359610
Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is
1 \(2: 1\)
2 \(2: 3\)
3 \(4: 3\)
4 \(3: 4\)
Explanation:
Given: \(R_{A}=R, \rho_{A}=\rho\) \(R_{B}=1.5 R, \rho_{B}=\rho / 2\) Acceleration due to gravity is given by \(g=\dfrac{G M}{R^{2}}=\dfrac{G}{R^{2}}\left(\rho \times \dfrac{4}{3} \pi R^{3}\right)\) \(\therefore g=\dfrac{4}{3} \pi G \rho R\) \(\Rightarrow g \propto \rho R\) \(g_{A}=\dfrac{4}{3} \pi G \rho_{A} R_{A}\) \(g_{B}=\dfrac{4}{3} \pi G \rho_{B} R_{B}\) \(\therefore \dfrac{g_{B}}{g_{A}}=\dfrac{\rho_{B} R_{B}}{\rho_{A} R_{A}}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{1.5}{2}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{3}{4}\)
JEE - 2023
PHXI08:GRAVITATION
359611
Four particles, each of mass \(m\), are placed at the corners of square and moving along a circle of radius \(r\) under the influence of mutual gravitational attraction. The speed of each particle will be -
359612
A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing \(W\) on earth will weigh on that planet
1 \({2\,W}\)
2 \({2^{2/3}}\,W\)
3 \({2^{1/3}}\,W\)\({2^{1/3}}\,W\)
4 \(W\)
Explanation:
\(M_{P}=2 M_{E}, W_{E}=W, W_{P}=\)? As, \(g_{E}=\dfrac{G M_{E}}{R_{E}^{2}}\) Mass of an object \(=\) Volume of the object \(\times\) density \(=\dfrac{4}{3} \pi R^{3} \times \rho\) \(\rho\) is same for both the planets \(\Rightarrow \dfrac{4}{3} \pi R_{P}^{3} \times \rho=2 \dfrac{4}{3} \pi R_{E}^{3} \times \rho\) Here, \(R_{P}\) and \(R_{E}\) are the radius of planet and Earth respectively Then, \(R_{P}=(2)^{1 / 3} R_{E}\) \(\Rightarrow g_{P}=\dfrac{G M_{P}}{R_{P}^{2}} \Rightarrow g_{P}=\dfrac{G \times 2 M_{E}}{\left(2^{1 / 3} R_{E}\right)^{2}} g_{P}\) \(g_{p}=\dfrac{2}{2^{2 / 3}} \times g_{E} \Rightarrow g_{p}=2^{1 / 3} \times g_{E}\) Weight, \(W=m g\) \(W_{P}=m g_{P} \Rightarrow W_{P}=m(2)^{1 / 3} \times g_{E}\) \( \Rightarrow {W_P} = {2^{1/3}} \times {W_E} = {2^{1/3}}\;W\)
JEE - 2023
PHXI08:GRAVITATION
359613
If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would
1 Remain unchanged
2 Decrease by \(0.5 \%\)
3 Be zero
4 Increase by \(2 \%\)
Explanation:
\(g=\dfrac{G M}{R^{2}}\) As \(M\) is constant \(\begin{aligned}& \dfrac{d g}{g}=-2 \dfrac{d R}{R} \\& \dfrac{d G}{G} \times 100=-\dfrac{1}{2} \dfrac{d R}{R} \times 100 \\& \dfrac{d g}{g} \times 100=2 \%\end{aligned}\) \(g\) will increase if \(R\) decreases.
PHXI08:GRAVITATION
359614
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
1 Will be directed towards the centre but not the same everywhere
2 Will have the same value everywhere but not directed towards the centre
3 Will be same everywhere in magnitude directed towards the centre
4 Cannot be zero at any point
Explanation:
If the earth is an approximate sphere of non-uniform density, then the centre of gravity of earth will not be situated at the centre of earth. The distance of different points on earth will be at different distances from the centre of gravity of earth. As, \(g \propto \dfrac{1}{r^{2}}\), so \(g\) is different for different points on the surface of earth but not zero.
