8. A uniform ring of mass m and radius a is placed directly
above a uniform sphere of mass M and of equal radius
The centre of the ring is at a distance 13 a from the centre
of the sphere. Find the gravitational force exerted by the
sphere on the ring.
find it any point on the ring
Answers
Answer:
A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance r3 as shown in the figure. The gravitational force exerted by the sphere on the ring will be.
Answer:
R.E.F image
On a differential part
of ring , the gravitational
force (dF) by sphere =
(2R
2
)(2R)
GMdm(2)
r
⇒d
F
=
4R
2
(R)
4Mdm
r
=
4R
3
GMdm
r
Total gravitational (F)=∫
0
m
4R
3
GMdm
r
r
is resolved into vectors are
with magnitude (
2
3
r
) directed perpendicular
to the plane of ring and
other radially with magnitude (
2
∣
r
∣
)
Let them be r
⊥
and
r
c
receptively.
Then,
r
=
r
⊥
+
rc
So,dF=
4R
3
GMdm
r
=
4R
3
GMdm
(
r
⊥
+
r
c
)
F
=∫
0
m
df
=∫
0
m
4R
3
GMdm(
r
⊥
)
+∫
0
m
4R
3
GMdm
r
c
F
=∫
0
m
4R
3
GMdm
(
2
3
)R
r
⊥
^
+∫
0
m
4R
3
GMdm
(
2
R
)
r
c
^
F
=
8R
2
3
GM
∫
0
m
dm
r
⊥
^
+
8R
2
GM
∫
0
m
dm
r
c
^
At every point
r
⊥
^
is same but
r
c
^
changes accordingly such that
∫
0
m
dm
r
⊥
^
=M
r
⊥
^
and ∫
0
m
r
c
^
=0
So,
F
=
8R
2
3
GM
m
r
⊥
^