Linear Density & Center of Mass

In summary, the conversation is about finding the mass and center of mass of a rod with a linear density function. The answer involves doing integration from 0 to 0.3 meters to find the mass and using a formula to find the center of mass.
  • #1
Ineedphysicshelp
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Would anyone be able to give me a hint for this question? Do I take the integral? I have possible answers but I am unsure.

A rod of length 30.0 cm has a linear density given by:

D = 50.0g/m + 20.0xg/m^2

where x is the distance from one end measured in meters. a. What is the mass of the rod. b. how far from the x=0 end is the center of mass?
 
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  • #2
Yes, you have to do the integration from 0 to 0.3 m.

Linear density (m') is mass/(unit length), so mass M = [tex]\int{m'(x)}\,dx[/tex].
 
Last edited:
  • #3
You'll just integrate to find the mass:

[tex]M = \int_0^L D(x') dx'[/tex]

and the center of mass is given by

[tex]x_{CM} = \frac {1}{M} \int_0^{L} x' D(x') dx'[/tex]
 

1. What is linear density and how is it calculated?

Linear density is a measure of the mass per unit length of an object. It is calculated by dividing the total mass of the object by its length. The units for linear density are typically kg/m or g/cm.

2. How does the linear density affect an object's center of mass?

The linear density of an object is directly related to its center of mass. The center of mass is the point at which an object's mass is evenly distributed. Objects with a higher linear density will have a center of mass closer to the heavier end of the object.

3. Can the linear density of an object change?

Yes, the linear density of an object can change if its mass or length changes. For example, if an object is cut in half, its linear density will be halved as well.

4. How does the center of mass affect an object's stability?

The center of mass plays a crucial role in an object's stability. If an object's center of mass is located above its base of support, it will be unstable and likely to topple over. On the other hand, if the center of mass is within the base of support, the object will be stable.

5. Can the center of mass be outside of an object?

No, the center of mass must always be located within the physical boundaries of an object. It is a point that represents the average location of an object's mass, so it cannot be outside of the object itself.

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