One mark questions. Gravitation. 01. What is the ratio of force of attraction between two bodies kept in air and the same distance apart in water? Ans- 1:1, because gravitational force is independent of the nature of the medium. 02. If the diameter of the earth becomes twice its present value but its mass remains unchanged,then how would be the weight of an object on the surface of the earth affected? Ans- w=mg=gmm/r2, W α 1/r2, W = W/4 03. A satellite does not require any fuel to circle around the earth. Why? Ans- gravitational force between earth and satellite provide necessary centripetal for circular motion. 04. What is the time period of a simple pendulum at the centre of the earth? Ans- g=0, T=α, pendulum does not oscillate 05. What is the value of gravitational potential energy at infinity? Ans- zero 06. Why is the gravitational potential energy negative? Ans- due to attractive force of gravitation 07. A satellite revolves close to the surface o f a planet. How is its orbital velocity related with velocity of escape from that planet? Ans- Ve= 2 Vo 08. How does the orbital velocity of a satellite depend on the mass of the satellite? Ans- Vo= GM/(R+h), independent of mass 09. A simple pendulum is mounted inside a spacecraft. What should be its time period of vibration? Ans- inside g=0, T=α 10. What will be the kinetic energy needed to project a body of mass m from the earth s surface of radius R to infinity? Ans- KE=1/2 m ( 2gR) 2 = mgr 11. What is the time period and height of a geostationary satellite above the surface of the earth? Ans-T=24 hrs, h=36000 Km 12. What is the basis of Kepler s law of area of planetary motion? Ans- law of conservation of angular momentum 13. The escape velocity from the earth for a body of 20gm is 11 km/s. What will be its value for a body of 100g? Ans- Ve= 11 Km/s 14. What will be our weight at the centre of earth, if the earth were a hollow sphere? Ans- zero. Because gravitational field strength inside a hollow sphere is zero. 15. If the acceleration due to gravity at a height h and depth d below the surface of earth are equal, how are h and d related?
Ans- d = 2h 16. How much is the torque due to gravity on a body about its centre of mass? Ans- zero 17. The gravitational force between two bodies is 1N. If the distance between them is doubled, what will be the force? Ans- F α 1/r2, F = 0.25 N 18. The mass and diameter of a planet are twice those of the earth. What will be acceleration due to gravity on the earth is g? Ans- g=gm/r 2, g =G2M/(2R) 2, g =g/4 19. Write the expression for acceleration due to gravity on the earth s surface? Ans- g= GMe/Re 2 20. The gravitational potential energy of a body at a distance r from the centre of the earth is U. What is the weight of the body at that point? Ans- W = mg = U/r 2MARK QUESTIONS... 1. State Newton s law of gravitation in vector form. 2. How can you find the mass of earth, starting from law of gravitation? 3. A planet reduces its radius by 1% with its mass remaining same. How acceleration due to gravity varies? 4. What is the gravitational force on a body inside a spherical shell? Why is it so? 5. Planet Mars has two moons phobos and deimos. Phobos has a period of 7 hours 39 minutes and orbital radius of 9.4 x 10³km. Calculate the mass of Mars. (G = 6.67 x 10ˉ¹¹Nm²kgˉ²) 6. What is the angular velocity at any point on the equator so that the body feels weightlessness? 7. What is gravitational potential energy at a point? How much of work is done in shifting a mass from the surface to a height equal to its radius? 8. If suddenly the gravitational force of attraction between the earth and a satellite revolving around it becomes zero, what will happen to the satellite? 9. Is it possible to place a satellite, so that it is always over New Delhi? Why? 10. What is escape velocity? Derive an expression for the same. 11. Define gravitational field strength. What is the field at a point distance r from a mass M?
12. Show graphically how gravitational field strength varies with distance from the centre of earth, onwards. Give the relation also. 13. The radius of the earth is reduced by 4%. The mass of the earth remains unchanged. What will be the change in escape velocity? 14. A high jumper can jump 1.5m on earth. With the same effort, how high will he be able to jump on a planet whose density is one-third and radius is one-fourth of that of the earth? 15. The mass and diameter of a planet are twice of those of the earth. What will be the period of oscillation of a pendulum on this planet, if it is a second s pendulum on the earth? 16. Satellite A is in a certain circular orbit about a planet, while satellite B is in a larger circular orbit. Which satellite has (i) the longer period and (ii) the greater speed? 17. If the radius of the earth were increased by a factor of 3, by what factor would its density have to be changed to keep g the same? 18. State Kepler s laws of planetary motion and deduce Newton s Law of gravitation from them. 19. Discuss the variation of g with depth. What happens to g at the centre of earth? 20. The radii of two planets are R and 2R respectively and their densities ρ and ρ/2 respectively. 3 marks What is the ratio of acceleration due to gravity at their surfaces? 1. Define gravitational potential energy of an object at a point in the gravitational field of the earth Derive an expression for it. What does the negative sign signify? Ans :- the gravitational potential energy of a body is the energy associated with it due to its position in the gravitational field of another body and is measured by the ammout of work done in bringing the body from infinity to a given point in the gravitational field of the other. Expression for gravitational potential energy Suppose the earth is a uniform sphere of M and radius R. To calculate the potential energy of a body of mass m located at a distance r from the centre of the earth,r >R. The total work done in bringing the body ftrom infinity(x= )to the point at x=r will be W = = =GMm = - The negative sign in the above equation indicate that the potential energy is due to gravitational attraction between earth and the body. 2. Give three differences between g and G. 3. Define orbital velocity. Derive the expression for orbital velocity of a satellite revolving around a planet. 4. State Kepler s laws of planetary motion.
