365771
In a system of two particles of masses \(m_{1}\) and \(m_{2}\) the first particle is moved by a distance \(d\)$ towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance
1 \(\dfrac{m_{2}}{m_{1}} d\) towards the centre of mass
2 \(\dfrac{m_{1}}{m_{2}} d\), away from the centre of mass
3 \(\dfrac{m_{1}}{m_{2}} d\), towards the centre of mass
4 \(\dfrac{m_{2}}{m_{1}} d\), away from the centre of mass
Explanation:
Let \(x_{1}\) and \(x_{2}\) be the position of masses \(m_{1}\) and \(m_{2}\) respectively. The position of centre of mass is \(x_{C M}=\dfrac{x_{1} m_{1}+x_{2} m_{2}}{m_{1}+m_{2}}\) If \(\Delta x_{1}\) and \(\Delta x_{2}\) be the changes in positions of \(m_{1}\) and \(m_{2}\) respectively, the change in the position of centre of mass, \(\Delta x_{C M}=\dfrac{\Delta x_{1} m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}}\) Given the centre of mass remains unchanged i.e. \(\Delta x_{C M}=0\) and \(\Delta x_{1}=d\) (towards centre of mass) \(\Rightarrow 0=\dfrac{d m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}} \Rightarrow \Delta x_{2}=-\dfrac{m_{1}}{m_{2}} d\) \(\therefore\) The displacement of the second mass should also be towards the centre of mass as these two masses are located on two opposite sides of their common centre of mass and their displacements have opposite signs.
MHTCET - 2020
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365772
Assertion : When a man jumps from a boat to the shore, the boat slightly moves away from the shore. Reason : The total momentum should remain conserved.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
When a man jumps from a boat to the shore, the boat does indeed slightly move away from the shore. This is because, according to the law of conservation of momentum, the total momentum of the system (man + boat) must remain conserved. When the man jumps, he gains momentum in the opposite direction, causing the boat to move slightly in the other direction. So correct option is (1)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365773
A uniform square plate \({A B C D}\) has a mass of \(5\,kg\). If two point masses of \(1\,kg\), each are placed at the corners \({C}\) and \({D}\), as shown in the figure, then the centre of mass shifts to the point which lies on
1 \({O C}\)
2 \({O D}\)
3 \({O Y}\)
4 \({O X}\)
Explanation:
As two equal masses are added on both side of X-axis, so C.O.M. will shifts towards the negative \({Y}\). Correct Option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365774
Assertion : Centre of mass of a system does not move under the action of internal forces. Reason : Internal forces can be conservative forces.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The center of mass of a system does not move under the action of internal forces. The reason is also true, as internal forces like spring force is conservative. So correct option is (2).
365771
In a system of two particles of masses \(m_{1}\) and \(m_{2}\) the first particle is moved by a distance \(d\)$ towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance
1 \(\dfrac{m_{2}}{m_{1}} d\) towards the centre of mass
2 \(\dfrac{m_{1}}{m_{2}} d\), away from the centre of mass
3 \(\dfrac{m_{1}}{m_{2}} d\), towards the centre of mass
4 \(\dfrac{m_{2}}{m_{1}} d\), away from the centre of mass
Explanation:
Let \(x_{1}\) and \(x_{2}\) be the position of masses \(m_{1}\) and \(m_{2}\) respectively. The position of centre of mass is \(x_{C M}=\dfrac{x_{1} m_{1}+x_{2} m_{2}}{m_{1}+m_{2}}\) If \(\Delta x_{1}\) and \(\Delta x_{2}\) be the changes in positions of \(m_{1}\) and \(m_{2}\) respectively, the change in the position of centre of mass, \(\Delta x_{C M}=\dfrac{\Delta x_{1} m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}}\) Given the centre of mass remains unchanged i.e. \(\Delta x_{C M}=0\) and \(\Delta x_{1}=d\) (towards centre of mass) \(\Rightarrow 0=\dfrac{d m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}} \Rightarrow \Delta x_{2}=-\dfrac{m_{1}}{m_{2}} d\) \(\therefore\) The displacement of the second mass should also be towards the centre of mass as these two masses are located on two opposite sides of their common centre of mass and their displacements have opposite signs.
MHTCET - 2020
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365772
Assertion : When a man jumps from a boat to the shore, the boat slightly moves away from the shore. Reason : The total momentum should remain conserved.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
When a man jumps from a boat to the shore, the boat does indeed slightly move away from the shore. This is because, according to the law of conservation of momentum, the total momentum of the system (man + boat) must remain conserved. When the man jumps, he gains momentum in the opposite direction, causing the boat to move slightly in the other direction. So correct option is (1)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365773
A uniform square plate \({A B C D}\) has a mass of \(5\,kg\). If two point masses of \(1\,kg\), each are placed at the corners \({C}\) and \({D}\), as shown in the figure, then the centre of mass shifts to the point which lies on
1 \({O C}\)
2 \({O D}\)
3 \({O Y}\)
4 \({O X}\)
Explanation:
As two equal masses are added on both side of X-axis, so C.O.M. will shifts towards the negative \({Y}\). Correct Option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365774
Assertion : Centre of mass of a system does not move under the action of internal forces. Reason : Internal forces can be conservative forces.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The center of mass of a system does not move under the action of internal forces. The reason is also true, as internal forces like spring force is conservative. So correct option is (2).
