365762
In the figure one fourth part of a uniform disc of radius \(R\) is shown. The distance of the centre of mass of the object from centre \(O'\) is
1 \(\dfrac{2 R}{2 \pi}\)
2 \(\dfrac{4 R}{3 \pi}\)
3 \(\sqrt{2} \dfrac{2 R}{3 \pi}\)
4 \(\sqrt{2} \dfrac{4 R}{3 \pi}\)
Explanation:
The \(CM\) has both \(x\) and \(y\) coordinates. By using papus theorem, rotate the material about \(x\)-axis. Volume covered \(=\) distance moved by \(CM\,\, \times \) Area of the material \(\frac{2}{3}\pi {R^3} = 2\pi {y_{CM}} \times \frac{{\pi {R^2}}}{4}{y_{CM}} = \frac{{4R}}{{3\pi }}\) Similarly \(x_{C M}=\dfrac{4 R}{3 \pi}\) \(r_{C M}=\sqrt{x_{C M}^{2}+y_{C M}^{2}}=\dfrac{4 \sqrt{2} R}{3 \pi}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365763
A thick straight wire of length \(\pi m\) is fixed at its midpoint and then bent in the form of a circle. The shift in its centre of mass is
NEET Test Series from KOTA - 10 Papers In MS WORD
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PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365762
In the figure one fourth part of a uniform disc of radius \(R\) is shown. The distance of the centre of mass of the object from centre \(O'\) is
1 \(\dfrac{2 R}{2 \pi}\)
2 \(\dfrac{4 R}{3 \pi}\)
3 \(\sqrt{2} \dfrac{2 R}{3 \pi}\)
4 \(\sqrt{2} \dfrac{4 R}{3 \pi}\)
Explanation:
The \(CM\) has both \(x\) and \(y\) coordinates. By using papus theorem, rotate the material about \(x\)-axis. Volume covered \(=\) distance moved by \(CM\,\, \times \) Area of the material \(\frac{2}{3}\pi {R^3} = 2\pi {y_{CM}} \times \frac{{\pi {R^2}}}{4}{y_{CM}} = \frac{{4R}}{{3\pi }}\) Similarly \(x_{C M}=\dfrac{4 R}{3 \pi}\) \(r_{C M}=\sqrt{x_{C M}^{2}+y_{C M}^{2}}=\dfrac{4 \sqrt{2} R}{3 \pi}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365763
A thick straight wire of length \(\pi m\) is fixed at its midpoint and then bent in the form of a circle. The shift in its centre of mass is
365762
In the figure one fourth part of a uniform disc of radius \(R\) is shown. The distance of the centre of mass of the object from centre \(O'\) is
1 \(\dfrac{2 R}{2 \pi}\)
2 \(\dfrac{4 R}{3 \pi}\)
3 \(\sqrt{2} \dfrac{2 R}{3 \pi}\)
4 \(\sqrt{2} \dfrac{4 R}{3 \pi}\)
Explanation:
The \(CM\) has both \(x\) and \(y\) coordinates. By using papus theorem, rotate the material about \(x\)-axis. Volume covered \(=\) distance moved by \(CM\,\, \times \) Area of the material \(\frac{2}{3}\pi {R^3} = 2\pi {y_{CM}} \times \frac{{\pi {R^2}}}{4}{y_{CM}} = \frac{{4R}}{{3\pi }}\) Similarly \(x_{C M}=\dfrac{4 R}{3 \pi}\) \(r_{C M}=\sqrt{x_{C M}^{2}+y_{C M}^{2}}=\dfrac{4 \sqrt{2} R}{3 \pi}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365763
A thick straight wire of length \(\pi m\) is fixed at its midpoint and then bent in the form of a circle. The shift in its centre of mass is
365762
In the figure one fourth part of a uniform disc of radius \(R\) is shown. The distance of the centre of mass of the object from centre \(O'\) is
1 \(\dfrac{2 R}{2 \pi}\)
2 \(\dfrac{4 R}{3 \pi}\)
3 \(\sqrt{2} \dfrac{2 R}{3 \pi}\)
4 \(\sqrt{2} \dfrac{4 R}{3 \pi}\)
Explanation:
The \(CM\) has both \(x\) and \(y\) coordinates. By using papus theorem, rotate the material about \(x\)-axis. Volume covered \(=\) distance moved by \(CM\,\, \times \) Area of the material \(\frac{2}{3}\pi {R^3} = 2\pi {y_{CM}} \times \frac{{\pi {R^2}}}{4}{y_{CM}} = \frac{{4R}}{{3\pi }}\) Similarly \(x_{C M}=\dfrac{4 R}{3 \pi}\) \(r_{C M}=\sqrt{x_{C M}^{2}+y_{C M}^{2}}=\dfrac{4 \sqrt{2} R}{3 \pi}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365763
A thick straight wire of length \(\pi m\) is fixed at its midpoint and then bent in the form of a circle. The shift in its centre of mass is