5. What are geostationary satellites? Obtain the expression for the height of geostationary satellite from the surface of the earth. 6. Define escape velocity. Derive the expression for it from the surface of the earth. 7. Define acceleration due to gravity. Derive a relation showing the variation of acceleration due to gravity g with height h. 8. What do you mean by free fall? How does acceleration due to gravity g vary with the depth d below the surface of the earth. Draw the related graph also. 9. What happens to a body when it is projected vertically upwards from the surface of the earth with a speed of 11200m/s. Compare the escape speeds of two planets of masses M and 4M and radius 2R and R respectively. 10. Derive an expression for the time period of a satellite orbiting around the earth. 11. Define gravitational potential energy of an object at a point in the gravitational field of the earth Derive an expression for it. What does the negative sign signify? 12. Give three differences between g and G. 13. Define orbital velocity. Derive the expression for orbital velocity of a satellite revolving around a planet. 5 MARK QUESTIONS 1.Obtain an expression for the acceleration due to gravity on the surface of earth in terms of mass and radius of earth. Discuss the variation of g with altitude depth and shape of earth. Hint Derivation of g=gm/r 2 b) derivation of g = g(1-2h/r) and g -= g(1-d/r) and g p>g E 2. What is escape velocity? Obtain an expression for escape velocity from the surface of earth. Why there is no atmosphere on the moon? Hint-Definition &Derivation of Ve=(2GM/R) 1/2.On moon Ve is less. 3. Define orbital velocity and establish an expression for it. Calculate the orbital velocity of an artificial satellite of earth orbiting at a height of 1000 km. Hint- Definition& derivation of v o =(GM/R+h) 1/2 By substituting V 0 = 7364m/s 4. State Kepler s laws of planetary motion. Deduce Newton s law of gravitation from Kepler s law. Hint-statements & deduction 5. Distinguish between gravitational potential and gravitational potential energy. Write their SI units. Obtain an expression for the gravitational potential enrgy of a body of mass m lying at a distance r from the centre of earth. Hint-Derivation of grav. Potential=-GM/r grav.pot.energy= -GMm/r 6. Derive an expression for the total energy of a satellite revolving round the earth. What is the significance of negative sign in the expression? What is the maximum value of gravitational potential energy? Hint- Derivation of K.E=GMm/2r, P.E= -GMm/r, T.E= -GMm/2r. Negative sign indicates that force is attractive. 7. What is a geostationary satellite? What is the expression for time period of a satellite orbiting round the earth? Calculate the height of the orbit above the surface of earth in which if a satellite is placed will appear stationary. Hint- Definition & derivation of T=2π(R+h)/v o Calculation of h=[t 2 R 2 g/4π] 1/3 R = 36000 km 8.Using Newton s law of gravitation prove Kepler s third law of planetary motion.