365771
In a system of two particles of masses \(m_{1}\) and \(m_{2}\) the first particle is moved by a distance \(d\)$ towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance
1 \(\dfrac{m_{2}}{m_{1}} d\) towards the centre of mass
2 \(\dfrac{m_{1}}{m_{2}} d\), away from the centre of mass
3 \(\dfrac{m_{1}}{m_{2}} d\), towards the centre of mass
4 \(\dfrac{m_{2}}{m_{1}} d\), away from the centre of mass
Explanation:
Let \(x_{1}\) and \(x_{2}\) be the position of masses \(m_{1}\) and \(m_{2}\) respectively. The position of centre of mass is \(x_{C M}=\dfrac{x_{1} m_{1}+x_{2} m_{2}}{m_{1}+m_{2}}\) If \(\Delta x_{1}\) and \(\Delta x_{2}\) be the changes in positions of \(m_{1}\) and \(m_{2}\) respectively, the change in the position of centre of mass, \(\Delta x_{C M}=\dfrac{\Delta x_{1} m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}}\) Given the centre of mass remains unchanged i.e. \(\Delta x_{C M}=0\) and \(\Delta x_{1}=d\) (towards centre of mass) \(\Rightarrow 0=\dfrac{d m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}} \Rightarrow \Delta x_{2}=-\dfrac{m_{1}}{m_{2}} d\) \(\therefore\) The displacement of the second mass should also be towards the centre of mass as these two masses are located on two opposite sides of their common centre of mass and their displacements have opposite signs.
MHTCET - 2020
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365772
Assertion : When a man jumps from a boat to the shore, the boat slightly moves away from the shore. Reason : The total momentum should remain conserved.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
When a man jumps from a boat to the shore, the boat does indeed slightly move away from the shore. This is because, according to the law of conservation of momentum, the total momentum of the system (man + boat) must remain conserved. When the man jumps, he gains momentum in the opposite direction, causing the boat to move slightly in the other direction. So correct option is (1)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365773
A uniform square plate \({A B C D}\) has a mass of \(5\,kg\). If two point masses of \(1\,kg\), each are placed at the corners \({C}\) and \({D}\), as shown in the figure, then the centre of mass shifts to the point which lies on
1 \({O C}\)
2 \({O D}\)
3 \({O Y}\)
4 \({O X}\)
Explanation:
As two equal masses are added on both side of X-axis, so C.O.M. will shifts towards the negative \({Y}\). Correct Option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365774
Assertion : Centre of mass of a system does not move under the action of internal forces. Reason : Internal forces can be conservative forces.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The center of mass of a system does not move under the action of internal forces. The reason is also true, as internal forces like spring force is conservative. So correct option is (2).
365771
In a system of two particles of masses \(m_{1}\) and \(m_{2}\) the first particle is moved by a distance \(d\)$ towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance
1 \(\dfrac{m_{2}}{m_{1}} d\) towards the centre of mass
2 \(\dfrac{m_{1}}{m_{2}} d\), away from the centre of mass
3 \(\dfrac{m_{1}}{m_{2}} d\), towards the centre of mass
4 \(\dfrac{m_{2}}{m_{1}} d\), away from the centre of mass
Explanation:
Let \(x_{1}\) and \(x_{2}\) be the position of masses \(m_{1}\) and \(m_{2}\) respectively. The position of centre of mass is \(x_{C M}=\dfrac{x_{1} m_{1}+x_{2} m_{2}}{m_{1}+m_{2}}\) If \(\Delta x_{1}\) and \(\Delta x_{2}\) be the changes in positions of \(m_{1}\) and \(m_{2}\) respectively, the change in the position of centre of mass, \(\Delta x_{C M}=\dfrac{\Delta x_{1} m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}}\) Given the centre of mass remains unchanged i.e. \(\Delta x_{C M}=0\) and \(\Delta x_{1}=d\) (towards centre of mass) \(\Rightarrow 0=\dfrac{d m_{1}+\Delta x_{2} m_{2}}{m_{1}+m_{2}} \Rightarrow \Delta x_{2}=-\dfrac{m_{1}}{m_{2}} d\) \(\therefore\) The displacement of the second mass should also be towards the centre of mass as these two masses are located on two opposite sides of their common centre of mass and their displacements have opposite signs.
MHTCET - 2020
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365772
Assertion : When a man jumps from a boat to the shore, the boat slightly moves away from the shore. Reason : The total momentum should remain conserved.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
When a man jumps from a boat to the shore, the boat does indeed slightly move away from the shore. This is because, according to the law of conservation of momentum, the total momentum of the system (man + boat) must remain conserved. When the man jumps, he gains momentum in the opposite direction, causing the boat to move slightly in the other direction. So correct option is (1)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365773
A uniform square plate \({A B C D}\) has a mass of \(5\,kg\). If two point masses of \(1\,kg\), each are placed at the corners \({C}\) and \({D}\), as shown in the figure, then the centre of mass shifts to the point which lies on
1 \({O C}\)
2 \({O D}\)
3 \({O Y}\)
4 \({O X}\)
Explanation:
As two equal masses are added on both side of X-axis, so C.O.M. will shifts towards the negative \({Y}\). Correct Option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365774
Assertion : Centre of mass of a system does not move under the action of internal forces. Reason : Internal forces can be conservative forces.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
The center of mass of a system does not move under the action of internal forces. The reason is also true, as internal forces like spring force is conservative. So correct option is (2).
