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Chapter 9: Problem 2

The moment of inertia of two spheres of equal masses about their diameters are equal. If one of them is solid and other is hollow, the ratio of their radii is (a) \(\sqrt{3}: \sqrt{5}\) (b) \(3: 5\) (c) \(\sqrt{5}: \sqrt{3}\) (d) \(5: 3\)

Short Answer

Expert verified
The ratio of their radii is \( \sqrt{5}: \sqrt{3} \) (option c).

Step by step solution

01

Identify Formulas

The moment of inertia of a solid sphere of mass \( m \) and radius \( r \) about its diameter is given by \( I_1 = \frac{2}{5}mr^2 \). For a hollow sphere (thin spherical shell) of the same mass \( m \) and radius \( R \), the moment of inertia about its diameter is given by \( I_2 = \frac{2}{3}mR^2 \).
02

Set Equations Equal

Since the moments of inertia are equal, we set \( I_1 = I_2 \). Therefore, we have:\[ \frac{2}{5}mr^2 = \frac{2}{3}mR^2 \]
03

Cancel Common Factors

We can cancel \( m \) and \( \frac{2}{x} \) from both sides of the equation due to the common factors, resulting in a simplified equation:\[ \frac{r^2}{5} = \frac{R^2}{3} \]
04

Solve for Radius Ratio

Cross-multiply to solve for the ratio of \( r^2 \) to \( R^2 \):\[ 3r^2 = 5R^2 \]Divide both sides by \( r^2R^2 \) and then take the square root to find the ratio of the radii:\[ \left(\frac{r}{R}\right)^2 = \frac{5}{3} \]\[ \frac{r}{R} = \sqrt{\frac{5}{3}} \]
05

Express the Ratio

The ratio of the radii \( \frac{r}{R} \) is equivalent to \( \sqrt{5}: \sqrt{3} \). Therefore, the answer to the given multiple-choice question is (c) \( \sqrt{5}: \sqrt{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solid Sphere
A solid sphere is a three-dimensional object where every point on its surface is equidistant from its center. Solid spheres are dense and filled throughout, making them heavier and influencing their rotational properties.

For physics calculations, especially concerning the moment of inertia, a solid sphere can be modeled to simplify understanding. The moment of inertia is crucial because it measures how difficult it is to rotate an object about an axis.

In the case of a solid sphere with mass \( m \) and radius \( r \), the formula to calculate the moment of inertia about its diameter is:
\[ I_1 = \frac{2}{5} mr^2 \]
This formula hints that the mass distribution inside the sphere plays a critical role in determining how much resistance the sphere will provide to rotational movement. It is lighter compared to objects of the same mass but with mass distributed farther from the axis.
Hollow Sphere
A hollow sphere, often referred to as a thin spherical shell, has all its mass concentrated on its surface, with an empty space inside. This characteristic significantly affects its moment of inertia. A hollow sphere is lighter internally but heavier rotationally than a solid sphere of the same mass.

When calculating the moment of inertia for a hollow sphere, we consider only the surface, as that is where the mass resides. For a hollow sphere of mass \( m \) and radius \( R \), the formula is:
\[ I_2 = \frac{2}{3} m R^2 \]
This formula shows that a hollow sphere resists changes in its rotational motion more than a solid sphere with the same mass and similar dimensions, due to more of its mass being located farther from the axis of rotation.
Radius Ratio
The radius ratio is an important factor to consider when comparing objects like solid and hollow spheres. It directly influences the relation between their moments of inertia, especially if these spheres share the same mass.

In problems where the moment of inertia of a solid and hollow sphere are equal, finding the ratio of their radii is essential. The step-by-step solution showed:
1. Begin by equalizing their moments of inertia: \( \frac{2}{5} mr^2 = \frac{2}{3} m R^2 \).
2. Simplify and cross-multiply to find:
\[ 3r^2 = 5R^2 \]
3. Solving for the radius ratio provides:
\[ \frac{r}{R} = \sqrt{\frac{5}{3}} \]
This gives the ratio \( \sqrt{5}: \sqrt{3} \), showing how the dimensions of the spheres need to be tuned to maintain rotational equivalency, even if their masses remain consistent.

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Most popular questions from this chapter

The spin driver of a washing machine revolving at 15 rps slow down to 5 rps, while making 50 revolutions. Angular acceleration of the driver is (a) \(-4 \pi \mathrm{rads}^{-2}\) (b) \(-4 \pi \mathrm{rads}^{-2}\) (c) \(8 \pi \mathrm{rads}^{-2}\) (d) \(-8 \pi \mathrm{rads}^{-2}\)

A thin uniform \(\operatorname{rod} A B\) of mass \(m\) and length \(L\) is hinged at one end \(A\) to the level floor. Initially, it stands vertically and is allowed to fall freely to the floor in the vertical plane. The angular velocity of the rod, when its end \(B\) strikes the floor is \((g\) is acceleration due to gravity) (a) \(\left(\frac{m g}{L}\right)\) (b) \(\left(\frac{m g}{3 L}\right)^{1 / 2}\) (c) \(\left(\frac{g}{L}\right)\) (d) \(\left(\frac{3 g}{L}\right)^{1 / 2}\)

Of the two eggs those have identical sizes, shapes and weights, one is raw, and other is half boiled. The ratio between the moment of inertia of the raw to the half boiled egg about central axis is (a) one (b) greater than one (c) less than one (d) not comparable

A cord is wound round the circumference of a wheel of radius \(r\). The axis of the wheel of horizontal and moment of inertia about it is \(I .\) A weight \(m g\) is attached to the end of the cord and falls from rest. After falling through a distance \(h\), the angular velocity of the wheel will be \((a)\left(\frac{2 g h}{1+m r}\right)^{1 / 2}\) (b) \(\left(\frac{2 m g r}{1+m r^{2}}\right)^{1 / 2}\) (c) \(\left(\frac{2 m g h}{1+2 m}\right)^{1 / 2}\) (d) \((2 g h)^{1 / 2}\)

Consider a uniform square plate of side \(a\) and mass \(m\). The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is (a) \(\frac{5}{6} m a^{2}\) (b) \(\frac{m a^{2}}{12}\) (c) \(\frac{7}{12} m a^{2}\) (d) \(\frac{2}{3} m a^{2}\)
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