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Q8.11:
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 8.12), the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow.
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 8.12), the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow.
Solution :
Step 1: Theoretical Foundation of Gravitational Intensity
The gravitational intensity $\vec{I}$ at any point $\vec{r}$ due to a continuous mass distribution is given by the vector integral: $$\vec{I} = -G \int \frac{dm}{|\vec{r} - \vec{r}'|^3} (\vec{r} - \vec{r}')$$ where $G$ is the universal gravitational constant, $dm$ is the elemental mass, $\vec{r}$ is the position vector of the point of interest, and $\vec{r}'$ is the position vector of the elemental mass.
Step 2: Symmetry Analysis of the Hemispherical Shell
For a hemispherical shell of uniform mass density $\sigma$ and radius $R$, we consider the superposition of gravitational fields generated by its constituent mass elements.
- At the Centre (Origin): Due to the hemispherical symmetry, for every elemental mass $dm$ at a position $(\theta, \phi)$ on the shell, there exists a vector contribution toward the shell. When integrating over the entire surface area, the horizontal components of the gravitational intensity cancel out due to axial symmetry [Per the Principle of Superposition]. The resulting net vector $\vec{I}_{centre}$ must be directed along the axis of symmetry, pointing directly downward toward the base of the hemisphere.
- At an Arbitrary Point P: At any point on the shell or within its proximity, the intensity vector is determined by the mass distribution relative to that specific point. Per Gauss's Law for Gravity, the intensity reflects the local flux distribution.
Step 3: Visual Representation of Intensity Vectors
Step 4: Logical Conclusion
Based on the vector sum calculation:
- The gravitational intensity at the centre of the hemispherical shell is directed along the axis of symmetry, pointing into the open face of the hemisphere (towards the centre of the hypothetical sphere).
- At any arbitrary point $P$ inside or on the shell, the gravitational intensity vector is the resultant of the mass elements, directed such that it accounts for the lack of symmetry compared to a complete spherical shell.
Final Solution: The gravitational intensity at the centre of the hemispherical shell is directed along the symmetry axis perpendicular to the base, and at point P, it is directed toward the local centre of mass of the shell assembly.
More Questions from Class 11 Physics Gravitation EXERCISES
- Q7.1(a): Answer the following : 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 ?
- Q7.1(b): Answer the following : 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 ?
- Q7.1(c): Answer the following : 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 ?
- Q7.10: 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)
- Q7.11: For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow
- Q7.12: 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\times10^{30}) kg, mass of the earth = (6\times10^{24}) kg. Neglect the effect of other planets etc. (orbital radius = (1.5 \times 10^{11}) m).
- Q7.13: How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is (1.5 \times 10^{8}) km.
- Q7.14: A saturn year is 29.5 times the earth year. How far is the saturn from the sun if the earth is (1.50 \times 10^{8}) km away from the sun ?
- Q7.15: 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 ?
- Q7.16: 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 N on the surface ?
- Q7.17: A rocket is fired vertically with a speed of (5 \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 \times 10^{24}) kg; mean radius of the earth = (6.4 \times 10^{6}) m; G = (6.67 \times 10^{-11} \text{ N m}^{2} \text{ kg}^{-2}).
- Q7.18: The escape speed of a projectile on the earth’s surface is (11.2 \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.
- Q7.19: 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\times10^{24}) kg; radius of the earth = (6.4 \times 10^{6}) m; G = (6.67 \times 10^{-11} \text{ N m}^{2} \text{ kg}^{-2}).
- Q7.2(a): Choose the correct alternative : Acceleration due to gravity increases/decreases with increasing altitude.
- Q7.2(b): Choose the correct alternative : Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).
- Q7.2(c): Choose the correct alternative : Acceleration due to gravity is independent of mass of the earth/mass of the body.
- Q7.2(d): Choose the correct alternative : The formula (–G M m (\frac{1}{r_{2}} – \frac{1}{r_{1}})) is more/less accurate than the formula (m g (r_{2} – r_{1})) for the difference of potential energy between two points (r_{2}) and (r_{1}) distance away from the centre of the earth.
- Q7.20: Two stars each of one solar mass (= (2\times10^{30}) 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 (10^{4}) km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
- Q7.21: 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 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 ?
