Change in the centre of mass of a leg

[Emethyst]

Homework Statement

To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0m long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0kg person, the mass of the upper leg would be 8.60kg, while that of the lower leg (including the foot) would be 5.25kg. a) Find the x-coordinate of the center of mass of this leg, relative to the hip joint, if it is stretched out horizontally. b) Find the y-coordinate of the center of mass of this leg, relative to the hip joint, if it is stretched out horizontally. c) Repeat parts a and b for the leg if it is bent at the knee to form a right angle with the upper leg remaining horizontal

Homework Equations

Centre of mass equation: c.m = m₁s₁ + m₂s₂ . . . / m₁ + m₂ . . .

The Attempt at a Solution

No idea how to do this one. I know the basic formula as shown above, but I can't seem to be able to find the right answer by plugging in the numbers given. I don't even know how to go about part c, so more or less I'm trying to figure out how to properly do this question. If anyone can be of assistance it would be greatly appreciated, thanks in advance.

 

[anathema]

Thank you for your post. I am happy to provide you with some guidance and assistance with this problem.

First, let's start by defining the variables in the problem. We have a human leg that is 92.0m long, with the upper leg and lower leg having equal lengths. Let's call the length of each leg L. The total mass of the leg is given as 70.0kg, with 8.60kg for the upper leg and 5.25kg for the lower leg (including the foot). We can call the masses of the upper and lower leg m1 and m2, respectively.

a) To find the x-coordinate of the center of mass, we need to use the formula for the center of mass equation that you have mentioned. In this case, we have two masses (m1 and m2) and two distances (s1 and s2) from the hip joint. The distance for the upper leg (s1) is simply L, as it is the full length of the leg. For the lower leg (s2), we need to take into account that it is attached to the upper leg, so the distance from the hip joint is only half of the total leg length, or L/2. Plugging in the values, we get:

c.m = (m1s1 + m2s2) / (m1 + m2)
= (8.60kg * L + 5.25kg * L/2) / (8.60kg + 5.25kg)
= (8.60kg * 92.0m + 5.25kg * 46.0m) / 13.85kg
= 92.2m

Therefore, the x-coordinate of the center of mass is 92.2m from the hip joint.

b) To find the y-coordinate of the center of mass, we need to take into account the fact that the leg is stretched out horizontally, which means that the center of mass will be at the same height as the hip joint. Therefore, the y-coordinate of the center of mass will be 0m.

c) To repeat parts a and b for the bent leg, we need to consider the new geometry of the leg. The upper leg remains horizontal, so the distance for the upper leg (s1) remains L. However, for the lower

 

[kerimek]

I can provide some guidance on how to approach this problem. First, it is important to understand the concept of center of mass. The center of mass is the point at which the mass of an object is evenly distributed, meaning that if you were to suspend the object at that point, it would balance perfectly. In this case, we are looking for the center of mass of a human leg, which is essentially a uniform rod with two masses (upper and lower leg) attached at each end.

To solve for the center of mass, we will use the equation you mentioned, which is the sum of the individual masses multiplied by their respective distances from the reference point (in this case, the hip joint) divided by the total mass of the leg. So, for part a, we can set up the equation as follows:

c.m = (8.60kg * 0.46m + 5.25kg * 0.46m) / (8.60kg + 5.25kg)

The distances used are half of the total length of the leg, since we are assuming both the upper and lower leg have equal lengths. This gives us a center of mass at 0.46m from the hip joint in the x-direction. Similarly, for the y-direction, the center of mass would be at the midpoint of the leg, which is 0.23m above the hip joint.

For part b, we can use the same equation but with the leg bent at a right angle. In this case, the upper leg remains horizontal, so the distance for the upper leg remains the same, but the lower leg now has a vertical component. So the equation becomes:

c.m = (8.60kg * 0.46m + 5.25kg * 0.23m) / (8.60kg + 5.25kg)

This gives us a center of mass at 0.39m from the hip joint in the x-direction and 0.23m above the hip joint in the y-direction.

I hope this helps to clarify the process for solving this problem. Keep in mind that this is a simplified model and in reality, the center of mass of a human leg may vary depending on factors such as muscle mass and bone structure. But this is a good exercise to understand the concept of center of mass and how it can be calculated.