The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity

As observed from the earth, the sun appears to move in an approximate circular orbit. For the motion of another planet like mercury as observed from the earth, this would

Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the CM (centre of mass) causing translation and a net torque at the CM causing rotation around an axis through the CM. For the earth-sun system (approximating the earth as a uniform density sphere).

Satellites orbitting the earth have finite life and sometimes debris of satellites fall to the earth. This is because

Both the earth and the moon are subject to the gravitational force of the sun. As observed from the sun, the orbit of the moon

In our solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They

Choose the wrong option.

Particles of masses $$2 M, m$$ and $$M$$ are respectively at points $$A, B$$ and $$C$$ with $$A B=\frac{1}{2}(B C) \cdot m$$ is much-much smaller than $$M$$ and at time $$t=0$$, they are all at rest as given in figure. At subsequent times before any collision takes place.

Which of the following options are correct?

If the law of gravitation, instead of being inverse square law, becomes an inverse cube law

If the mass of the sun were ten times smaller and gravitational constant $$G$$ were ten times larger in magnitude. Then,

If the sun and the planets carried huge amounts of opposite charges,

There have been suggestions that the value of the gravitational constant $$G$$ becomes smaller when considered over very large time period (in billions of years) in the future. If that happens, for our earth,

Supposing Newton's law of gravitation for gravitation forces $$\mathbf{F}_1$$ and $$\mathbf{F}_2$$ between two masses $$m_1$$ and $$m_2$$ at positions $$\mathbf{r}_1$$ and $$\mathbf{r}_2$$ read

$$\mathbf{F}_1=-\mathbf{F}_2=-\frac{\mathbf{r}_{12}}{r_{12}^3} \mathrm{GM}^2 0\left(\frac{m_1 m_2}{M_0^2}\right)^n$$

where $$M_0$$ is a constant of dimension of mass, $$\mathbf{r}_{12}=\mathbf{r}_1-\mathbf{r}_2$$ and $$n$$ is a number. In such a case,

Which of the following are true?

The centre of mass of an extended body on the surface of the earth and its centre of gravity

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.

Give one example each of central force and non-central force.

Draw areal velocity versus time graph for mars.

What is the direction of areal velocity of the earth around the sun?

How is the gravitational force between two point masses affected when they are dipped in water keeping the separation between them the same?

Is it possible for a body to have inertia but no weight?

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?

An astronaut inside a small spaceship orbitting around the earth cannot detect gravity. If the space station orbitting around the earth has a large size, can he hope to detect gravity?

The gravitational force between a hollow spherical shell (of radius $$R$$ and uniform density) and a point mass is $$F$$. Show the nature of $$F$$ versus $$r$$ graph where $$r$$ is the distance of the point from the centre of the hollow spherical shell of uniform density.

Out of aphelion and perihelion, where is the speed of the earth more and why?

What is the angle between the equatorial plane and the orbital plane of

(a) polar satellite?

(b) geostationary satellite?

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 the earth's spin and orbital motion, show that mean solar day is 4 min longer than the sidereal day. In other words, distant stars would rise 4 min early every successive day.

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.

Show the nature of the following graph for a satellite orbitting the earth.

(a) KE versus orbital radius $$R$$

(b) PE versus orbital radius $$R$$

(c) TE versus orbital radius $$R$$

Shown are several curves [fig. (a), (b), (c), (d), (e), (f)]. Explain with reason, which ones amongst them can be possible trajectories traced by a projectile (neglect air friction).

An object of mass $m$ is raised from the surface of the earth to a height equal to the radius of the earth, that is, taken from a distance $R$ to $2 R$ from the centre of the earth. What is the gain in its potential energy?

A mass $m$ is placed at $P$ a distance $h$ along the normal through the centre 0 of a thin circular ring of mass $M$ and radius $r$ (figure).

If the mass is moved further away such that $O P$ becomes $2 h$, by what factor the force of gravitation will decrease, if $h=r$ ?

A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let $r$ be the distance of the body from the centre of the star and let its linear velocity be $v$, angular velocity $\omega$, kinetic energy $K$, gravitational potential energy $U$, total energy $E$ and angular momentum $l$. As the radius $r$ of the orbit increases, determine which of the above quantities increase and which ones decrease.

Six point masses of mass $m$ each are at the vertices of a regular hexagon of side $l$. Calculate the force on any of the masses.

A satellite is to be placed in equatorial geostationary orbit around the earth for communication.

(a) Calculate height of such a satellite.

(b) Find out the minimum number of satellites that are needed to cover entire earth, so that atleast one satellite is visible from any point on the equator.

$$\left[M=6 \times 10^{24} \mathrm{~kg}, R=6400 \mathrm{~km}, T=24 \mathrm{~h}, G=6.67 \times 10^{-11} \mathrm{SI} \text { unit }\right]$$

Earth's orbit is an ellipse with eccentricity 0.0167 . Thus, the earth's distance from the sun and speed as it moves around the sun varies from day-to-day. This means that the length of the solar day is not constant through the year. Assume that the earth's spin axis is normal to its orbital plane and find out the length of the shortest and the longest day. A day should be taken from noon to noon. Does this explain variation of length of the day during the year?

A satellite is in an elliptic orbit around the earth with aphelion of $6 R$ and perihelion of $2 R$ where $R=6400 \mathrm{~km}$ is the radius of the earth. Find eccentricity of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred to a circular orbit of radius $6 R$ ? $$ \left[G=6.67 \times 10^{-11} \text { SI unit and } M=6 \times 10^{24} \mathrm{~kg}\right] $$