# How Does a Student's Position Affect Scale Readings and Center of Gravity?

• Soojin
In summary, the conversation discusses a problem involving a uniform plank and a student lying on top of it. The student's weight and the distance of their center of gravity from their feet are to be determined. The equations w=mg and center of mass = mr2/sum of masses are mentioned, but the correct equation for center of mass is stated to be M_{tot}\boldsymbol{x}_{cm} = \sum_i m_i \boldsymbol{x}_i. The equation L1W1=L2W2 is also mentioned, but it is incomplete as it does not include the total length of the plank. Additional information is needed to determine the location of the student's center of gravity.
Soojin

## Homework Statement

For some reason I can't get my picture to show up, but here is the link to it:
http://img.photobucket.com/albums/v302/Robi41035/Picture1.jpg

"The plank is uniform and 2.2 m long. Initially the scales each read 100 N. A 1.60 m tall student then lies on top of the plank, with the soles of his feet directly above scale B. Now scale A reads 394.0 N and scale B reads 541 N.

a) What is the student's weight?

b) How far is his center of gravity from the soles of his feet?

c) When standing, how far above the floor is his center of gravity, expressed as a fraction of his height?"

## Homework Equations

a) w = mg
Possibly L1W1=L2W2.

b)center of gravity = mr2/sum of masses

c) I think this one is just the answer for b/1.60.

## The Attempt at a Solution

a) I know that weight = mg. I also thought I might have to use the equation L1W1=L2W2, but I'm not sure how to set this up.

b) I know that the center of gravity = mr2/sum of masses, but I'm not sure what I should be using as "m" and "r".

I know this is simple, but I'm having a hard time grasping the concepts. If anyone can help me out, I would appreciate it a lot. Thanks!

Last edited:
Soojin said:
b)center of gravity = mr2/sum of masses
That is not the equation for the center of mass. One way to check: Look at the units. The center of mass is a position vector: It should have units of length. Your equation has units of length squared. The correct equation is
$$M_{tot}\boldsymbol{x}_{cm} = \sum_i m_i \boldsymbol{x}_i$$

a) I know that weight = mg. I also thought I might have to use the equation L1W1=L2W2, but I'm not sure how to set this up.
You are missing that L1+L2=L=2.2 meters. What this will give you is the center of mass of the plank+person. You will need to use some additional information to get the location of the center of mass of the person.

Firstly, it is important to note that the terms "center of gravity" and "weight" are often used interchangeably, but they actually have different meanings. The center of gravity is the point at which an object's weight is evenly distributed, while weight is a measure of the force of gravity acting on an object.

Now, let's address the given problem. To determine the student's weight, we can use the formula w = mg, where w is weight, m is mass, and g is the acceleration due to gravity (9.8 m/s^2). We know that the scale A reads 394.0 N, so that is the weight of the plank and the student combined. We also know that the scale B reads 541 N, which is the weight of the plank and the student minus the weight of the plank alone (100 N). So, we can set up the equation 394.0 N + 541 N = m(9.8 m/s^2), where m is the mass of the plank and the student combined. Solving for m, we get a mass of approximately 98.5 kg. Therefore, the student's weight is 98.5 kg.

To find the center of gravity, we can use the formula center of gravity = mr2/sum of masses. In this case, the "m" represents the mass of the student (since that is the only variable mass in the problem) and "r" represents the distance from the soles of his feet to his center of gravity. We can set up two equations using the information given: 394.0 N = (98.5 kg)(9.8 m/s^2)r and 541 N = (98.5 kg)(9.8 m/s^2)(2.2 m - r). Solving for r, we get a value of approximately 1.04 m. This means that the student's center of gravity is 1.04 m from the soles of his feet.

Finally, to determine how far above the floor the student's center of gravity is, we can use the formula h = r/1.60, where h is the height above the floor and r is the distance from the soles of his feet to his center of gravity. Plugging in the value we found for r (1.04 m), we get a height of approximately 0.65 m above the floor

## What is the difference between center of gravity and weight?

The center of gravity is the point at which an object's weight is evenly distributed, while weight is the measurement of the force of gravity acting on an object. Weight is expressed in units of mass, such as pounds or kilograms, while center of gravity is a location.

## How is center of gravity determined?

The center of gravity can be determined by finding the balance point of an object, either by physical measurement or through mathematical calculations based on the object's shape and mass distribution. It can also be affected by external forces, such as an additional weight being added to the object.

## Why is center of gravity important?

The center of gravity is important because it affects an object's stability and balance. Objects with a lower center of gravity are more stable and harder to tip over, while objects with a higher center of gravity are more prone to tipping. It is also important in the design and engineering of structures and vehicles to ensure their stability and safety.

## Can the center of gravity change?

Yes, the center of gravity can change depending on the position and orientation of an object. For example, when a person bends their knees, their center of gravity moves lower, making them more stable. External forces such as weight distribution, movement, or added weight can also cause a change in an object's center of gravity.

## How does center of gravity affect motion?

The center of gravity affects an object's motion by determining its stability and balance. If an object's center of gravity is not aligned with its base of support, it will tip or fall over. In terms of rotational motion, the center of gravity also plays a role in determining an object's moment of inertia, which affects its ability to rotate or spin.

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