359610
Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is
1 \(2: 1\)
2 \(2: 3\)
3 \(4: 3\)
4 \(3: 4\)
Explanation:
Given: \(R_{A}=R, \rho_{A}=\rho\) \(R_{B}=1.5 R, \rho_{B}=\rho / 2\) Acceleration due to gravity is given by \(g=\dfrac{G M}{R^{2}}=\dfrac{G}{R^{2}}\left(\rho \times \dfrac{4}{3} \pi R^{3}\right)\) \(\therefore g=\dfrac{4}{3} \pi G \rho R\) \(\Rightarrow g \propto \rho R\) \(g_{A}=\dfrac{4}{3} \pi G \rho_{A} R_{A}\) \(g_{B}=\dfrac{4}{3} \pi G \rho_{B} R_{B}\) \(\therefore \dfrac{g_{B}}{g_{A}}=\dfrac{\rho_{B} R_{B}}{\rho_{A} R_{A}}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{1.5}{2}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{3}{4}\)
JEE - 2023
PHXI08:GRAVITATION
359611
Four particles, each of mass \(m\), are placed at the corners of square and moving along a circle of radius \(r\) under the influence of mutual gravitational attraction. The speed of each particle will be -
359612
A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing \(W\) on earth will weigh on that planet
1 \({2\,W}\)
2 \({2^{2/3}}\,W\)
3 \({2^{1/3}}\,W\)\({2^{1/3}}\,W\)
4 \(W\)
Explanation:
\(M_{P}=2 M_{E}, W_{E}=W, W_{P}=\)? As, \(g_{E}=\dfrac{G M_{E}}{R_{E}^{2}}\) Mass of an object \(=\) Volume of the object \(\times\) density \(=\dfrac{4}{3} \pi R^{3} \times \rho\) \(\rho\) is same for both the planets \(\Rightarrow \dfrac{4}{3} \pi R_{P}^{3} \times \rho=2 \dfrac{4}{3} \pi R_{E}^{3} \times \rho\) Here, \(R_{P}\) and \(R_{E}\) are the radius of planet and Earth respectively Then, \(R_{P}=(2)^{1 / 3} R_{E}\) \(\Rightarrow g_{P}=\dfrac{G M_{P}}{R_{P}^{2}} \Rightarrow g_{P}=\dfrac{G \times 2 M_{E}}{\left(2^{1 / 3} R_{E}\right)^{2}} g_{P}\) \(g_{p}=\dfrac{2}{2^{2 / 3}} \times g_{E} \Rightarrow g_{p}=2^{1 / 3} \times g_{E}\) Weight, \(W=m g\) \(W_{P}=m g_{P} \Rightarrow W_{P}=m(2)^{1 / 3} \times g_{E}\) \( \Rightarrow {W_P} = {2^{1/3}} \times {W_E} = {2^{1/3}}\;W\)
JEE - 2023
PHXI08:GRAVITATION
359613
If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would
1 Remain unchanged
2 Decrease by \(0.5 \%\)
3 Be zero
4 Increase by \(2 \%\)
Explanation:
\(g=\dfrac{G M}{R^{2}}\) As \(M\) is constant \(\begin{aligned}& \dfrac{d g}{g}=-2 \dfrac{d R}{R} \\& \dfrac{d G}{G} \times 100=-\dfrac{1}{2} \dfrac{d R}{R} \times 100 \\& \dfrac{d g}{g} \times 100=2 \%\end{aligned}\) \(g\) will increase if \(R\) decreases.
PHXI08:GRAVITATION
359614
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
1 Will be directed towards the centre but not the same everywhere
2 Will have the same value everywhere but not directed towards the centre
3 Will be same everywhere in magnitude directed towards the centre
4 Cannot be zero at any point
Explanation:
If the earth is an approximate sphere of non-uniform density, then the centre of gravity of earth will not be situated at the centre of earth. The distance of different points on earth will be at different distances from the centre of gravity of earth. As, \(g \propto \dfrac{1}{r^{2}}\), so \(g\) is different for different points on the surface of earth but not zero.
