91.
A spherically symmetric gravitational system of particles has a mass density \[\rho = \left\{ \begin{array}{l}
{\rho _0}\,{\rm{for}}\,r \le R\\
0\,{\rm{for}}\,r > R
\end{array} \right.\]
where $${\rho _0}$$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed $$v$$ as a function of distance $$r \left( {0 < r < \infty } \right)$$ from the centre of the system is represented by-
A
B
C
D
Answer :
92. If the distance of earth is halved from the sun, then the no. of days in a year will be
A
365
B
182.5
C
730
D
129
Answer :
129
93. The mean radius of the earth is $$R,$$ its angular speed on its own axis is $$\omega $$ and the acceleration due to gravity at the earth’s surface is $$g.$$ What will be the radius of the orbit of a geostationary satellite?
A
$${\left( {\frac{{{R^2}g}}{{{\omega ^2}}}} \right)^{\frac{1}{3}}}$$
B
$${\left( {\frac{{Rg}}{{{\omega ^2}}}} \right)^{\frac{1}{3}}}$$
C
$${\left( {\frac{{{R^2}{\omega ^2}}}{g}} \right)^{\frac{1}{3}}}$$
D
$${\left( {\frac{{{R^2}g}}{\omega }} \right)^{\frac{1}{3}}}$$
Answer :
$${\left( {\frac{{{R^2}g}}{{{\omega ^2}}}} \right)^{\frac{1}{3}}}$$
94. The mass of a spaceship is $$1000 \,kg.$$ It is to be launched from the earth's surface out into free space. The value of $$g$$ and $$R$$ (radius of earth) are $$10\,m/{s^2}$$ and $$6400 \,km$$ respectively. The required energy for this work will be-
A
$$6.4 \times {10^{11}}\,Joules$$
B
$$6.4 \times {10^{8}}\,Joules$$
C
$$6.4 \times {10^{9}}\,Joules$$
D
$$6.4 \times {10^{10}}\,Joules$$
Answer :
$$6.4 \times {10^{10}}\,Joules$$
95. What will be the formula of the mass in terms of $$g, R$$ and $$G$$ ? ($$R$$ = radius of the earth)
A
$${g^2}\frac{R}{G}$$
B
$$G\frac{{{R^2}}}{g}$$
C
$$G\frac{R}{g}$$
D
$$g\frac{{{R^2}}}{G}$$
Answer :
$$g\frac{{{R^2}}}{G}$$
96. Assuming the radius of the earth as $$R,$$ the change in gravitational potential energy of a body of mass $$m,$$ when it is taken from the earth's surface to a height $$3R$$ above its surface, is
A
$$3\,mg\,R$$
B
$$\frac{3}{4}\,mg\,R$$
C
$$1\,mg\,R$$
D
$$\frac{3}{2}\,mg\,R$$
Answer :
$$\frac{3}{4}\,mg\,R$$
97. The value of $$'g'$$ at a particular point is $$9.8\,m/{s^2}.$$ Suppose the earth suddenly shrinks uniformly to half its present size without losing any mass. The value of $$'g'$$ at the same point (assuming that the distance of the point from the centre of the earth does not shrink) will now be
A
$$4.9\,m/{\sec ^2}$$
B
$$3.1\,m/{\sec ^2}$$
C
$$9.8\,m/{\sec ^2}$$
D
$$19.6\,m/{\sec ^2}$$
Answer :
$$9.8\,m/{\sec ^2}$$
98. A mass $$M$$ is split into two parts m and $$\left( {M - m} \right),$$ which are then separated by a certain distance. What ratio of $$\frac{m}{M}$$ maximizes the gravitational force between the two parts?
A
$$\frac{1}{3}$$
B
$$\frac{1}{2}$$
C
$$\frac{1}{4}$$
D
$$\frac{1}{5}$$
Answer :
$$\frac{1}{2}$$
99. A compass needle which is allowed to move in a horizontal plane is taken to a geomagnetic pole. It
A
will become rigid showing no movement
B
will stay in any position
C
will stay in North-South direction only
D
will stay in East-West direction only
Answer :
will stay in North-South direction only
100. Two stars of mass $${m_1}$$ and $${m_2}$$ are parts of a binary system. The radii of their orbits are $${r_1}$$ and $${r_2}$$ respectively, measured from the $$C.M.$$ of the system. The magnitude of gravitational force $${m_1}$$ exerts on $${m_2}$$ is
A
$$\frac{{{m_1}{m_2}G}}{{{{\left( {{r_1} + {r_2}} \right)}^2}}}$$
B
$$\frac{{{m_1}G}}{{{{\left( {{r_1} + {r_2}} \right)}^2}}}$$
C
$$\frac{{{m_2}G}}{{{{\left( {{r_1} + {r_2}} \right)}^2}}}$$
D
$$\frac{{\left( {{m_1} + {m_2}} \right)}}{{{{\left( {{r_1} + {r_2}} \right)}^2}}}$$
Answer :
$$\frac{{{m_1}{m_2}G}}{{{{\left( {{r_1} + {r_2}} \right)}^2}}}$$