Question

The linear density of a thin rod of length lm varies 2 = (1 + 2x ), where x is the distance from its one end. Find the distance of its centre of mass from this end
​

Answer 1

Explanation:

λ=2

meter

kg

+

meter

2

2kg

.x

λ=(2+2x)kg/m ; x is in meter

suppose there is an element of length dx

& at a distance x from left must end.

mass of this element

is dm=λdx=(2+2x)dx

⇒ mass of rod = sum of marvels of these elements

M=∫

x=0

1

dm=∫

0

1

(2+2x)dx=[2x+

2

2x

]

0

1

m=3kg

x

c.m

=

∫dm

∫

0

1

xdm

=

∫

0

1

(2+2x)dx

∫

0

1

(2+2x)xdx

x

cm

=

3

∫

0

1

(2x+2x

2

)

dx=

3

[

2

2x

2

+

3

2x

3

]

0

1

x

cm

=

3

1+

3

2

meter =

9

5

meter

from are end (having less density)