359610
Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is
1 \(2: 1\)
2 \(2: 3\)
3 \(4: 3\)
4 \(3: 4\)
Explanation:
Given: \(R_{A}=R, \rho_{A}=\rho\) \(R_{B}=1.5 R, \rho_{B}=\rho / 2\) Acceleration due to gravity is given by \(g=\dfrac{G M}{R^{2}}=\dfrac{G}{R^{2}}\left(\rho \times \dfrac{4}{3} \pi R^{3}\right)\) \(\therefore g=\dfrac{4}{3} \pi G \rho R\) \(\Rightarrow g \propto \rho R\) \(g_{A}=\dfrac{4}{3} \pi G \rho_{A} R_{A}\) \(g_{B}=\dfrac{4}{3} \pi G \rho_{B} R_{B}\) \(\therefore \dfrac{g_{B}}{g_{A}}=\dfrac{\rho_{B} R_{B}}{\rho_{A} R_{A}}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{1.5}{2}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{3}{4}\)
JEE - 2023
PHXI08:GRAVITATION
359611
Four particles, each of mass \(m\), are placed at the corners of square and moving along a circle of radius \(r\) under the influence of mutual gravitational attraction. The speed of each particle will be -
359612
A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing \(W\) on earth will weigh on that planet
1 \({2\,W}\)
2 \({2^{2/3}}\,W\)
3 \({2^{1/3}}\,W\)\({2^{1/3}}\,W\)
4 \(W\)
Explanation:
\(M_{P}=2 M_{E}, W_{E}=W, W_{P}=\)? As, \(g_{E}=\dfrac{G M_{E}}{R_{E}^{2}}\) Mass of an object \(=\) Volume of the object \(\times\) density \(=\dfrac{4}{3} \pi R^{3} \times \rho\) \(\rho\) is same for both the planets \(\Rightarrow \dfrac{4}{3} \pi R_{P}^{3} \times \rho=2 \dfrac{4}{3} \pi R_{E}^{3} \times \rho\) Here, \(R_{P}\) and \(R_{E}\) are the radius of planet and Earth respectively Then, \(R_{P}=(2)^{1 / 3} R_{E}\) \(\Rightarrow g_{P}=\dfrac{G M_{P}}{R_{P}^{2}} \Rightarrow g_{P}=\dfrac{G \times 2 M_{E}}{\left(2^{1 / 3} R_{E}\right)^{2}} g_{P}\) \(g_{p}=\dfrac{2}{2^{2 / 3}} \times g_{E} \Rightarrow g_{p}=2^{1 / 3} \times g_{E}\) Weight, \(W=m g\) \(W_{P}=m g_{P} \Rightarrow W_{P}=m(2)^{1 / 3} \times g_{E}\) \( \Rightarrow {W_P} = {2^{1/3}} \times {W_E} = {2^{1/3}}\;W\)
JEE - 2023
PHXI08:GRAVITATION
359613
If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would
1 Remain unchanged
2 Decrease by \(0.5 \%\)
3 Be zero
4 Increase by \(2 \%\)
Explanation:
\(g=\dfrac{G M}{R^{2}}\) As \(M\) is constant \(\begin{aligned}& \dfrac{d g}{g}=-2 \dfrac{d R}{R} \\& \dfrac{d G}{G} \times 100=-\dfrac{1}{2} \dfrac{d R}{R} \times 100 \\& \dfrac{d g}{g} \times 100=2 \%\end{aligned}\) \(g\) will increase if \(R\) decreases.
PHXI08:GRAVITATION
359614
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
1 Will be directed towards the centre but not the same everywhere
2 Will have the same value everywhere but not directed towards the centre
3 Will be same everywhere in magnitude directed towards the centre
4 Cannot be zero at any point
Explanation:
If the earth is an approximate sphere of non-uniform density, then the centre of gravity of earth will not be situated at the centre of earth. The distance of different points on earth will be at different distances from the centre of gravity of earth. As, \(g \propto \dfrac{1}{r^{2}}\), so \(g\) is different for different points on the surface of earth but not zero.
