Advertisements
Advertisements
Question
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).
Advertisements
Solution 1
Mass of the Sun, Ms = 2 × 1030 kg
Mass of the Earth, Me = 6 × 10 24 kg
Orbital radius, r = 1.5 × 1011 m
Mass of the rocket = m
Let x be the distance from the centre of the Earth where the gravitational force acting on satellite P becomes zero.
From Newton’s law of gravitation, we can equate gravitational forces acting on satellite P under the influence of the Sun and the Earth as:
`(GmM_s)/(r-x)^2 = GmM_e/x^2`
((r-x)/x)^2 = `M_s/M_e`
`(r-x)/r = ((2xx10^(30))/(60xx10^(24)))^(1/2) = 577.35`
`1.5 xx 10^(11)-x=577.35x`
`x = (1.5xx10^11)/578.35` = `2.59 xx 10^8` m
Solution 2
Mass of Sun, M = 2 x 1030 kg; Mass of Earth, m = 6 x 1024 kg Distance between Sim and Earth, r = 1.5 x 1011 m
Let at the point P, the gravitational force on the rocket due to Earth = gravitional force on the rocket due to sun
Let x = distance of the point { from the Earth
Then `(Gm)/x^2 = (GM)/(r-x)^2`
`=>(r-x)^2/x^2 = M/m = (2xx10^(30))/(6xx10)^24 = 10^6/3`
or `(r-x)/x = 10^3/sqrt3 => r/x = 10^3/sqrt3 + 1 =~ 10^3/sqrt3`
or `x = (sqrt3r)/10^3 = (1.732xx1.5xx10^11)/10^3 = 2.6 xx 10^8` m
RELATED QUESTIONS
What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface? (Mass of the earth is 6 × 1024 kg and radius of the earth is 6.4 × 106 m).
What happens to the force between two objects, if the masses of both objects are doubled?
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.
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?
Which of the following quantities remain constant in a planetary motion (consider elliptical orbits) as seen from the sun?
How will the force of gravitation between two objects change if the distance between them is:
Halved
Where will you weigh more: at the moon's surface or at the earth's surface?
What is meant by the equation :
`g= Gxxm/r^2`
where the symbols have their usual meanings.
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 pull the skin of your arm
What does a force do in the following case?
You twist a piece of rubber.
Name and state the action and reaction in the following case:
A book lying on a table.
Name and state the action and reaction in the following case:
A person walking on the ground.
State Newton's law of gravitation. What is the difference between:
Gravity and 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 ______.
Molecules in air in the atmosphere are attracted by gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree.
Two particles of equal mass 'm' go around a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle with respect to its centre of mass is ______.
For an object outside a uniform solid sphere, from where is the sphere's mass considered to act?
The gravitational force between two objects acts in which direction?
