365741
Two particles of mass \(1\;kg\) and \(3\;kg\) have position vectors \(2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(-2 \hat{i}+3 \hat{j}-4 \hat{k}\) respectively. The centre of mass has a position vector
1 \(-\hat{i}-3 \hat{j}-2 \hat{k}\)
2 \(\hat{i}+3 \hat{j}-2 \hat{k}\)
3 \(-\hat{i}+3 \hat{j}-2 \hat{k}\)
4 \(-\hat{i}+3 \hat{j}+2 \hat{k}\)
Explanation:
Here, \({m_1} = 1\,\,kg,{m_2} = 3\,\,kg\) \(\vec{r}_{1}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{r}_{2}=-2 \hat{i}+3 \hat{j}-4 \hat{k}\) The position vector of the centre of mass is \(\begin{aligned}& \vec{r}_{1}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}} \\& =\dfrac{(1)(2 \hat{i}+3 \hat{j}+4 \hat{k})+(3)(-2 \hat{i}+3 \hat{j}-4 \hat{k})}{1+3} \\& =\dfrac{-4 \hat{i}+12 \hat{j}-8 \hat{k}}{4}=-\hat{i}+3 \hat{j}-2 \hat{k}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365742
Two bodies of masses \(1\;kg\,\,{\rm{and}}\,\,3\;kg\) have position vectors \(\hat{i}+2 \hat{j}+\hat{k}\) and \(-3 \hat{i}-2 \hat{j}+\hat{k}\), respectively. The centre of mass of this system has a position vector.
1 \(-\hat{i}+\hat{j}+\hat{k}\)
2 \(-2 \hat{i}+2 \hat{k}\)
3 \(-2 \hat{i}-\hat{j}+\hat{k}\)
4 \(-2 \hat{i}-\hat{j}-2 \hat{k}\)
Explanation:
The position vector of centre of mass \(\vec{r}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}\) \(=\dfrac{1(\hat{i}+2 \hat{j}+\hat{k})+3(-3 \hat{i}-2 \hat{j}+\hat{k})}{1+3}\) \(\dfrac{1}{4}(-8 \hat{i}-4 \hat{j}+4 \hat{k})=-2 \hat{i}-\hat{j}+\hat{k}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365743
The centre of mass of a system of two particles of masses \(m_{1}\) and \(m_{2}\) is at a distance \(d_{1}\) from \(m_{1}\) and at a distance \(d_{2}\) from mass \(m_{2}\) such that
Refer figure, The distance of \(CM\) from masses \(m_{1}\) and \(m_{2}\) are \(\begin{aligned}& d_{1}=\dfrac{m_{2} d}{m_{1}+m_{2}} \\& d_{2}=\dfrac{m_{1} d}{m_{1}+m_{2}} \\& \therefore \dfrac{d_{1}}{d_{2}}=\dfrac{m_{2}}{m_{1}}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365744
Assertion : Position of centre of mass is independent of the reference frame. Reason : Centre of mass is same for all bodies.
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 centre of mass \({\rm{(COM)}}\) is an intrinsic property of a system and is independent of the reference frame. It is calculated based on the positions and masses of the particles within the system. \({\rm{COM}}\) of different bodies is different. So correct option is (3).
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PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365741
Two particles of mass \(1\;kg\) and \(3\;kg\) have position vectors \(2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(-2 \hat{i}+3 \hat{j}-4 \hat{k}\) respectively. The centre of mass has a position vector
1 \(-\hat{i}-3 \hat{j}-2 \hat{k}\)
2 \(\hat{i}+3 \hat{j}-2 \hat{k}\)
3 \(-\hat{i}+3 \hat{j}-2 \hat{k}\)
4 \(-\hat{i}+3 \hat{j}+2 \hat{k}\)
Explanation:
Here, \({m_1} = 1\,\,kg,{m_2} = 3\,\,kg\) \(\vec{r}_{1}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{r}_{2}=-2 \hat{i}+3 \hat{j}-4 \hat{k}\) The position vector of the centre of mass is \(\begin{aligned}& \vec{r}_{1}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}} \\& =\dfrac{(1)(2 \hat{i}+3 \hat{j}+4 \hat{k})+(3)(-2 \hat{i}+3 \hat{j}-4 \hat{k})}{1+3} \\& =\dfrac{-4 \hat{i}+12 \hat{j}-8 \hat{k}}{4}=-\hat{i}+3 \hat{j}-2 \hat{k}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365742
Two bodies of masses \(1\;kg\,\,{\rm{and}}\,\,3\;kg\) have position vectors \(\hat{i}+2 \hat{j}+\hat{k}\) and \(-3 \hat{i}-2 \hat{j}+\hat{k}\), respectively. The centre of mass of this system has a position vector.
