# Centre of mass of non-uniform rod

• carlyn medona
Won't happen again.In summary, the problem is asking to find the distance of center of mass from the lighter end of a non-uniform rod whose mass increases linearly with distance. The linear density at the lighter end is given by λ= kx+a, where k is a constant and x is the distance from the lighter end. The user attempted to solve the problem by integrating the equation m = ∫(kx+a)dx, but had difficulty eliminating k from the equation.
carlyn medona

## Homework Statement

The mass of non uniform rod increases linearly with distance from lighter end. If m is mass of the rod and l it's total length a the linear density at lighter end, then found the distance of centre of mass from lighter end

## Homework Equations

I put λ= kx+a where lands is linear density k a constant and x the distance from lighter end and integrated but can't get the answer in terms of m l and a

## The Attempt at a Solution

Please show us what you did.

So how do I eliminate k from the equation

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• 15115830849729529855.jpg
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Remember that m = ∫(kx+a)dx

I think I made a mistake in my first calculation , is this one right?

#### Attachments

• 1511584864433740238330.jpg
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carlyn medona said:
I think I made a mistake in my first calculation , is this one right?
Yes this looks right.

Thanks for checking

carlyn medona said:
So how do I eliminate k from the equation

For future reference: you should not post images of handwritten work; actually take the time and trouble to type things out. Most helpers will not even look at all at such posted images, but you were lucky in this case to get a sympathetic reader. Please consult the post "Guidelines for students and helpers" by Vela, pinned to the start of this forum's file list.

Oops sorry for that.

## 1. What is the definition of centre of mass of a non-uniform rod?

The centre of mass of a non-uniform rod is the point where the entire mass of the rod can be considered to be concentrated, while still accounting for the varying distribution of mass along the length of the rod.

## 2. How is the centre of mass of a non-uniform rod calculated?

The centre of mass can be calculated by dividing the total mass of the rod by the total length of the rod, and then multiplying this value by the distance from one end of the rod to the centre of mass of each individual segment. The sum of these values for all segments will give the coordinates of the centre of mass.

## 3. Why is it important to know the centre of mass of a non-uniform rod?

Knowing the centre of mass of a non-uniform rod is important in determining its stability, as well as predicting its motion when subjected to external forces. It also helps in designing structures or objects that need to maintain a particular balance.

## 4. Can the centre of mass of a non-uniform rod be outside the physical boundaries of the rod?

Yes, the centre of mass of a non-uniform rod can be located outside the physical boundaries of the rod. This can happen if the distribution of mass is not symmetric or if there are different densities along the length of the rod.

## 5. How does the centre of mass of a non-uniform rod differ from that of a uniform rod?

The centre of mass of a non-uniform rod is calculated by taking into account the varying distribution of mass along the length of the rod, while the centre of mass of a uniform rod is located at its geometrical centre. This means that the centre of mass of a non-uniform rod can be located at a different position compared to that of a uniform rod with the same mass and length.

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