Kepler's Law

Q.

If the earth is at one-fourth of its present distance from the sun, the duration of the year will be

1. Half the present year

2. One-eighth the present year

3. One-fourth the present year

4. One-sixth the present year

Q.

The angular momentum of a planet of mass $M$ moving around the sun in an elliptical orbit is $\overrightarrow{L}$. The magnitude of the areal velocity of the planet is

1. $\frac{L}{M}$

2. $\frac{2L}{M}$

3. $\frac{L}{2M}$

4. $\frac{4L}{M}$

Q.

For the planet-sun system identify the correct statement

1. The angular momentum of the planet is conserved

2. The total kinetic energy of the system is conserved

3. The momentum of the planet is conserved

4. All of the above

Q.

Two satellites $S_1$ and $S_2$ revolve round a planet in coplanar circular orbits in the same sense. Their periods of revolution are 1 hour and 8 hours respectively. The radius of the orbit of $S_1={10}^4$km When $S_2$ is closest to $S_1$ find speed of $S_2$ relative to $S_2$ and the angular speed of $S_1$ actually observed by an astronaut at $S_1$.

1. -4.91 $\times {10}^{-4}$ rad $s^{-1}$

2. -2.91 $\times {10}^{-4}$ rad $s^{-1}$

3. 2.3 $\times {10}^{-4}$ rad $s^{-1}$

4. -6.00 $\times {10}^{-8}$ rad $s^{-1}$

Q. The orbit of a planet around a star is an ellipse.

1. True
2. False

Q.

Two planets of equal mass orbit a much more massive star (figure). Planet $m_1$ moves in a circular orbit of radius $1\times {10}^8$ km with period 2 year. Planet $m_2$ moves in an elliptical orbit with closest distance $r_1=1\times {10}^8$ km and farthest distance $r_{2}1.8\times {10}^8$km, as shown. Using the fact that mean radius of an elliptical orbit is the length of the semi-major axis, find the period of $m_2^{{}^{\prime }}$ s orbit.

1. 2.1 year

2. 3.31 year

3. 4.12 year

4. 5.24 year

Q.

Venus looks brighter than other planets because

1. It is heavier than other planets

2. It has higher density than other planets

3. It is closer to the earth than other planets

4. It has no atmosphere

Q. A mass of $50\text{kg}$ is placed at the center of a uniform spherical shell of mass $100\text{kg}$ and radius $50\text{m}$. If the gravitational potential at a point, $25\text{m}$ from the center is $V\text{kg/m}$. The value of $V$ is:

1. $-4G$
2. $-60G$
3. $-20G$
4. $+2G$

Q. The earth moves around the sun in an elliptical orbit as shown in figure. The ratio $\frac{OA}{OB}=x$. The ratio of the speed of the earth at B to that at A is nearly

1. $\sqrt{x}$
2. $x$
3. $x\sqrt{x}$
4. $x^2$

Q. Two satellites $S_1$ and $S_2$ revolve around a planet in coplanar circular orbits in the same sense. Their periods of revolution are $1\text{hour}$ and $8\text{hours}$ respectively. The radius of the orbit of $S_1$ is ${10}^4\text{km}$. When $S_1$ is closest to $S_2$, the angular speed of $S_2$ as observed by an astronaut in $S_1$ is :

1. $\pi \text{rad/hr}$
2. $\pi /3\text{rad/hr}$
3. $2\pi \text{rad/hr}$
4. $\pi /2\text{rad/hr}$

Q. The escape velocity for a body projected vertically upwards from the surface of earth is 11 km/s. If the body is projected at an angle of ${45}^{\circ }$ with the vertical, the escape velocity will be

1. $11\sqrt{2}km{s}^{-1}$
2. $22km{s}^{-1}$
3. $11km{s}^{-1}$
4. $\frac{11}{\sqrt{2}}km{s}^{-1}$

Q. The orbit of a planet around a star is an ellipse.

1. True
2. False

Q. The planet mercury is revolving in an elliptical orbit around the sun as shown in the figure. The kinetic energy of mercury will be greater at

1. $A$
2. $B$
3. $C$
4. $D$

Q.

The period of revolution of an earth satellite close to surface of earth is 90 min. The time period of another satellite in an orbit at a distance of three times the radius of earth from its surface will be

1. $90\sqrt{80}$min

2. 360 min

3. 720 min

4. 270 min

Q. An earth satellite has angular momentum $\alpha$ and its areal velocity about earth is $\beta$, then mass of satellite is

1. $\frac{\alpha }{\beta }$
2. $\frac{\alpha }{2\beta }$
3. $\frac{\alpha }{3\beta }$
4. $\frac{2\alpha }{\beta }$

Q.