1 \(-\hat{i}+\hat{j}+\hat{k}\)
2 \(-2 \hat{i}+2 \hat{k}\)
3 \(-2 \hat{i}-\hat{j}+\hat{k}\)
4 \(-2 \hat{i}-\hat{j}-2 \hat{k}\)
Explanation:
The position vector of centre of mass \(\vec{r}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}\) \(=\dfrac{1(\hat{i}+2 \hat{j}+\hat{k})+3(-3 \hat{i}-2 \hat{j}+\hat{k})}{1+3}\) \(\dfrac{1}{4}(-8 \hat{i}-4 \hat{j}+4 \hat{k})=-2 \hat{i}-\hat{j}+\hat{k}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365743
The centre of mass of a system of two particles of masses \(m_{1}\) and \(m_{2}\) is at a distance \(d_{1}\) from \(m_{1}\) and at a distance \(d_{2}\) from mass \(m_{2}\) such that
Refer figure, The distance of \(CM\) from masses \(m_{1}\) and \(m_{2}\) are \(\begin{aligned}& d_{1}=\dfrac{m_{2} d}{m_{1}+m_{2}} \\& d_{2}=\dfrac{m_{1} d}{m_{1}+m_{2}} \\& \therefore \dfrac{d_{1}}{d_{2}}=\dfrac{m_{2}}{m_{1}}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365744
Assertion : Position of centre of mass is independent of the reference frame. Reason : Centre of mass is same for all bodies.
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 centre of mass \({\rm{(COM)}}\) is an intrinsic property of a system and is independent of the reference frame. It is calculated based on the positions and masses of the particles within the system. \({\rm{COM}}\) of different bodies is different. So correct option is (3).
365741
Two particles of mass \(1\;kg\) and \(3\;kg\) have position vectors \(2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(-2 \hat{i}+3 \hat{j}-4 \hat{k}\) respectively. The centre of mass has a position vector
1 \(-\hat{i}-3 \hat{j}-2 \hat{k}\)
2 \(\hat{i}+3 \hat{j}-2 \hat{k}\)
3 \(-\hat{i}+3 \hat{j}-2 \hat{k}\)
4 \(-\hat{i}+3 \hat{j}+2 \hat{k}\)
Explanation:
Here, \({m_1} = 1\,\,kg,{m_2} = 3\,\,kg\) \(\vec{r}_{1}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{r}_{2}=-2 \hat{i}+3 \hat{j}-4 \hat{k}\) The position vector of the centre of mass is \(\begin{aligned}& \vec{r}_{1}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}} \\& =\dfrac{(1)(2 \hat{i}+3 \hat{j}+4 \hat{k})+(3)(-2 \hat{i}+3 \hat{j}-4 \hat{k})}{1+3} \\& =\dfrac{-4 \hat{i}+12 \hat{j}-8 \hat{k}}{4}=-\hat{i}+3 \hat{j}-2 \hat{k}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365742
Two bodies of masses \(1\;kg\,\,{\rm{and}}\,\,3\;kg\) have position vectors \(\hat{i}+2 \hat{j}+\hat{k}\) and \(-3 \hat{i}-2 \hat{j}+\hat{k}\), respectively. The centre of mass of this system has a position vector.
