GRAVITATION
Very Short Question Answers
Q1. State the unit and dimension of the universal gravitational constant (G).
Ans. S.I Unit- $\,\,Nm^{2}/kg^{2}$ Dimensional Formula :$\,\,[M^{-1}\,L^{3}\,T^{-2}\,]$
Q2. State the vector form of Newton's law of gravitation.
Ans. Gravitational force $F=-G \frac {m_{1}m_{2}}{r^{2}}\hat{r} \:\:(or) \:\:F=-G \frac {m_{1}m_{2}}{r^{3}}\vec{r}$
Q3. If the gravitational force of Earth on the Moon is F. what is the gravitational force of moon on earth? Do these forces form an action-reaction pair?
Ans.- The gravitational force of the moon on the earth = -F
- These two forces are equal and opposite direction. Hence, they form action and reaction pair
Q4. What would be the change in acceleration due to gravity (g) at the surface, if the radius of Earth decreases by 2% keeping the mass of Earth constant?
Ans. $We\, know \, that\, \frac{\bigtriangleup R}{R}=-2%$
$Change\, in\, acceleration \,due\,to\,gravity\,$
$\frac{\bigtriangleup g}{g}=-2\times \frac{\bigtriangleup R}{R}=-2\times-2=4$
Q5. As we go from one planet to another, how will a) the mass and b) the weight of a body change?
Ans.- The mass of the body is always constant.
- Weight of the body also changes.
Q6. Keeping the length of a simple pendulum constant, will the time period be the same on all planets? Support your answer with reason.
Ans. $Time\,Period \,\: T \propto \frac{1} {\sqrt g} $
Hence, the gravitational force varies from planet to planet, so the time period also changes from planet to planet.
Q8. What are the factors that make 'g the least at the equator and maximum at the poles?
Ans. At Equator: i) equatorial radius of the earth is maximum. ii) latitude, φ=0°
At Poles: i) polar radius of the earth is minimum ii) latitude φ=90°
(Or) i) shape of the earth ii) rotation of the earth.
Q9. Hydrogen is in abundance around the sun but not around earth. Explain.
Ans. The escape velocity on the sun is greater than r.m.s velocity of hydrogen.But the escape velocity on earth is less than the r.m.s velocity of hydrogen. So, hydrogen is in abundance around the sun and less around the earth.
Q10. What is the time period of revolution of a geostationary satellite? Does it rotate from West to East or from East to West?.
Ans. Time period of revolution of a geostationary satellite is 24 hours. It rotates from west to east.
Q11. What are polar satellites?
Ans. A satellite that revolves in a polar orbit is called a polar satellite.A polar orbit passes over north and south poles of the earth.Time period (T) = 100 min.
Short Question Answers
Q1. State Kepler's laws of planetary motion
Ans. Law of orbits: All the planets moves in elliptical orbits with the sun situated at one of the foci.
Law of areas: The line that joins any planet to the sun sweeps equal area in equal intervals of time.
Law of periods: The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet. $$T^{2}\propto a^{3}$$
Q2. Derive the relation between acceleration due to gravity(g) at the surface of a planet and Gravitational constant(G)
Ans. Consider a body of mass $ m $ situated on the surface of the planet of mass $M$and radius $R$. According Newton's Universal law of gravitation the force of attraction between mass and planet is
$$F=G\frac{Mm}{R^{2}}$$
Where force $F=mg$
$$mg=G\frac{Mm}{R^{2}}$$ $$\\ \therefore g=\frac{GM}{R^{2}}$$
The above equation shows the relation between acceleration due to gravity$(g) $at the surface of a planet and Gravitational constant$(G)$.
