Advertisements
Advertisements
प्रश्न
Write the three laws given by Kepler. How did they help Newton to arrive at the inverse square law of gravity?
Advertisements
उत्तर
The orbit of a planet moving around the sun
Kepler’s first law: The orbit of a planet is an ellipse with the Sun at one of the foci.
In the above figure, the elliptical orbit of a planet revolving around the sun. The position of the Sun is indicated by S.
Kepler’s second law: The line joining the planet and the sun sweeps equal areas in equal intervals of time.
A → B, C → D and E → F are the displacements of the planet in equal intervals of time.
The straight lines AS and CS sweep equal areas in equal intervals of time, i.e., area ASB and CSD are equal.
Kepler’s third law: The square of its period of revolution around the Sun is directly proportional to the cube of the mean distance of a planet from the Sun.
Thus, if r is the average distance of the planet from the Sun and T is its period of revolution, then
T2 α r3 i.e., `(T^2)/(r^3)` = constant = K
For simplicity, we shall assume the orbit to be a circle.
Circular motion of planet around the sun
S denotes the position of the Sun, P denotes the position of a planet at a given instant, and r denotes the radius of the orbit (= the distance of the planet from the Sun). Here, the speed of the planet is uniform.
v = `"distance travelled"/"time taken"`
= `(2pir)/T`
If m is the mass of the planet, the centripetal force exerted on the planet by the Sun (= gravitational force),
F = `(mv^2)/r`
F = `(m ((2pir)/T)^2)/r`
F = `(4 m pi^2r)/T^2`
Multiplying and dividing by r2 we get,
F = `(4 m pi^2)/r^2`
F = `(r^3/T^2)` ...(i)
According to Kepler’s third law,
`T^2/r^3 = K` ...(ii) [∵ K = constant]
Substituting (ii) in (i)
F = `(4m pi^2)/(K) xx 1/r^2`
F = constant × `1/r^2` [∵ where, `(4 m pi^2)/K` = constant]
∴ `F prop 1/r^2`
APPEARS IN
संबंधित प्रश्न
How does the force of gravitation between two objects change when the distance between them is reduced to half?
The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth? Why?
What happens to the force between two objects, if the mass of one object is doubled?
Choose the correct alternative:
Acceleration due to gravity is independent of mass of the earth/mass of the body.
The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12) (i) a, (ii) b, (iii) c, (iv) 0.
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.
Which of the Kepler’s laws of planetary motion led Newton to establish the inverse-square rule for gravitational force between two bodies ?
At noon, the sun and the earth pull the objects on the earth's surface in opposite directions. At midnight, the sun and the earth pull these objects in same direction. Is the weight of an object, as measured by a spring balance on the earth's surface, more at midnight as compared to its weight at noon?
The mass of moon is about 0.012 times that of earth and its diameter is about 0.25 times that of earth. The value of G on the moon will be:
How will the force of gravitation between two objects change if the distance between them is:
Infinite
Why does a ball moving on a table top eventually stops?
Explain why:
The atmosphere does not escape.
The ______ force is much weaker than other forces in nature.
State Newton’s universal law of gravitation. Express it with the mathematical form of force of gravitation?
Give the applications of universal law gravitation.
Particles of masses 2M, m and M are respectively at points A, B and C with AB = ½ (BC). m is much-much smaller than M and at time t = 0, they are all at rest (Figure). At subsequent times before any collision takes place ______.
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.
We can shield a charge from electric fields by putting it inside a hollow conductor. Can we shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?
Who measured the value of the Universal Gravitational Constant G using the Cavendish balance?
A hollow spherical shell of uniform density exerts what gravitational force on a point mass placed inside it?
