Gravitation Notes Exercise - I

GRAVITATION Exercise - I Page # 10 UNSOLVED PROBLEMS Q.1 Answer the following : NCERT XI_8.1 (a) Among the known types of forces in nature, the gravitational force is the weakest. Why then does it play a dominant role for motion of bodies on the terrestrial, astronomical and cosmological scale (b) Do the forces of friction and other contact forces arise due to gravitational attraction ? If not, what is the origin of these forces ? (c) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means ? (d) An astronaut inside a small space ship 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 ? (e) If you compare the gravitational force on the Earth due to the Sun to that due to the moon, you would find that the Sun's pull is greater than the moon's pull is greater than the tidal effect of Sun. Why? Q.2 Choose the correct alternative :NCERT XI_8.2 (a) Ac c e le r a t ion d ue t o gr a v it y in cr ea s e s / decreases with increasing altitude. (b) Ac c e le r a t ion d ue t o gr a v it y in cr ea s e s / decreases with increasing depth (assume the Earth to be a sphere of uniform density). (c) The effect of rotation on the effective value of acceleration due to gravity is greatest at the equator/poles. (d) Acceleration due to gravity is independent of mass of the Earth/mass of the body. (e) The formula – G Mm (1/r2 – 1/r 1) is more/less accurate than the formula mg (r 2 – r 1) for the difference of potential energy between two points r 2 and r1 distance away from the centre of the Earth. Q.3 Choose the correct alternative : NCERT XI_8.3 (a) If the gravitational potential energy of two mass points infinite distance away is taken to be is (positive/negative/zero) (b) The universe on the large scale is shaped by (gravitational/electromagnetic) forces, on the atomic scale by (gravitational/electromagnetic) forces, on the nuclear scale by (gravitational/ electromagnetic/strong nuclear) forces. (c) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy. (d) The energy required to rocket an orbiting satellite out of Earth's gravitational influence is more/ less than the energy requires to project a stationary object at the same height (as the satellite) out of Earth's influence. Q.4 Does the escape speed of a body from the Earth depend on (a) the mass of the body, (b) the location from where it is projected, (c) the direction of projection, (d) the height of the location from where the body is launched ? Explain your answer. NCERT XI_8.4 Q.5 A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed, (b) angular speed, (c) angular momentum, (d) kinetic energy, (e) potential energy, (f) total energy throughout its orbit ? Neglect any mass loss of the comet when it comes very close to the Sun. Q.6 Which of the following observations point to the equivalence of inertial and gravitational mass : (a) Two spheres of different masses dropped from the top of a long evacuated tube reach the bottom of the tube at the same time. (b) The time-period of a simple pendulum is independent of its mass. (c) The gravitational force on a particle inside a hollow isolated sphere is zero. (d) For a man in closed cabin that is falling freely under gravity, gravity disappears'. (e) An astronaut inside a spaceship orbiting around the earth feels weightless. (f) Planets orbiting around the sun obey Kepler's Third Law (approximately). (g) The gravitational force on a body due to the Earth is equal and opposite to the gravitational force on the Earth due to the body. Q.7 Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem. GRAVITATION Page # 11 Q.8 The gravitation intensity at the centre of the drumhead defined by a hemispherical shell has the direction indicated by the arrow (see figure) (i) a, (ii) a, (iii) C, (iv) zero. g p d e f b c a P C Q.9 For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow (i) d, (ii) e, (iii) f, (iv) g. Q.10 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 × 10 30 kg, mass of the Earth = 6 × 10 24 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 10 11 m). Q.11 How will you 'weight the Sun', that is estimate its mass ? You will need to know the period of one of its planets and the radius of the planetary orbit. The mean orbital radius of the Earth around the Sun is 1.5 × 10 8 km. Estimate the mass of the sun. Q.12 A Saturn year is 29.5 times the Earth year. How far is the Saturn from the Sun if the Earth is 1.50 × 108 km away from the Sun ? Q.13 A body weighs 63 N on the surface of the Earth. What is the gravitational force on it due to the Earth at a height equal to half the radius of the Earth ? Q.14 Assuming the Earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the Earth if it weighed 250 N on the surface ? Q.15 A rocket is fired vertically with a speed of 5 km s–1 from the Earth's surface. How far from the Earth does the rocket go before returning to the Earth ? Mass of the earth = 6.0 × radius of the Earth = 6.4 × 10 6 m ; G = 6.67 × 10–11 N m 2 kg–2. 