Q3. How does the acceleration due to gravity(g) change for the same and depth(d)
Ans. Let $'g'$ be the acceleration due to gravity on the surface of the earth, $\rho$ is density of the earth
$$g=\frac{GM_{e}}{R^{2}}=\frac{4}{3}\pi\rho GR$$
$g_{h}$ be the acceleration due to gravity at a height \('h'\)
$$g_h=\frac{GM_e}{(R+h)^2}$$
$$\begin{aligned}
&\frac{g_{_h}}{g}=\frac{GM_{e}}{(R+h)^{2}}\frac{R^{2}}{GM_{e}} \\
&\frac{g_{_h}}{g}=\frac{R^{2}}{(R+h)^{2}} \\
&g_{_h}=g\left(1-\frac{2h}{R}\right) \\
\end{aligned}$$
and $g_d$ be the acceleration due to gravity in a depth \('d'\)
$$\begin{gathered}
g_{_d}=\frac{4}{3}\pi\rho G(R-d) \\
\frac{g_{_d}}{g}=\frac{R-d}{R}=1-\frac{d}{R} \\
g_{_d}=g\left(1-{\frac{d}{R}}\right)
\end{gathered}$$
Therefore, if $h=d$ ,than $g_{_h}<g_{_d}$
If $d=R$ than $g_{_d}=0$ it means at center of theearth acceleration due to gravity is zero.
Q4. What is orbital velocity? Obtain an expression for it
Ans. Orbital Velocity: The horizontal velocity of an object to revolve around the planet in a. circular orbit is called Orbital Velocity,
Expression:Consider a satellite of mass $m$ revolving around the Earth at a height $h$. Let M be the. mass and $R$ be the radius of the Earth.Let' $V_0 $ be the orbital velocity on the surface oftheEarth The gravitational force between Earth and satellite provides necessary centripetal force
Earth's Gravitational force on the body
$$F=\frac{GMm}{(R+h)^{2}}$$While satellite moves in orbits the centripetal force $$F=\frac{mV_o^2}{(R+h)}$$ From the above equations
$$\begin{gathered}
\frac{mV_{o}^{2}}{(R+h)}=\frac{GMm}{(R+h)^{2}} \\
V_{o}^{2}=\frac{GM}{(R+h)} \\
\end{gathered}$$ Where $R >> h, \: GM=gR^2$ $$V_{o} =\sqrt{gR} $$
Q5. What is escape velocity? Obtain an expression for it.
Ans. Definition : The minimum velocity required for an object to escape from the gravitational. influence of the planet is called Escape Velocity
Expression for escape velocity :Consider a body of mass m thrown with a velocity $V_{e}$ from a planet of mass $M$ and radius $R$.
Kinetic energy of theobject is equal to potential energy and it is in opposite direction.
$$\begin{gathered}
\frac{1}{2}mV_{e}^{2}=-\left(-\frac{GMm}{R}\right) \\
V_{e}=\sqrt{\frac{2GM}{R}} \\
V_{e}=\sqrt{2gR}
\end{gathered}$$
Earth escape velocity $V_{e}= 11. 2$ $km/ s$
Q6. What is a geostationary satellite? State its uses.
Ans. Defination:A satellite whose time period of rotation is $24$ hours and rotates along the direction of rotation of earth in equatorial plane is called geostationary satellite.
Uses:
- Study the upper layers of atmosphere.
- Forecast the changes in the atmosphere
- Know the shape and size of the earth.
- Identify theminerals and natural resourcespresent inside and on the surface of the earth
- Transmit the T.V.programmes to distant places
Q7. If two places are at the same height from the mean sea level: One is a mountain and other is in air. At which place will 'g' be greater? State the reason for your answer.
Ans. Acceleration due to gravity $(g) $is more on mountain due to the presence of mass of mountain.
Q8. The weight of an object is more at the poles than at the equator. At which of these can we get more sugar for the same weight? State the reason for your answer.
Ans. Acceleration due to gravity $(g)$at pole is greater than at equator. Hence, more sugar is obtained due to at equator for same weight $(W=mg).$
Q9. If a nut becomes loose and gets detached from a satellite revolving around the earth.will it fall down to earth or will it revolve around earth? Give reasons for your answer.
Ans. It will revolves in the same orbit due to inertia of motion.
Q10. An object projected with a velocity greater than or equal to 11.2 km/s will not return to earth. Explain the reason.
Ans. The escape velocity of the earth is$ 11.2 km/s$. Hence, An object projected with a velocity greater than or equal to $11.2 km/s$ will not return to earth.
Essay Question Answers
Problems Question Answers
Q2. The mass of a ball is four times the mass of another ball. When these balls are separated by a distance of 10 cm. the force of gravitation between them is $6.67\times10^{7}$ N. Find the masses of the two balls.
Ans. the