Question

A rod of length 3 m and its mass per unit length is directl
proportional to the distance x from its one end. The cente
of gravity of the rod from that end will be at
(1) 1.5 m
(2) 2 m
(3) 2.5 m
(4) 3.0 m

Answer 1

Answer:

/////////////////////////////////3.0

Answer 2

Solution:
Length of rod = 3 m
mass per unit length ραx
ρ
α
x
ρ=kx
ρ
=
k
x
Let dx be small mass element
xcm=∫xdm∫dm
x
c
m
=
∫
x
d
m
∫
d
m
dm=kxdx
d
m
=
k
x
d
x
xcm=∫03x2dx∫03xdx
x
c
m
=
∫
0
3
x
2
d
x
∫
0
3
x
d
x
=27392
=
27
3
9
2
=2