1024 kg; mean Q.16 The escape speed of a projectile on the Earth's surface is 11.2 km s –1. A body is projected out with thrice this speed. What is the speed of the body far away from the Earth ? Ignore the presence of the Sun and other planets. Q.17 A satellite orbits the Earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the Earth's gravitational influence ? Mass of the satellite = 200 kg; mass of the Earth = 6.0 × 10 24 kg; radius of the Earth = 6.4 × 10 6 m; G = 6.67 × 10–11 N m2 kg–2. Q.18 Two stars each of one solar mass (= 2 × 1030 kg) are approaching each other for a head on collision. When they are a distance 10 9 km, their speeds are negligible. What is the speed with which they collide ? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G). Q.19 Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational field and potential at the mid point of the line joining the centres of the spheres ? Is an object placed at that point in equilibrium ? If so, is the equilibrium stable or unstable? What are the factors on which it depend. Q.20 As you have learnt in the text, a geostationary satellite orbits the Earth at a height of nearly 36,000 km from the surface of the Earth. What is the potential due to Earth's gravity at the site of this satellite ? (Take the potential energy at infinity to be zero). Mass of the Earth = 6.0 × 1024 kg, radius = 6400 km. Q.21 In a two-stage launch of a satellite, the first stage brings the satellite to a height of 150 km and the second stage gives it the necessary critical speed to put it in a circular orbit around the Earth. Which stage requires more expenditure of fuel ? (Neglect damping due to air resistance, especially in the first stage). Mass of the Earth = 6.0 × 10 24 kg; radius = 6400 km; G = 6.67 ×10 –11 N m2 kg–2. Q.22 In an imaginary planetary system, the central star has the same mass as our Sun, but is much brighter so that only a planet twice the distance GRAVITATION Page # 12 between the Earth and the Sun can support life. Q.28 A non-homogeneous sphere of radius R has Assuming biological evolution (including aging the following density variation : processes etc.) on that planet similar to ours, what would be the average life span of a 'human' on that Q.29 Light from a massive star suffers 'gravitational planet in terms of its natural year " The average red-shift' i.e., its wavelength changes towards the life span of a human on the Earth may be taken to red end due to the gravitational attraction of the be 70 years. star. Obtain the formula for this gravitational red- Q.23 Imagine a tunnel dug along a diameter of the earth. Show that a particle dropped from one end of the tunnel executes simple harmonic motion. What shift using the simple consideration that a photon of frequency v has energy hv (h is the Planck's constant) and mass hv/c2. Estimate the magnitude is the time (equal to its known average density = of the red-shift for light of wavelength 5000 Å from 5520 kg m–3) and G = 6.67 × 10 –11 N m2 kg–2. a star of mass 10 32 kg and radius 106 km. Use the Neglect all damping forces. known values of G and c. (G = 6.67 × 10 –11 N m2 Q.24 If the Earth were a perfect sphere of radius 6.37 × 10 6 m, rotating about its axis with a period of one day (= 8.64 × 10 4 s) how much would the acceleration due to a gravity (g) differ from the poles to equator ? kg–2, c = 3.00 × 108 m s–1). Q.30 The following problem is calculator/ computer based : We have studied the depth dependence of the acceleration due to gravity of the Earth (g(d)), Q.25 A star 2.5 times the mass of the sun and collapsed to a size of 12 km rotates with the speed assuming the Earth to be a sphere of constant density. Dziewonski and Anderson have proposed a of 1.2 rev. per second. (Extremely compact stars sophisticated five-tier model outlined in the table of this kind are known as neutron stars. Certain below. Plot the variation of g(r) with r where r is observed stellar objects called pulsars are believed the distance from the centre of the Earth to belong to this category). Will an object placed (0 < r < RE). on its equator remain stuck to its surface due to Tier Inner radius (km) Outer radius (km) Average density 5 8 (10 kg/m ) Q.26 A spaceship is stationed on Mars. How much Inner Core 0.0 1221.5 12.893 energy must be expended on the spaceship to rocket Outer Core 1221.5 3480.0 10.901 it out of the solar system ? mass of the space ship Low er Mantle 2480.0 5701.0 4.904 = 1000 kg; mass of the sun = 2 × 1030 kg; mass of Upper Mantle 5701.0 6346.6 3.605 Crust/Ocean 6346.6 6371.0 2.395 Total 0.0 6371.0 5.513 gravity ? (Mass of the Sun = 2 × 10 30 kg). Mars = 6.4 × 1023 kg; radius of Mars = 3395 km; radius of the orbit of Mars = 2.28 × 10 8 km; G = 6.67 × 10 –11 N m2 kg–2. Q.27 A rocket is fired 'vertically' from the surface of Mars with a speed of 2 km s–1. If 20% of its initial energy is lost due to Martian atmospheric resistance, how far will the rocket go from the surface of Mars before returning to it ? Mass of Mars = 6.4 × 10 23 kg; radius of Mars = 3395 km; G = 6.67 × 10 –11 N m2 kg–2. 1