Two uniform solid spheres of equal radii R but masses M and 4M have a centre-to-centre separation 6R as shown in figure. The two spheres are held fixed. A projectile of mass 
is projected from the surface of the sphere of mass M directly towards the centre of the second sphere. N is the point where net gravitational is zero.
With reference to the above situation, match the Column I (quantities) with Column II (mathematical expressions).
Column I | Column II | ||
A. | Distance r of neutral point N | I. |
|
B. | Minimum speed of the projectile, , so that it reaches the surface of the second sphere. | II. |
|
C. | The speed with which the projectile hits the seconds sphere, if projected with . | III. | |
D. | Gravitational potential energy due to | IV. |
Choose the correct answer from the options given below.
Text Solution
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The projectile is acted upon by two mutually opposing gravitational forces of the two
spheres.
Hence, there must be a point on the line 
, where 
, i.e. net external force due to gravitational attraction force vanishes. The point is called neutral point N. Let it be at a distance r from O.
At N, F 1 = F 2
or 
The neutral point 
is not relevant.
Thus, 
.
To project the projectile with minimum speed from M, it is sufficient to project the particle with a speed which would enable it to reach N. After N, projectile will be attracted by 4M.
At the neutral point, speed approaches zero, i.e. 
Total energy of the particle at 
GPE due to 
GPE due to 
From the principle of conservation of mechanical energy,
Therefore, minimum speed of the projectile with which it reaches the surface of second sphere is 
.
Let 
be the speed with which projectile hits the second sphere. Applying principle of conservation of energy, we get
Hence, 
and 
.
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