Two uniform solid sphere of equal radii R ,but mass m and 4 m have a centre to centre separation 6R,as shown in fig8.10.The two spheres are held fixed.A projectile of mass m is projected from the surface of the sphere of mass m directly towards the centre of the second sphere obtain an experience for the minimum speed v of the projectile so that it reaches the surface of the second sphere.
Answers
Answer:
ANSWER
Le the projectile of mass m is fired with minimum velocity, v from the surface of the sphere of mass M to reach the surface of the sphere of mass 4M. Let N be a neutral point at a distance r from the center of the sphere of mass M. At neutral point N,
r
2
GM
=
(6R−r)
2
G(4M)m
(6R−r)
2
=4r
2
6R−r=±2r or r=2R or −6R
The point r=−6R does not concern us.
Thus, ON=r=2R
It is sufficient to project the projectile with a speed that would enable it to reach N. Thereafter, the greater gravitational pull of 4M would suffice.
The mechanical energy at the surface of M is E
i
=
2
1
mv
2
−
R
GMm
−
5R
G(4M)m
At the neutral point N, speed approaches zero.
∴ The mechanical energy at N is E
N
=−
2R
GMm
4R
G(4M)m
=−
2R
GMm
−
R
GMm
According to law of conservation of mechanical energy, E
i
−E
N
2
1
mv
2
−
R
GMm
5R
G(4M)m
=−
2R
GMm
−
R
GMm
v
2
=
R
2GM
[
5
4
−
2
1
]=
5
3
R
GM
or v=(
5
3
R
GM
)
1/2