1 \(-\hat{i}+\hat{j}+\hat{k}\)
2 \(-2 \hat{i}+2 \hat{k}\)
3 \(-2 \hat{i}-\hat{j}+\hat{k}\)
4 \(-2 \hat{i}-\hat{j}-2 \hat{k}\)
Explanation:
The position vector of centre of mass \(\vec{r}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}\) \(=\dfrac{1(\hat{i}+2 \hat{j}+\hat{k})+3(-3 \hat{i}-2 \hat{j}+\hat{k})}{1+3}\) \(\dfrac{1}{4}(-8 \hat{i}-4 \hat{j}+4 \hat{k})=-2 \hat{i}-\hat{j}+\hat{k}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365743
The centre of mass of a system of two particles of masses \(m_{1}\) and \(m_{2}\) is at a distance \(d_{1}\) from \(m_{1}\) and at a distance \(d_{2}\) from mass \(m_{2}\) such that
Refer figure, The distance of \(CM\) from masses \(m_{1}\) and \(m_{2}\) are \(\begin{aligned}& d_{1}=\dfrac{m_{2} d}{m_{1}+m_{2}} \\& d_{2}=\dfrac{m_{1} d}{m_{1}+m_{2}} \\& \therefore \dfrac{d_{1}}{d_{2}}=\dfrac{m_{2}}{m_{1}}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365744
Assertion : Position of centre of mass is independent of the reference frame. Reason : Centre of mass is same for all bodies.
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 centre of mass \({\rm{(COM)}}\) is an intrinsic property of a system and is independent of the reference frame. It is calculated based on the positions and masses of the particles within the system. \({\rm{COM}}\) of different bodies is different. So correct option is (3).
365741
Two particles of mass \(1\;kg\) and \(3\;kg\) have position vectors \(2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(-2 \hat{i}+3 \hat{j}-4 \hat{k}\) respectively. The centre of mass has a position vector
1 \(-\hat{i}-3 \hat{j}-2 \hat{k}\)
2 \(\hat{i}+3 \hat{j}-2 \hat{k}\)
3 \(-\hat{i}+3 \hat{j}-2 \hat{k}\)
4 \(-\hat{i}+3 \hat{j}+2 \hat{k}\)
Explanation:
Here, \({m_1} = 1\,\,kg,{m_2} = 3\,\,kg\) \(\vec{r}_{1}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{r}_{2}=-2 \hat{i}+3 \hat{j}-4 \hat{k}\) The position vector of the centre of mass is \(\begin{aligned}& \vec{r}_{1}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}} \\& =\dfrac{(1)(2 \hat{i}+3 \hat{j}+4 \hat{k})+(3)(-2 \hat{i}+3 \hat{j}-4 \hat{k})}{1+3} \\& =\dfrac{-4 \hat{i}+12 \hat{j}-8 \hat{k}}{4}=-\hat{i}+3 \hat{j}-2 \hat{k}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365742
Two bodies of masses \(1\;kg\,\,{\rm{and}}\,\,3\;kg\) have position vectors \(\hat{i}+2 \hat{j}+\hat{k}\) and \(-3 \hat{i}-2 \hat{j}+\hat{k}\), respectively. The centre of mass of this system has a position vector.
1 \(-\hat{i}+\hat{j}+\hat{k}\)
2 \(-2 \hat{i}+2 \hat{k}\)
3 \(-2 \hat{i}-\hat{j}+\hat{k}\)
4 \(-2 \hat{i}-\hat{j}-2 \hat{k}\)
Explanation:
The position vector of centre of mass \(\vec{r}=\dfrac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}\) \(=\dfrac{1(\hat{i}+2 \hat{j}+\hat{k})+3(-3 \hat{i}-2 \hat{j}+\hat{k})}{1+3}\) \(\dfrac{1}{4}(-8 \hat{i}-4 \hat{j}+4 \hat{k})=-2 \hat{i}-\hat{j}+\hat{k}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365743
The centre of mass of a system of two particles of masses \(m_{1}\) and \(m_{2}\) is at a distance \(d_{1}\) from \(m_{1}\) and at a distance \(d_{2}\) from mass \(m_{2}\) such that
Refer figure, The distance of \(CM\) from masses \(m_{1}\) and \(m_{2}\) are \(\begin{aligned}& d_{1}=\dfrac{m_{2} d}{m_{1}+m_{2}} \\& d_{2}=\dfrac{m_{1} d}{m_{1}+m_{2}} \\& \therefore \dfrac{d_{1}}{d_{2}}=\dfrac{m_{2}}{m_{1}}\end{aligned}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365744
Assertion : Position of centre of mass is independent of the reference frame. Reason : Centre of mass is same for all bodies.
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 centre of mass \({\rm{(COM)}}\) is an intrinsic property of a system and is independent of the reference frame. It is calculated based on the positions and masses of the particles within the system. \({\rm{COM}}\) of different bodies is different. So correct option is (3).
