A person of mass M lies on the floor doing leg raises. His leg is 0.85m long and pivots at the hip. Treat his legs (including feet) as uniform cylinders, with both legs comprising 34.5% of body mass, and the rest of his body as a uniform cylinder comprising the rest of his mass. He raises both legs 40∘ above the horizontal.

1)How far does the center of mass of each leg rise?
2)How far does the entire body's center of mass rise?

To answer these questions, we need to understand the concept of center of mass and use some basic principles of physics.

1) How far does the center of mass of each leg rise?

The center of mass of an object is the point at which the object can be balanced perfectly. In this case, since we are considering a uniform cylinder, the center of mass of each leg would be at its geometric center.

The first step is to find the mass of each leg. We are given that both legs (including feet) comprise 34.5% of the body mass. So, the mass of each leg can be calculated as:

Mass of each leg (m_leg) = 0.345 * Mass of body (M)

Next, we can calculate the height by which the center of mass of each leg rises when the person raises his legs. This height depends on the length of the leg (l_leg) and the angle at which it is raised (θ).

The height can be calculated using the formula:

Height = l_leg * sin(θ)

Substituting the given values, we can calculate the height to which the center of mass of each leg rises.

2) How far does the entire body's center of mass rise?

To calculate the height to which the entire body's center of mass rises, we need to consider both legs and the rest of the body. Since the legs make up 34.5% of the body mass, the remaining 65.5% of the body mass would be in the rest of the body.

The height to which the entire body's center of mass rises can be calculated as:

Height = (Height of each leg * Mass of each leg) + (Height of the rest of the body * Mass of the rest of the body)

Substituting the values obtained from the previous calculations, we can find the height to which the entire body's center of mass rises when the person raises his legs.

By following these steps and performing the necessary calculations, you can find the answers to both questions.

To calculate the vertical displacement of the center of mass of each leg and the entire body, we can use the principles of rotational equilibrium.

1) To find the displacement of the center of mass of each leg, we can use the formula:

Δh = r * sin(θ)

Where:
Δh = vertical displacement of the center of mass
r = distance from the pivot to the center of mass
θ = angle of rotation

It is given that the person's leg is 0.85m long and pivots at the hip. We assume that the center of mass is located at half of the leg length. Therefore, the distance from the pivot to the center of mass of each leg is 0.85m/2 = 0.425m.

To find the vertical displacement, Δh, we plug in the values into the formula:

Δh = 0.425m * sin(40°)
Δh ≈ 0.273m

Therefore, the center of mass of each leg rises approximately 0.273 meters.

2) To find the displacement of the entire body's center of mass, we need to consider the proportion of each component's mass. It is given that both legs (including feet) comprise 34.5% of the body mass. Thus, the rest of the body (including the torso, arms, and head) comprises the remaining 65.5% of the mass.

Since the center of mass of each leg rises 0.273 meters as calculated above, the vertical displacement of the entire body's center of mass would be the weighted average of the displacement of each leg's center of mass.

Weighted average displacement = (Proportion of leg mass * Displacement of leg center of mass)

= (0.345 * 0.273m) + (0.345 * 0.273m)
≈ 0.188m

Therefore, the entire body's center of mass rises approximately 0.188 meters.