The time period of a satellite in a circular orbit around a planet is independent of______.

1. the mass of the planet

2. the radius of the planet

3. the mass of the satellite

4. density of the planet

Q.

Mass M is divided into two parts xM and (1 − x)M. For a given separation, the value of x for which the gravitational attraction between the two pieces becomes maximum is

1. $\frac{1}{2}$

2. $\frac{3}{5}$

3. $1$

4. $2$

Q. The earth E moves in an elliptical orbit with the Sun S at one of the foci as shown in figure. Its speed of motion will be maximum at the point

1. C
2. A
3. B
4. D

Q.

The distance of neptune and saturn from sun are nearly ${10}^{13}$ and ${10}^{12}$ meters respectively. Assuming that they move in circular orbits, their periodic times will be in the ratio

1. $\sqrt{10}$

2. 100

3. $10\sqrt{10}$

4. $\frac{1}{\sqrt{10}}$

Q.

A planet moves around the sun. At a given point P, it is closest from the sun at a distance $d_1$ and has a speed $v_1$. At another point Q, when it is farthest from the sun at a distance $d_2$, its speed will be

1. $\frac{d_1^2{v}_1}{d_2^2}$

2. $\frac{d_2{v}_1}{d_1}$

3. $\frac{d_1{v}_1}{d_2}$

4. $\frac{d_2^2{v}_1}{d_1^2}$

Q. A satellite can be in a geostationary orbit around earth at a distance $r$ from the center. If the angular velocity of earth about its axis doubles, a satellite can now be in a geostationary orbit around earth, if its distance from the center is

1. $\frac{r}{2}$
2. $\frac{r}{2\sqrt{2}}$
3. $\frac{r}{(4{)}^{1/3}}$
4. $\frac{r}{(2{)}^{1/3}}$

Q.

The period of a satellite in a circular orbit of radius R is T, the period of another satellite in a circular orbit of radius 4R is

1. 4T

2. $\frac{T}{4}$

3. 8T

4. $\frac{T}{8}$

Q.

The distance of neptune and saturn from sun are nearly ${10}^{13}$ and ${10}^{12}$ meters respectively. Assuming that they move in circular orbits, their periodic times will be in the ratio

1. $\sqrt{10}$

2. 100

3. $10\sqrt{10}$

4. $\frac{1}{\sqrt{10}}$

Q. The earth moves around the sun in an elliptical orbit as shown in figure. The ratio $\frac{OA}{OB}=x$. The ratio of the speed of the earth at B to that at A is nearly

1. $\sqrt{x}$
2. $x$
3. $x\sqrt{x}$
4. $x^2$

Q. The planet mercury is revolving in an elliptical orbit around the sun as shown in the figure. The kinetic energy of mercury will be greater at

1. $A$
2. $B$
3. $C$
4. $D$

Q. A comet revolves around the Sun, then which of the following is incorrect?

1. Speed is greatest when comet is closest to the Sun
2. In half motion, comet is retarded and in the other half motion, comet is accelerated
3. Angular momentum of comet about any point is constant
4. Motion is in plane

Q. Which of the following are true for Planetary system in the solar system

1. Conservation of kinetic energy
2. Conservation of linear momentum
3. Conservation of angular momentum
4. None of these

Q.

If the distance between the earth and the sun were reduced to half its present value, Then the number of days in one year would have been

1. 65

2. 129

3. 183

4. 730

Q. Two satellites $S_1$ and $S_2$ revolve around a planet in coplanar circular orbits in the same sense. Their periods of revolution are $1\text{hour}$ and $8\text{hours}$ respectively. The radius of the orbit of $S_1$ is ${10}^4\text{km}$. When $S_1$ is closest to $S_2$, the angular speed of $S_2$ as observed by an astronaut in $S_1$ is :

1. $\pi \text{rad/hr}$
2. $\pi /3\text{rad/hr}$
3. $2\pi \text{rad/hr}$
4. $\pi /2\text{rad/hr}$

Q.

In planetary motion the areal velocity of position vector of a planet depends on angular velocity $\left(\omega \right)$ and the distance of the planet from sun (r). If so the correct relation for areal velocity is

1. $\frac{dA}{dt}\propto \omega r$

2. $\frac{dA}{dt}\propto {\omega }^{2}r$

3. $\frac{dA}{dt}\propto \omega r^2$

4. $\frac{dA}{dt}\propto \sqrt{\omega r}$