405 N and the one near her feet reads 360 N. Find the center of her mass.
To determine the center of mass, we can use the principle of balancing torques. The center of mass is the point where all the weight of an object can be considered to act.
In this scenario, we have a person lying on a plank resting on two scales. The scales measure the normal force exerted by the person's body. Since there are two scales, one near the person's head and the other near their feet, we can use the difference in the readings to find the distance from the center of mass to the scales.
Let's assume the distance from the center of mass to the scale near the person's head is x, and the distance from the center of mass to the scale near the person's feet is (2 - x) since the person is 2m tall.
Now, let's calculate the torques exerted by the person's body on the plank, about the scale near their head and the scale near their feet.
Torque about the head scale = (lying person's weight) * x
Torque about the feet scale = (lying person's weight) * (2 - x)
Since the scales are in equilibrium, the torques exerted by the body must be equal.
Thus, we can write:
(lying person's weight) * x = (lying person's weight) * (2 - x)
By canceling out the weight factor, we get:
x = 2 - x
Simplifying, we can add "x" to both sides:
2x = 2
Dividing both sides by 2:
x = 1
Therefore, the center of mass of the person lies at a distance of 1m from the scale near their head and at a distance of (2 - 1) = 1m from the scale near their feet.