Calculate the period of revolution of Neptune around the sun given that diameter of its orbit is 30 times the diameter of earth s orbit around the sun. Hint- T N 2 =T E 2 (r N /r E ) 3 =(1) 2 X (300) 3 =27000,T N =164.3 years. 9. Prove Kepler s second law of planetary motion. Distinguish between geostationary and polar satellites. Hint-proof of Da/dt=L/2m,a constant. Difference between geostationary and polar satellites 10. Discuss the variation of acceleration due to gravity with 1) height 2)depth from the surface of earth. Draw a graph showing the variation of g with distance r from the centre of earth. At what height above the surface of earth the value of g is the same as in a mine 80km deep? Hint- Derivation of g =g(1-2h/r) g =g(1-d/r) Graph g(1-2h/r)= g(1-d/r) 2h/R=d/R h=d/r= 40km. HOTS S.NO QUESTIONS MARKS Q1 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. A1 Q2 A2 Q3 A3 Q4 Molecules experience the vertically downward force due to gravity just like an apple falling from a tree. Due to thermal motion, which is random, their velocity is not in the vertical direction. The downward force of gravity causes the density of air in the atmosphere close to earth higher than the density as we go up. Give one example each of central force and non-central force Central forc:-- gravitational force of a point mass, electrostatic force due to a point charge. Non-central force:-- spin-dependent nuclear forces, magnetic force between two current carrying loops. What is the direction of areal velocity of the earth around the sun? It is normal to the plane containing the earth and the sun as areal Velocity How is the gravitational force between two point masses
A4 Q5 A5 Q6 A6 Q7 A7 Q8 A8 affected when they are dipped in water keeping the separation between them the same? It remains same as the gravitational force is independent of the medium separating the masses. Is it possibe for a body to have inertia but no weight? Yes, a body will always have mass but the gravitational force on it can be zero; for example, when it is kept at the centre of the earth. 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? No An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity? Yes, if the size of the spaceship is large enough for him to detect the variation in g. 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. F Q9 A9 R Out of aphelion and perihelion, where is the speed of the earth more and why? At perihelion because the earth has to cover greater linear
Q10 distance to keep the areal velocity constant What is the angle between the equatorial plane and the orbital plane of (a) (b) Polar satellite? Geostationary satellite? A11 Q12 a)90,b)0 Mean solar day is the time interval between two successive noon when sun passes through zenith point (meridian). Sidereal day is the time interval between two successive transit of a distant star through the zenith point (meridian). By drawing appropriate diagram showing earth s spin and orbital motion, show that mean solar day is four minutes longer than the sidereal day. In other words, distant stars would rise 4 minutes early every successive day. (HINT: you may assume circular orbit for the earth). A12 Every day the earth advances in the orbit by approximately 1o. Then, it will have to rotate by 361 (which we define as 1 day) to have sun at zenith point again. Since 361 corresponds to 24 hours; extra 1 corresponds to approximately 4 minute [3 min 59 seconds]. Q13 Two identical heavy spheres are separated by a distance 10 times their radius. Will an object placed at the mid point of the line joining their centres be in stable equilibrium or unstable equilibrium? Give reason for your answer.
A13 Consider moving the mass at the middle by a small amount GMm /(R-h) 2 to the right GMm/(R+h) 2 second. Hence the net force will also be towards the right. Hence the equilibrium is unstable. NUMERICALS Gravitation 1 1 Write the value of g near a planet with a mass equal to that of earth and radius 1 twice that of earth. Ans :g=gm/r 2 ;g =GM/4R 2 =g/4=10/4 m/s 2 =2.5m/s 2 ) 2 Find the weight of a body of mass 50kg near earth s surface. Ans :W=mg=5x10N=50N 3 What is the value of g at a point at depth of half the radius of earth below the surface of earth. Ans :g =g(1-d/r)=10x1/2=5m/s 2 4 Find the escape speed of an object near a planet whose mass to radius ratio is 10 11 :1 1 1 1 Ans :Ve= 2Gx10 11 )m/s= (2x6.67)m/s=3.65m/s 5 Find the approximate value of g and at a height equal to 1/10 th of the radius of earth.(from the ground) 1 Ans:g =g(1-2x1/10)m/s 2 =10x4/5 m/s 2 =8m/s 2 6 Calculate the correct value of g at a height of 4800km above the ground. g =g(r+h) 2 /R 2 )=10x112 2 /64 2 = 7 Calculate the decrease in the value of g at a depth of 4800km below the ground compared to g value at the surface of earth. 2 2
g =g(1-x4800/6400)=10x1/4=2.5 m/s 2 8 A body of mass 500kg is thrown upwards with a speed of 20m/s.calculate its kinetic and potential energies at the maximum height where it reaches. 2 Equate the total energy at the starting point and the total energy at the highest point to calculate required values :P.E=-GMm/r 9 A body weighs 70 kgf on the surface of earth.how much will it weigh on the surface of another planet whose mass is 1/9 and radius is 1/2 that of earth. 3 10 Ans :g =gm /mx(r/r ) 2 =10x (1/9x(2) 2) =20/3=6.7m/s 2 w =mg =70x6.7=469N=46.9kgf State Kepler s third law.find the time periods of revolution of of a planet whose distance from the sun is 4 times that of earth. Ans :Statement T 2= T 1 (R 2 /R 1 ) 3/2 =1x (4 3 )ysr=8yrs. 3 Value Based Questions. 01. Roshan and Ravi were class XI students. They were studying Gravitation chapter. Suddenly Roshan asked Ravi that a person sitting in a satellite feels weightlessness, but for a person on moon it is not. Why? Both were thinking for the answer. Then they went to their physics teacher for help. The teacher gave the answer with reason in a very easy and simple manner. (a) What can be the possible answer and reason for that question? (b) What are the values being displayed by the students? (c ) What are the values displayed by the teacher? 02. Two students Anjali and Anaga of class XI science were discussing about certain concepts in gravitation chapter. Anjali said that my weight is 50N on earth. Then Anaga asked what could be your weight on moon? After a while Anjali said that I will discuss with my teacher and give you the answer. The teacher explained the answer with suitable mathematical calculation. (a) What could be the correct answer given by the teacher to Anjali? (b) What are the values displayed by the students? (c ) What is he value displayed by the teacher?