Linear Mass Density - Bars (Center of Mass)

In summary, the problem asks to find the coordinate of the center of mass for 4 black bars with constant linear density. The solution involves using the equation M0 X = summation of (MiXi) and finding the length of each smaller bar along the diagonals, which is L/√2. The answer is given as M/sqrt(2) at (L/4, L/4) and at (L/4, 3L/4). This is because the smaller bars have a mass of M/√2, making the mass less in the middle of the bar even though it is uniform.
  • #1
azukibean
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1. Homework Statement
The 4 black bars have constant linear density. In terms of L, find the coordinate of the center of mass

Homework Equations


M0 X = summation of (MiXi)

The Attempt at a Solution


I understand most of the solution, except the answer key says one thing I don't get
M/sqrt(2) at (L/4, L/4)
and at (L/4, 3L/4).
Why is there less mass in the middle of bar though the bar is uniform?
 
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  • #2
M is the mass of a bar of length L. The diagonal of the square is L√2. Half the diagonal is L/√2. That's the length of each of the smaller little bars along the diagonals. So their masses must be M/√2.

Chet
 
  • #3
Thanks! That made a lot of sense.
 

Related to Linear Mass Density - Bars (Center of Mass)

1. What is linear mass density?

Linear mass density, also known as linear density or linear mass, is a measure of the mass per unit length of a one-dimensional object, such as a bar or wire. It is typically denoted by the symbol λ (lambda) and is expressed in units of kilograms per meter (kg/m).

2. How is linear mass density calculated?

Linear mass density is calculated by dividing the total mass of an object by its length. For example, if a bar has a mass of 2 kilograms and a length of 1 meter, its linear mass density would be 2 kg/m.

3. What is the center of mass for a bar with linear mass density?

The center of mass for a bar with linear mass density is the point at which the mass of the bar is evenly distributed along its length. It can be calculated by finding the weighted average of the positions of all the individual mass elements along the bar.

4. How does the linear mass density affect the center of mass of a bar?

The linear mass density directly affects the position of the center of mass of a bar. A higher linear mass density means that there is more mass per unit length, which results in the center of mass being closer to the heavier end of the bar.

5. What are some real-life applications of linear mass density in bars?

Linear mass density is used in various fields such as engineering, physics, and materials science. Some examples of its applications include determining the strength and stability of structures, designing cables and wires for suspension bridges, and studying the behavior of materials under tension or compression.

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