# Center of Mass bounded by Equations

• nat1
In summary: M and dm are the same as above.(y1 + y2)/2 gives you the y-position of the cm for the given strip. The strip's mass is still dm, for which you previously obtained a formula. The ycm of the object is still given by the weighted sum:##y_{cm} = \frac{1}{M} \int y\;dm ##where you use your new y-position in place of y, and use your dm expression as before.
nat1

## Homework Statement

I have equations that are y1= 2sin($\frac{3}{2}$x) and y2= $\frac{1}{3}$x the point where they intersect is called "a" (about x≈1.88). Find the center of mass where M is the total mass of the object.

## Homework Equations

xcm= $\frac{1}{M}$∫x dM

## The Attempt at a Solution

I found the vertical center of mass by using the area between the curves and setting

dm= density*thickness*(y1 - y2)dx

and setting

M=density*thickness*area

area is between the curves. As I am trying to find the CM along the horizontial of the object i can't figure out dm because when y>$\frac{1}{3}$(a) y1 has two values, how can i correct for this? my first thought is to rearrange the equations to find f(y)=x and subtract the x values but i still run into the same problem.

Nat

nat1 said:

## Homework Statement

I have equations that are y1= 2sin($\frac{3}{2}$x) and y2= $\frac{1}{3}$x the point where they intersect is called "a" (about x≈1.88). Find the center of mass where M is the total mass of the object.

## Homework Equations

xcm= $\frac{1}{M}$∫x dM

## The Attempt at a Solution

I found the vertical center of mass by using the area between the curves and setting

dm= density*thickness*(y1 - y2)dx

and setting

M=density*thickness*area

area is between the curves. As I am trying to find the CM along the horizontial of the object i can't figure out dm because when y>$\frac{1}{3}$(a) y1 has two values, how can i correct for this? my first thought is to rearrange the equations to find f(y)=x and subtract the x values but i still run into the same problem.

Nat

You should be able to find both the x and y centers of mass by integrating along the x-axis. Consider that your dm "strips" bounded by y2 and y1 in addition to having a position along the x-axis, will also each have a vertical center where the actual center of mass of that strip lies.

im still not sure i understand %100:

So each dm will have a vertical center of (y1-y2)/2 and this is where the center of mass for each strip lies. so then i would just need to take the average of these over the interval [0, a]

Thanks!

nat1 said:
im still not sure i understand %100:

So each dm will have a vertical center of (y1-y2)/2 and this is where the center of mass for each strip lies. so then i would just need to take the average of these over the interval [0, a]

Thanks!

Careful, the midpoint between two numbers y1 and y2 is (y1 + y2)/2. And you still need to do the weighted sum of these positions (the dm strips all have different masses).

Good luck!

gneill said:
Careful, the midpoint between two numbers y1 and y2 is (y1 + y2)/2. And you still need to do the weighted sum of these positions (the dm strips all have different masses).

Good luck!

oops, my fault! i would do the integral of $\frac{1}{area between the curves}$∫(y1+y2)/2dx is this right? assuming M and dm are the same as above.

(y1 + y2)/2 gives you the y-position of the cm for the given strip. The strip's mass is still dm, for which you previously obtained a formula. The ycm of the object is still given by the weighted sum:

##y_{cm} = \frac{1}{M} \int y\;dm ##

where you use your new y-position in place of y, and use your dm expression as before.

## What is the definition of center of mass?

The center of mass is the point at which the entire mass of an object is evenly distributed, and is the point where the object will balance perfectly.

## How is the center of mass calculated for a bounded object?

The center of mass for a bounded object is calculated by taking the average of the positions of all the individual mass elements within the object, weighted by their respective masses.

## What are the factors that affect the center of mass of an object?

The center of mass of an object is affected by the distribution of mass within the object and the external forces acting on it.

## How does the center of mass change when an object is divided?

The center of mass of an object will remain the same regardless of its shape or division, as long as the mass and its distribution within the object remain unchanged.

## Why is the center of mass an important concept in physics?

The center of mass is an important concept in physics because it helps us understand the stability and balance of objects, as well as their motion and how they respond to external forces.

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