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
The weight of an object is more at the poles than at the equator. Is it beneficial to purchase goods at equator and sell them at the pole? Does it matter whether a spring balance is used or an equal-beam balance is used?
Define one Newton. How much maximum acceleration can it produce in a mass of 1 kg?
The acceleration produced by a force in an object is directly proportional to the applied _________ And inversely proportional to the _________ Of the object.
How will the force of gravitation between two objects change if the distance between them is:
Infinite
Why don't you feel the force of attraction between your friend sitting close to you and yourself?
A force can produce ________, In an object at rest. It can __________ an object and change its __________ of motion.
What does a force do in the following case?
You apply brakes to a running car.
Explain the difference between g and G.
The distance-time values for an object moving along straight line are given below:
| Time (s) | Distance (m) |
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
Show that gravity decreases at higher altitudes.
What do you mean by a gravitational constant?
The value of universal gravitational constant (G) in the SI unit is ______.
The _______ force is much weaker than other forces in nature.
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).
Different points in earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the c.m. (centre of mass) causing translation and a net torque at the c.m. causing rotation around an axis through the c.m. For the earth-sun system (approximating the earth as a uniform density sphere).
Choose the wrong option.
The gravitational force between a hollow spherical shell (of radius R and uniform density) and a point mass is F. Show the nature of F vs r graph where r is the distance of the point from the centre of the hollow spherical shell of uniform density.
Write the answer of the question with reference to laws of gravitation.
Write the value of the universal gravitational constant.
Newton's universal law of gravitation applies to ______.