359610
Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is
1 \(2: 1\)
2 \(2: 3\)
3 \(4: 3\)
4 \(3: 4\)
Explanation:
Given: \(R_{A}=R, \rho_{A}=\rho\) \(R_{B}=1.5 R, \rho_{B}=\rho / 2\) Acceleration due to gravity is given by \(g=\dfrac{G M}{R^{2}}=\dfrac{G}{R^{2}}\left(\rho \times \dfrac{4}{3} \pi R^{3}\right)\) \(\therefore g=\dfrac{4}{3} \pi G \rho R\) \(\Rightarrow g \propto \rho R\) \(g_{A}=\dfrac{4}{3} \pi G \rho_{A} R_{A}\) \(g_{B}=\dfrac{4}{3} \pi G \rho_{B} R_{B}\) \(\therefore \dfrac{g_{B}}{g_{A}}=\dfrac{\rho_{B} R_{B}}{\rho_{A} R_{A}}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{1.5}{2}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{3}{4}\)
JEE - 2023
PHXI08:GRAVITATION
359611
Four particles, each of mass \(m\), are placed at the corners of square and moving along a circle of radius \(r\) under the influence of mutual gravitational attraction. The speed of each particle will be -
359612
A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing \(W\) on earth will weigh on that planet
1 \({2\,W}\)
2 \({2^{2/3}}\,W\)
3 \({2^{1/3}}\,W\)\({2^{1/3}}\,W\)
4 \(W\)
Explanation:
\(M_{P}=2 M_{E}, W_{E}=W, W_{P}=\)? As, \(g_{E}=\dfrac{G M_{E}}{R_{E}^{2}}\) Mass of an object \(=\) Volume of the object \(\times\) density \(=\dfrac{4}{3} \pi R^{3} \times \rho\) \(\rho\) is same for both the planets \(\Rightarrow \dfrac{4}{3} \pi R_{P}^{3} \times \rho=2 \dfrac{4}{3} \pi R_{E}^{3} \times \rho\) Here, \(R_{P}\) and \(R_{E}\) are the radius of planet and Earth respectively Then, \(R_{P}=(2)^{1 / 3} R_{E}\) \(\Rightarrow g_{P}=\dfrac{G M_{P}}{R_{P}^{2}} \Rightarrow g_{P}=\dfrac{G \times 2 M_{E}}{\left(2^{1 / 3} R_{E}\right)^{2}} g_{P}\) \(g_{p}=\dfrac{2}{2^{2 / 3}} \times g_{E} \Rightarrow g_{p}=2^{1 / 3} \times g_{E}\) Weight, \(W=m g\) \(W_{P}=m g_{P} \Rightarrow W_{P}=m(2)^{1 / 3} \times g_{E}\) \( \Rightarrow {W_P} = {2^{1/3}} \times {W_E} = {2^{1/3}}\;W\)
JEE - 2023
PHXI08:GRAVITATION
359613
If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would
1 Remain unchanged
2 Decrease by \(0.5 \%\)
3 Be zero
4 Increase by \(2 \%\)
Explanation:
\(g=\dfrac{G M}{R^{2}}\) As \(M\) is constant \(\begin{aligned}& \dfrac{d g}{g}=-2 \dfrac{d R}{R} \\& \dfrac{d G}{G} \times 100=-\dfrac{1}{2} \dfrac{d R}{R} \times 100 \\& \dfrac{d g}{g} \times 100=2 \%\end{aligned}\) \(g\) will increase if \(R\) decreases.
PHXI08:GRAVITATION
359614
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
1 Will be directed towards the centre but not the same everywhere
2 Will have the same value everywhere but not directed towards the centre
3 Will be same everywhere in magnitude directed towards the centre
4 Cannot be zero at any point
Explanation:
If the earth is an approximate sphere of non-uniform density, then the centre of gravity of earth will not be situated at the centre of earth. The distance of different points on earth will be at different distances from the centre of gravity of earth. As, \(g \propto \dfrac{1}{r^{2}}\), so \(g\) is different for different points on the surface of earth but not zero.
