PYQ Vault

Physics · Textbook solutions

Gravitation

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 35 questions

Worked Examples

8 q

Solved Examples

Worked · 8
  1. Eg 7.1
    Let the speed of the planet at the perihelion PP in Fig. 7.1(a) be vPv_P and the Sun-planet distance SP be rPr_P. Relate {rP, vP}\{r_P,\ v_P\} to the corresponding quantities at the aphelion {rA, vA}\{r_A,\ v_A\}. Will the planet take equal times to traverse BACBAC and CPBCPB?
  2. Eg 7.2
    Three equal masses of mm kg each are fixed at the vertices of an equilateral triangle ABC. (a) What is the force acting on a mass 2m2m placed at the centroid G of the triangle? (b) What is the force if the mass at the vertex A is doubled? Take AG=BG=CG=1 mAG = BG = CG = 1\,\text{m} (see Fig. 7.5)
  3. Eg 7.3
    Find the potential energy of a system of four particles placed at the vertices of a square of side ll. Also obtain the potential at the centre of the square.
  4. Eg 7.4
    Two uniform solid spheres of equal radii RR, but mass MM and 4M4M have a centre to centre separation 6R6R, as shown in Fig. 7.10. The two spheres are held fixed. A projectile of mass mm is projected from the surface of the sphere of mass MM directly towards the centre of the second sphere. Obtain an expression for the minimum speed vv of the projectile so that it reaches the surface of the second sphere.
  5. Eg 7.5
    The planet Mars has two moons, phobos and delmos. (i) phobos has a period 7 hours, 39 minutes and an orbital radius of 9.4×103 km9.4 \times 10^3\,\text{km}. Calculate the mass of mars. (ii) Assume that earth and mars move in circular orbits around the sun, with the martian orbit being 1.52 times the orbital radius of the earth. What is the length of the martian year in days?
  6. Eg 7.6
    Weighing the Earth : You are given the following data: g=9.81 m s−2g = 9.81\,\text{m s}^{-2}, RE=6.37×106 mR_E = 6.37 \times 10^6\,\text{m}, the distance to the moon R=3.84×108 mR = 3.84 \times 10^8\,\text{m} and the time period of the moon's revolution is 27.3 days. Obtain the mass of the Earth MEM_E in two different ways.
  7. Eg 7.7
    Express the constant kk of Eq. (7.38) in days and kilometres. Given k=10−13 s2 m−3k = 10^{-13}\,\text{s}^2\,\text{m}^{-3}. The moon is at a distance of 3.84×105 km3.84 \times 10^5\,\text{km} from the earth. Obtain its time-period of revolution in days.
  8. Eg 7.8
    A 400 kg satellite is in a circular orbit of radius 2RE2R_E about the Earth. How much energy is required to transfer it to a circular orbit of radius 4RE4R_E? What are the changes in the kinetic and potential energies?

Exercises

27 q
  1. Ex 7.1(a)
    Answer the following : (a) 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 ?
  2. Ex 7.1(b)
    Answer the following : (b) 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 ?
  3. Ex 7.1(c)
    Answer the following : (c) 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. (you can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the moon's pull is greater than the tidal effect of sun. Why ?
  4. Ex 7.2(a)
    Choose the correct alternative : (a) Acceleration due to gravity increases/decreases with increasing altitude.
  5. Ex 7.2(b)
    Choose the correct alternative : (b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).
  6. Ex 7.2(c)
    Choose the correct alternative : (c) Acceleration due to gravity is independent of mass of the earth/mass of the body.
  7. Ex 7.2(d)
    Choose the correct alternative : (d) The formula −G Mm(1/r2−1/r1)-G\,Mm\left(1/r_2 - 1/r_1\right) is more/less accurate than the formula mg(r2−r1)mg\left(r_2 - r_1\right) for the difference of potential energy between two points r2r_2 and r1r_1 distance away from the centre of the earth.
  8. Ex 7.3
    Suppose there existed a planet that went around the Sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?
  9. Ex 7.4
    Io, one of the satellites of Jupiter, has an orbital period of 1.7691.769 days and the radius of the orbit is 4.22×108 m4.22 \times 10^8\,\text{m}. Show that the mass of Jupiter is about one-thousandth that of the sun.
  10. Ex 7.5
    Let us assume that our galaxy consists of 2.5×10112.5 \times 10^{11} stars each of one solar mass. How long will a star at a distance of 50,000 ly50{,}000\,\text{ly} from the galactic centre take to complete one revolution ? Take the diameter of the Milky Way to be 105 ly10^5\,\text{ly}.
  11. Ex 7.6(a)
    Choose the correct alternative: (a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
  12. Ex 7.6(b)
    Choose the correct alternative: (b) The energy required to launch an orbiting satellite out of earth's gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth's influence.
  13. Ex 7.7
    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?
  14. Ex 7.8
    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.
  15. Ex 7.9
    Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem.
  16. Ex 7.10
    In the following two exercises, choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 7.11)
    1. A.
      a
    2. B.
      b
    3. C.
      c
    4. D.
      0
  17. Ex 7.11
    For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow
    1. A.
      d
    2. B.
      e
    3. C.
      f
    4. D.
      g
  18. Ex 7.12
    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= 2 \times 10^{30}\,\text{kg}, mass of the earth =6×1024 kg= 6 \times 10^{24}\,\text{kg}. Neglect the effect of other planets etc. (orbital radius =1.5×1011 m= 1.5 \times 10^{11}\,\text{m}).
  19. Ex 7.13
    How will you 'weigh the sun', that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5×108 km1.5 \times 10^8\,\text{km}.
  20. Ex 7.14
    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 km1.50 \times 10^8\,\text{km} away from the sun ?
  21. Ex 7.15
    A body weighs 63 N63\,\text{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 ?
  22. Ex 7.16
    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 N250\,\text{N} on the surface ?
  23. Ex 7.17
    A rocket is fired vertically with a speed of 5 km s−15\,\text{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 =6.0×1024 kg= 6.0 \times 10^{24}\,\text{kg}; mean radius of the earth =6.4×106 m= 6.4 \times 10^6\,\text{m}; G=6.67×10−11 N m2 kg−2G = 6.67 \times 10^{-11}\,\text{N m}^2\,\text{kg}^{-2}.
  24. Ex 7.18
    The escape speed of a projectile on the earth's surface is 11.2 km s−111.2\,\text{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.
  25. Ex 7.19
    A satellite orbits the earth at a height of 400 km400\,\text{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= 200\,\text{kg}; mass of the earth =6.0×1024 kg= 6.0 \times 10^{24}\,\text{kg}; radius of the earth =6.4×106 m= 6.4 \times 10^6\,\text{m}; G=6.67×10−11 N m2 kg−2G = 6.67 \times 10^{-11}\,\text{N m}^2\,\text{kg}^{-2}.
  26. Ex 7.20
    Two stars each of one solar mass (=2×1030 kg)(= 2 \times 10^{30}\,\text{kg}) are approaching each other for a head on collision. When they are a distance 109 km10^9\,\text{km}, their speeds are negligible. What is the speed with which they collide ? The radius of each star is 104 km10^4\,\text{km}. Assume the stars to remain undistorted until they collide. (Use the known value of GG).
  27. Ex 7.21
    Two heavy spheres each of mass 100 kg100\,\text{kg} and radius 0.10 m0.10\,\text{m} are placed 1.0 m1.0\,\text{m} apart on a horizontal table. What is the gravitational force 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 ?