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Chapter 11: Problem 3

A disc of radius \(R\) has a hole of radius \(r\) in it. The disc is heated then (a) \(R\) and \(r\) both increase (b) \(R\) increases, \(r\) decreases (c) \(R\) decreases, \(r\) increases (d) \(\mathrm{R}\) and \(r\) both will decrease

Short Answer

Expert verified
(a) Both \( R \) and \( r \) increase.

Step by step solution

01

Understanding Thermal Expansion

When a material is heated, it tends to expand. For a circular disc, this means that both the outer radius and any inner holes (such as the radius of the hole) increase. This expansion applies uniformly across the entire disc because the material expands equally in all directions due to its physical properties.
02

Initial Conditions

We know from thermal expansion that both the radius of the disc, denoted as \( R \), and the radius of the hole, denoted as \( r \), are subject to increase when the disc is heated. There is no condition where parts of the disc warp or shrink when uniformly heated.
03

Applying the Concept

Given that both the outer boundary (\( R \)) and the hole (\( r \)) are part of the same homogeneous disc material, the heating effect will cause both to expand. Therefore, option (a) where both \( R \) and \( r \) increase, aligns with the principles of thermal expansion.

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Key Concepts

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

Circular Disc
A circular disc is a two-dimensional object with a round shape. It is characterized by a single, uniform face and a precise circular boundary. The disc can have different feature points like a radius or a central point. When discussing the physics of a disc, its radius, and sometimes a central hole, parameters are crucial.
Any modifications to a circular disc such as heating will generally impact all points across its structure. In this context, if a disc is said to have a hole at its center, it further complicates its geometric simplicity. Both the outer boundary and the boundaries of the hole must be considered when studying effects like thermal expansion.
Radius Expansion
When an object heats up, its molecules gain kinetic energy and move away from each other, leading to expansion. This principle is notably applicable in structures like circular discs.
  • The expansion is directly influenced by the initial size of the radius.
  • Not only does the outer radius increase, but also any internal features, such as holes, expand as well.
For a disc with initial radius \( R \), this means both \( R \) and the radius of any hole, \( r \), will grow upon heating. Since the material of the disc uniformly expands, no differential expansion occurs. This results in both radii increasing proportionally.
Material Properties
The thermal expansion of a disc is intimately linked to its material properties. Different materials respond in varying degrees to changes in temperature. The key properties include:
  • Thermal Expansion Coefficient: This determines how much the material expands per degree of temperature change.
  • Homogeneity: Uniform material ensures equal expansion throughout.
  • Elasticity: Dictates if the material returns to its original shape after expansion and contracting.
In homogeneous materials, like those composing a disc with a hole, the thermal expansion is consistent across the entire object. This results in synchronized changes in dimensions, such as the outer radius \( R \) and the hole's radius \( r \). When considering this, any calculations on expansion should account for these material less.
Uniform Heating
Uniform heating means applying the same amount of heat energy across the whole structure. This is crucial for consistent thermal expansion.
  • Ensures that no part of the disc expands or contracts more than another.
  • Prevents distortions and maintains the integrity and symmetry of the disc.
With uniform heating, each section of the disc will experience the same thermal energy increase, ensuring that all parts of the disc, including the central hole, will expand evenly. This is why both \( R \) and \( r \) grow together when uniformly heated.

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Most popular questions from this chapter

\begin{aligned} &\text { The difference in the length of a lead rod and a zinc rod remains same at all temperatures, then calcu- }\\\ &\text { late ratio of their original length at } \left.0^{\circ} \mathrm{C} . \mid \alpha_{\mathrm{T} \text { ead }}=2.8 \times 10^{-5} \mathrm{q} \mathrm{C}^{-1} \text { and } \alpha_{\text {7inc }}=2.6 \times 10^{-5} \mathrm{u} \mathrm{C}^{-1}\right] \text {. } \end{aligned}

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Which of the following statements is not correct for an ideal gas? (a) Internal energy of an ideal gus does not depend on its volume. (b) Specific heat of any ideal gas is equal to \(\frac{3}{2} R\). (c) If any gas obeys Boyle's law, Charle's law, then it must always be an ideal gas. (d) Ideal gas molecules have finite mass and so small volume.

Which of the following laws is not delinod correctly? (a) Temperature remaining constant, the volume occupied by gas is inversely proportional to the pressure exerted by it according to Boyle's law (b) Equal volume of all gases at same temperature and pressure contains equal number of molecules according to Avogadro's law (c) Energy associated with cach degrce of freedom of a gas molecule is \(\frac{1}{2} K T\) according to law of equipartition of energy (d) For non reactive gases the total pressure exerted by all gases mixed together is equal to the sum of their individual pressures

A Camot cngine is made to work first betwecn \(200 \mathrm{~K}\) and \(100 \mathrm{~K}\) and then hetwecn \(400 \mathrm{~K}\) and \(200 \mathrm{~K}\). The ratio of clficiencies \(\eta_{2} / \eta_{1}\) in two cases is (a) \(1: 15\) (b) \(1: 1\) (c) \(1: 2\) (d) \(1.73: 1\)
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