359610
Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is
1 \(2: 1\)
2 \(2: 3\)
3 \(4: 3\)
4 \(3: 4\)
Explanation:
Given: \(R_{A}=R, \rho_{A}=\rho\) \(R_{B}=1.5 R, \rho_{B}=\rho / 2\) Acceleration due to gravity is given by \(g=\dfrac{G M}{R^{2}}=\dfrac{G}{R^{2}}\left(\rho \times \dfrac{4}{3} \pi R^{3}\right)\) \(\therefore g=\dfrac{4}{3} \pi G \rho R\) \(\Rightarrow g \propto \rho R\) \(g_{A}=\dfrac{4}{3} \pi G \rho_{A} R_{A}\) \(g_{B}=\dfrac{4}{3} \pi G \rho_{B} R_{B}\) \(\therefore \dfrac{g_{B}}{g_{A}}=\dfrac{\rho_{B} R_{B}}{\rho_{A} R_{A}}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{1.5}{2}\) \(\dfrac{g_{B}}{g_{A}}=\dfrac{3}{4}\)
JEE - 2023
PHXI08:GRAVITATION
359611
Four particles, each of mass \(m\), are placed at the corners of square and moving along a circle of radius \(r\) under the influence of mutual gravitational attraction. The speed of each particle will be -
359612
A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing \(W\) on earth will weigh on that planet
1 \({2\,W}\)
2 \({2^{2/3}}\,W\)
3 \({2^{1/3}}\,W\)\({2^{1/3}}\,W\)
4 \(W\)
Explanation:
\(M_{P}=2 M_{E}, W_{E}=W, W_{P}=\)? As, \(g_{E}=\dfrac{G M_{E}}{R_{E}^{2}}\) Mass of an object \(=\) Volume of the object \(\times\) density \(=\dfrac{4}{3} \pi R^{3} \times \rho\) \(\rho\) is same for both the planets \(\Rightarrow \dfrac{4}{3} \pi R_{P}^{3} \times \rho=2 \dfrac{4}{3} \pi R_{E}^{3} \times \rho\) Here, \(R_{P}\) and \(R_{E}\) are the radius of planet and Earth respectively Then, \(R_{P}=(2)^{1 / 3} R_{E}\) \(\Rightarrow g_{P}=\dfrac{G M_{P}}{R_{P}^{2}} \Rightarrow g_{P}=\dfrac{G \times 2 M_{E}}{\left(2^{1 / 3} R_{E}\right)^{2}} g_{P}\) \(g_{p}=\dfrac{2}{2^{2 / 3}} \times g_{E} \Rightarrow g_{p}=2^{1 / 3} \times g_{E}\) Weight, \(W=m g\) \(W_{P}=m g_{P} \Rightarrow W_{P}=m(2)^{1 / 3} \times g_{E}\) \( \Rightarrow {W_P} = {2^{1/3}} \times {W_E} = {2^{1/3}}\;W\)
JEE - 2023
PHXI08:GRAVITATION
359613
If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would
1 Remain unchanged
2 Decrease by \(0.5 \%\)
3 Be zero
4 Increase by \(2 \%\)
Explanation:
\(g=\dfrac{G M}{R^{2}}\) As \(M\) is constant \(\begin{aligned}& \dfrac{d g}{g}=-2 \dfrac{d R}{R} \\& \dfrac{d G}{G} \times 100=-\dfrac{1}{2} \dfrac{d R}{R} \times 100 \\& \dfrac{d g}{g} \times 100=2 \%\end{aligned}\) \(g\) will increase if \(R\) decreases.
PHXI08:GRAVITATION
359614
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
1 Will be directed towards the centre but not the same everywhere
2 Will have the same value everywhere but not directed towards the centre
3 Will be same everywhere in magnitude directed towards the centre
4 Cannot be zero at any point
Explanation:
If the earth is an approximate sphere of non-uniform density, then the centre of gravity of earth will not be situated at the centre of earth. The distance of different points on earth will be at different distances from the centre of gravity of earth. As, \(g \propto \dfrac{1}{r^{2}}\), so \(g\) is different for different points on the surface of earth but not zero.
