Advertisements
Advertisements
Question
Suppose the gravitational potential due to a small system is k/r2 at a distance r from it. What will be the gravitational field? Can you think of any such system? What happens if there were negative masses?
Advertisements
Solution
The gravitational potential due to the system is given as \[V = \frac{k}{r^2}\]
Gravitational field due to the system :
\[E = - \frac{dV}{dr}\]
\[ \Rightarrow E = - \frac{d}{dr}\left( \frac{k}{r^2} \right) = - \left( - \frac{2k}{r^3} \right)\]
\[ \Rightarrow E = \frac{2k}{r^3}\]
We can see that for this system , \[E \propto \frac{1}{r^3}\]
This type of system is not possible because \[F_g\] is always proportional to inverse of square of distance(experimental fact).
If there were negative masses, then this type of system is possible.
This system is a dipole of two masses, i.e., two masses, one positive and the other negative, separated by a small distance.
In this case, the gradational field due to the dipole is proportional to \[\frac{1}{r^3}\]
APPEARS IN
RELATED QUESTIONS
Choose the correct answer from among the given ones:
For the problem 8.10, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow (i) d, (ii) e, (iii) f, (iv) g.
A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero? Mass of the sun = 2 ×1030 kg, mass of the earth = 6 × 1024 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 1011 m).
State the universal law of gravitation. Name the scientist who gave this law.
Let V and E be the gravitational potential and gravitational field at a distance r from the centre of a uniform spherical shell. Consider the following two statements :
(A) The plot of V against r is discontinuous.
(B) The plot of E against r is discontinuous.
A body is suspended from a spring balance kept in a satellite. The reading of the balance is W1 when the satellite goes in an orbit of radius R and is W2 when it goes in an orbit of radius 2 −R.
Two small bodies of masses 10 kg and 20 kg are kept a distance 1.0 m apart and released. Assuming that only mutual gravitational forces are acting, find the speeds of the particles when the separation decreases to 0.5 m.
A uniform metal sphere of radius a and mass M is surrounded by a thin uniform spherical shell of equal mass and radius 4a (In the following figure). The centre of the shell falls on the surface of the inner sphere. Find the gravitational field at the points P1 and P2 shown in the figure.
Write an expression for the gravitational force of attraction between two bodies of masses m1 and m2 separated by a distance r.
How will the force of gravitation between two objects change if the distance between them is:
Made four times
The force of attraction between any two material objects is called __________.
Why don't you feel the force of attraction between your friend sitting close to you and yourself?
A ball is thrown up with a speed of 4.9 ms-1.
Calculate the time it takes to reach this height.
What does a force do in the following case?
You twist a piece of rubber.
State Newton's law of gravitation. What is the difference between:
g and G?
Mahendra and Virat are sitting at a distance of 1 m from each other.Their masses are 75 kg and 80 kg respectively. What is the gravitational force between them? (G = 6.67 × 10-11 Nm2/kg2)
To project the rockets which of the following principle(s) is /(are) required?
For the weight of body of mass 5 kg to be zero on equator of the earth, angular velocity of the earth must be (The radius of earth = 6400 km, acceleration due to gravity = 10 m/s2).
As observed from earth, the sun appears to move in an approximate circular orbit. For the motion of another planet like mercury as observed from earth, this would ______.
If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law- ______.
- planets will not have elliptic orbits.
- circular orbits of planets is not possible.
- projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.
- there will be no gravitational force inside a spherical shell of uniform density.
Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be ______.
