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Chapter 5: Problem 562

A ring of mass \(\mathrm{M}\) and radius \(\mathrm{r}\) is melted and then molded in to a sphere then the moment of inertia of the sphere will be..... \(\\{\mathrm{A}\\}\) more than that of the ring \\{B \\} Less than that of the ring \(\\{\mathrm{C}\\}\) Equal to that of the ring \\{D\\} None of these

Short Answer

Expert verified
The moment of inertia of the sphere is less than that of the ring. The correct answer is (B) Less than that of the ring.

Step by step solution

01

Calculate the moment of inertia of the ring

The moment of inertia of a ring with mass \(M\) and radius \(r\) is given by the formula: \(I_\text{ring} = Mr^2\) In our case, we have the mass (\(M\)) and radius (\(r\)) of the ring, so we can directly apply this formula: \(I_\text{ring} = M r^2\)
02

Calculate the moment of inertia of the sphere

When the ring is melted and molded into a sphere, its moment of inertia will be given by the formula: \(I_\text{sphere} = \frac{2}{5}M R^2\) Since the mass and radius of the sphere are both equal to the mass and radius of the ring, we can substitute those values: \(I_\text{sphere} = \frac{2}{5}M r^2\)
03

Compare the moments of inertia

Now we have both moments of inertia and can compare them: \(I_\text{ring} = M r^2\) and \(I_\text{sphere} = \frac{2}{5}M r^2\) Start by dividing both sides by \(M r^2\): \(\frac{I_\text{ring}}{M r^2} = 1\) and \(\frac{I_\text{sphere}}{M r^2} = \frac{2}{5}\) Now we compare these two ratios, noticing that \(\frac{2}{5} < 1\) This means that \(I_\text{sphere} = \frac{2}{5}M r^2 < M r^2 = I_\text{ring}\) Hence, the moment of inertia of the sphere is less than that of the ring.
04

Answer

The correct answer is (B) Less than that of the ring.

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Most popular questions from this chapter

A circular disc of radius \(\mathrm{R}\) and thickness \(\mathrm{R} / 6\) has moment of inertia I about an axis passing through its centre and perpendicular to its plane. It is melted and re-casted in to a solid sphere. The moment of inertia of the sphere about its diameter as axis of rotation is \(\ldots\) \(\\{\mathrm{A}\\} \mathrm{I}\) \(\\{\mathrm{B}\\}(2 \mathrm{I} / 8)\) \(\\{\mathrm{C}\\}(\mathrm{I} / 5)\) \(\\{\mathrm{D}\\}(\mathrm{I} / 10)\)

If the earth were to suddenly contract so that its radius become half of it present radius, without any change in its mass, the duration of the new day will be... \(\\{\mathrm{A}\\} 6 \mathrm{hr}\) \\{B \(12 \mathrm{hr}\) \(\\{\mathrm{C}\\} 18 \mathrm{hr}\) \\{D \(\\} 30 \mathrm{hr}\)

From a circular disc of radius \(\mathrm{R}\) and mass \(9 \mathrm{M}\), a small disc of radius \(\mathrm{R} / 3\) is removed from the disc. The moment of inertia of the remaining portion about an axis perpendicular to the plane of the disc and passing through \(\mathrm{O}\) is.... \(\\{\mathrm{A}\\} 4 \mathrm{MR}^{2}\) \(\\{\mathrm{B}\\}(40 / 9) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\} 10 \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(37 / 9) \mathrm{MR}^{2}\)

Two point masses of \(0.3 \mathrm{~kg}\) and \(0.7 \mathrm{~kg}\) are fixed at the ends of a rod of length \(1.4 \mathrm{~m}\) and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of ..... \(\\{\mathrm{A}\\} 0.4 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\) \\{B \(0.98 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\) \\{C\\} \(0.7 \mathrm{~m}\) from mass of \(0.7 \mathrm{~kg}\) \\{D \(\\} 0.98 \mathrm{~m}\) from mass of \(0.7 \mathrm{~kg}\)

Two discs of the same material and thickness have radii \(0.2 \mathrm{~m}\) and \(0.6 \mathrm{~m}\) their moment of inertia about their axes will be in the ratio \(\\{\mathrm{A}\\} 1: 81\) \(\\{\mathrm{B}\\} 1: 27\) \(\\{C\\} 1: 9\) \(\\{\mathrm{D}\\} 1: 3\)
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