- Q7.3: 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 ?
- Q7.4: Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is (4.22 \times 10^{8}) m. Show that the mass of Jupiter is about one-thousandth that of the sun.
- Q7.5: Let us assume that our galaxy consists of (2.5 \times 10^{11}) stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milky Way to be (10^{5}) ly.
- Q7.6(a): Choose the correct alternative: If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
- Q7.6(b): Choose the correct alternative: 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.
- Q7.7(a): Does the escape speed of a body from the earth depend on the mass of the body,
- Q7.7(b): Does the escape speed of a body from the earth depend on the location from where it is projected,
- Q7.7(c): Does the escape speed of a body from the earth depend on the direction of projection,
- Q7.7(d): Does the escape speed of a body from the earth depend on the height of the location from where the body is launched?
- Q7.8(a): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant linear speed,
- Q7.8(b): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant angular speed,
- Q7.8(c): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant angular momentum,
- Q7.8(d): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant kinetic energy,
- Q7.8(e): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant potential energy,
- Q7.8(f): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q7.9: Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem.
- Q8.1(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 ?
- Q8.1(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 ?
- Q8.1(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 ?
- Q8.10: 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 8.12)
- Q8.12: 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 = $2 \times 10^{30}$ kg, mass of the earth = $6 \times 10^{24}$ kg. Neglect the effect of other planets etc. (orbital radius = $1.5 \times 10^{11}$ m).
- Q8.13: How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is $1.5 \times 10^8$ km.
- Q8.14: A saturn year is 29.5 times the earth year. How far is the saturn from the sun if the earth is $1.50 \times 10^8$ km away from the sun ?
- Q8.15: 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 ?
- Q8.16: 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 N on the surface ?
- Q8.17: 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 = $6.0 \times 10^{24}$ kg; mean radius of the earth = $6.4 \times 10^6$ m; G = $6.67 \times 10^{–11}$ N m$^2$ kg$^{–2}$.
- Q8.18: 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.
- Q8.19: 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 = $6.0 \times 10^{24}$ kg; radius of the earth = $6.4 \times 10^6$ m; G = $6.67 \times 10^{–11}$ N m$^2$ kg$^{–2}$.
- Q8.2(a): Choose the correct alternative : (a) Acceleration due to gravity increases/decreases with increasing altitude.
- Q8.2(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).
- Q8.2(c): Choose the correct alternative : (c) Acceleration due to gravity is independent of mass of the earth/mass of the body.
- Q8.2(d): Choose the correct alternative : (d) The formula $-G Mm(\frac{1}{r_2} – \frac{1}{r_1})$ is more/less accurate than the formula $mg(r_2 – r_1)$ for the difference of potential energy between two points $r_2$ and $r_1$ distance away from the centre of the earth.
- Q8.20: Two stars each of one solar mass (= $2 \times 10^{30}$ 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 $10^4$ km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
- Q8.21: 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 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 ?
- Q8.3: 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 ?
- Q8.4: Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is $4.22 \times 10^8$ m. Show that the mass of Jupiter is about one-thousandth that of the sun.
- Q8.5: Let us assume that our galaxy consists of $2.5 \times 10^{11}$ stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milky Way to be $10^5$ ly.
- Q8.6(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.
- Q8.6(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.
- Q8.7(a): Does the escape speed of a body from the earth depend on (a) the mass of the body,
- Q8.7(b): Does the escape speed of a body from the earth depend on (b) the location from where it is projected,
- Q8.7(c): Does the escape speed of a body from the earth depend on (c) the direction of projection,
- Q8.7(d): Does the escape speed of a body from the earth depend on (d) the height of the location from where the body is launched?
- Q8.8(a): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(b): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (b) angular speed, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(c): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (c) angular momentum, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(d): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (d) kinetic energy, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(e): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (e) potential energy, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(f): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.9(a): Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet,
- Q8.9(b): Which of the following symptoms is likely to afflict an astronaut in space (b) swollen face,
- Q8.9(c): Which of the following symptoms is likely to afflict an astronaut in space (c) headache,
- Q8.9(d): Which of the following symptoms is likely to afflict an astronaut in space (d) orientational problem.
CBSE Solutions for Class 11 Physics Gravitation
Chapters in CBSE - Class 11 Physics
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