two balls of masses 5m and m have radii 2r and r.their center of masses are separated by 10r.they move towards each other under their gravitational force. the distance moved by the centre of smaller sphere when the sheres touch each other?
center of gravity does not move.
5 kg and 1 kg balls
radii 2 and 1
call x = 0 at center of 5 kg ball
Original:
then find CG
5(0) + 1 (10) = 6 (Xcg)
Xcg = 5/3
Final
5 Xbig + 1 Xsmall = 6 (5/3) = 10
Xsmall - Xbig = 3 or Xsmall = Xbig+3
so
5 Xbig + Xbig+ 3 = 10
6 Xbig = 7
Xbig = 7/6
Xsmall = 7/6 + 18/6 = 25/6
so small ball moved
10 - 25/6 = (60-25)/6 = 35/6
To find the distance moved by the center of the smaller sphere when the spheres touch each other, we need to determine the point at which they will make contact.
First, let's consider the gravitational force between the spheres. The gravitational force between two objects can be derived using the formula:
F = (G * m1 * m2) / d^2
Where:
F is the gravitational force,
G is the gravitational constant,
m1 and m2 are the masses of the two spheres, and
d is the distance between their centers.
In this case, we have two spheres with masses 5m and m. Their centers of mass are separated by 10r, which means the distance between their centers (d) is 10r.
Next, we need to determine the point at which the gravitational force between the spheres is enough to overcome any other forces and make them touch each other. In this case, both the spheres are moving towards each other under the influence of their mutual gravitational force. As they approach each other, the force of gravity between them increases.
To find the point of contact, we need to equate the gravitational force to the other forces involved. In this case, one of the forces is the force required to move the smaller sphere.
Now, the force required to move the smaller sphere is given by Newton's second law:
F = m * a
Where:
F is the force required,
m is the mass of the smaller sphere, and
a is the acceleration.
Since the spheres touch each other, we can assume that their relative acceleration approaches zero when the spheres come into contact. Therefore, the force required to move the smaller sphere will also be zero.
Now we set up the equation:
(G * m1 * m2) / d^2 = 0
Since the force required to move the smaller sphere is zero, the gravitational force between the spheres must also be zero. Solving this equation, we find that either m1 or m2 must be zero.
However, since both spheres have non-zero masses, this situation is not possible. Therefore, the spheres cannot touch each other under the influence of gravity alone.
In summary, the center of the smaller sphere will not move when the spheres touch each other, as there is no force required to move it in this scenario.