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You are given a rod of length L. The lin...

You are given a rod of length L. The linear mass density is `lambda` such that `lambda=a+bx`. Here a and b are constants and the mass of the rod increases as x decreases. Find the mass of the rod

Text Solution

Verified by Experts

We are not calculated the mass of the rod simply multiplying linear mass density `lambda` will the length of the rod as `lambda` is not constant.

Hence, we have to divide entire rod length into a number of elements and add the mass of the elements.
`M=dm_1+dm_2+dm_3+...`
This step can be written in terms of integration.
`M=intdm`
If we take an element `dx` on the rod at a distance `x` from the left end of the rod.

Mass of this element `dm=lambda.dx`
`dm=(a+bx)dx`
Hence, total mass,
`M=intdm=underset0oversetLint(a+bx)dx`
`=underset0oversetLintadx+underset0oversetLintbxdx`
`=aunderset0oversetLintdx+bunderset0oversetLintxdx`
`=a[x]_0^L+b[x^2/2]_0^L=aL+(bL^2)/(2)`
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