A person whose height is 1.88 m is lying on a light board placed on two scales so that scale 1 is under the person’s head and scale 2 is under the person’s feet. Scale 1 reads 48.3 kg and scale 2 reads 39.3 kg .
Where is the center of mass of the person (the distance from his head)?
The Correct Answer and Explanation is :
To determine the center of mass of the person, we can use the concept of moments, which is the principle that the sum of the moments about a point (in this case, the person’s head) must equal the total weight of the person. This can be calculated using the readings from the two scales under the person.
Step-by-Step Solution:
- Total Weight of the Person:
The total weight (force) of the person can be obtained by summing the readings of the two scales.
[
\text{Total Weight} = 48.3 \, \text{kg} + 39.3 \, \text{kg} = 87.6 \, \text{kg}
]
Since 1 kg corresponds to 9.8 N of force (the acceleration due to gravity), we can calculate the total force:
[
\text{Total Force} = 87.6 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 858.48 \, \text{N}
] - Moment Equation:
The center of mass is located at a point where the sum of the moments (torques) about the head equals the total weight of the person. The moment (torque) produced by the weight at each scale is the product of the force and the distance from the point where we are calculating the moment (the head).
Let ( d ) be the distance from the person’s head to their center of mass. The total moment equation around the head is:
[
\text{Moment from Scale 1} = \text{Force from Scale 1} \times 0 \, (\text{since it’s at the head})
]
[
\text{Moment from Scale 2} = \text{Force from Scale 2} \times d
]
Thus, the total moment balance about the head is:
[
\text{Moment from Scale 1} + \text{Moment from Scale 2} = \text{Total Force} \times \text{Distance from Head to Center of Mass}
]
[
0 + 39.3 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times d = 858.48 \, \text{N} \times d
]
The distance ( d ) can be calculated by solving:
[
\text{Moment from Scale 2} = 39.3 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times d = 48.3 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times (1.88 \, \text{m} – d)
] - Distance from Head to Center of Mass:
Using the above setup, we can now calculate the center of mass position. Let’s go ahead and compute it.
The center of mass of the person is approximately 1.04 meters from their head.
Explanation:
In this problem, we used the principle of moments to calculate the location of the center of mass. The total weight of the person is distributed between the two scales, one under the head and one under the feet. By considering the forces at each scale and balancing the moments (torques) about the head, we determined the center of mass distance from the head.
The force from each scale acts through the point where the scale is located. By equating the moments on both sides of the person’s body, we could solve for the position of the center of mass. The result, approximately 1.04 meters, tells us how far from the head the center of mass is located.
This approach assumes that the body is essentially rigid and that the scales provide accurate readings of the forces acting on the body.
Now, let me generate an image to visually represent this situation.
Here is the visual representation of the situation, showing the person lying on the light board with the two scales and the center of mass marked approximately 1.04 meters from their head. Let me know if you’d like to explore anything else!
