Thermal Expansion

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. On many railway lines, between one steel rail and the next, there is a small gap, left on purpose. Why would engineers leave gaps in a railway line? Think about a hot summer afternoon. Today: thermal expansion, how solids and liquids grow when heated, and what happens when they cannot. Questions on this come up often in JEE, so let's begin.

0:29Solids expand: length, area, volume

So, heat a rod of length L, and it grows by delta L, which equals L times alpha times delta T, okay? The little triangle is the Greek capital delta, meaning change in. So delta L is the change in length, and delta T, the change in temperature. And alpha, the Greek letter after L, is the coefficient of linear expansion; it depends on the material. Its unit is K to the power minus one, read as per kelvin. With me? Now think. Heat a square plate instead. Both its length and its breadth grow by the fraction alpha delta T, so how fast does its area grow? Twice as fast! Yes, exactly! Both sides stretch, so the coefficient of area expansion, beta, the Greek letter beta, is two alpha. Clear? And for volume, three directions stretch. So the coefficient of volume expansion, gamma, the Greek letter gamma, is three alpha. This holds for isotropic solids, which expand equally in every direction. Got it?

1:37Trap: a hole in a heated plate

Now, a famous trap. Take a metal plate with a round hole in it, and heat the plate. The metal expands into the hole, so the hole gets smaller, right? No, no, careful! The hole gets bigger. Every distance in the plate grows, including the distance across the hole. Okay? So, the rule: any hole or cavity expands exactly as if it were made of the surrounding material. Imagine the hole filled with a disc of the same metal; that disc would grow by the same formula. So for a hole, use the alpha of the plate. Right?

2:13Liquids: real and apparent expansion

Now liquids. Their true expansion is called real expansion, with coefficient gamma r: gamma with a small r below it, for real. Okay? But a liquid sits in a vessel, and the vessel expands too. So the rise in level that you see: is it more, or less, than the real expansion? Less! Correct! The vessel makes extra room. So gamma apparent equals gamma real, minus gamma of the vessel; the small words below each gamma say whose coefficient it is. Clear? And as a liquid expands, its mass stays the same but its volume grows, so its density falls. Rho, the Greek letter, is the density, and rho prime, with the little dash, is the new density. Rho prime equals rho, over one plus gamma delta T, with me? Now a strange one. Heat water from zero degrees Celsius to four degrees Celsius; the small circle means degrees. Does it expand? No! Water is unusual: it contracts from zero to four degrees Celsius. So water is densest at four degrees Celsius. Got it? That is why lakes freeze from the top. Surface water cooled below four degrees is lighter, so it stays on top and freezes, while the water below stays near four degrees Celsius.

3:36No gap? Thermal stress in a clamped rod

Right? Now, back to our puzzle. What if there were no gap? Clamp a rod between two rigid walls and heat it. It wants to grow by L alpha delta T, but the walls do not let it. So what does it do to the walls? It pushes on them, hard! And the walls squeeze it back by that same amount, so the rod has a strain, delta L over L, equal to alpha delta T, okay? Stress equals Young's modulus Y times strain. So the thermal stress, sigma, is Y alpha delta T, where sigma, the Greek letter, means stress, not surface tension. This uses Hooke's law, so it holds within the elastic limit. Right? And force is stress times area, so the force on the walls is F equals Y A alpha delta T, with capital A the area of the rod. Now a topper's trick: look, there is no L in this formula! Long rod or short rod, the same force. Clear?

4:40Pendulum clocks in summer

One more case: the pendulum clock. Its time period, capital T here, is proportional to root L, the square root of the length; the sign after T means proportional to. In summer the rod is longer, so the clock runs slow. With me? So, when it is warmer than its calibration temperature by delta theta, where theta, the Greek letter, stands for temperature here, so delta theta is a change in temperature, it loses half alpha delta theta, times eighty six thousand four hundred seconds, each day. The half comes from the square root, and eighty six thousand four hundred is the number of seconds in a day. Got it?

5:19Example 1: expansion gap for 12.5 m steel rails

Now, Example one: our railway puzzle. Steel rails twelve point five metres long are laid at ten degrees Celsius. What gap is needed for forty degrees Celsius? Alpha is one point two times ten to the power minus five per kelvin. Pause the video, and try it first. Done? Okay, let's check. The rails get as hot as forty degrees, so delta T is forty, right? No, no, wait! Delta T is the change: forty minus ten, thirty degrees Celsius. And a change of thirty degrees Celsius is a change of thirty kelvin, so it matches the unit of alpha. Okay? Now substitute: twelve point five, times one point two times ten to the power minus five, times thirty. Clear? Twelve point five times one point two is fifteen, and fifteen times thirty is four hundred and fifty. So delta L is four point five times ten to the power minus three metre: four point five millimetres. Puzzle solved! Each rail needs a gap of four point five millimetres to grow into on a hot day. Got it?

6:30Example 2: force on walls from a clamped heated steel rod

Now Example two: no gap at all. Here is a steel rod of area one square centimetre, clamped between rigid walls and heated by forty kelvin. Young's modulus Y is two times ten to the power eleven pascal, that is, newton per square metre, and alpha is one point two times ten to the power minus five per kelvin. Find the force on the walls. Pause and try. Hint: S I units first. Done? Okay, let's check. Area first. One centimetre is ten to the power minus two metre, so one square centimetre is ten to the power minus two square metre, right? Oh no, careful! Square the conversion too. Ten to the power minus two, squared, is ten to the power minus four, so capital A is ten to the power minus four square metre. With me? Now the formula: F equals Y A alpha delta T, taking the rod to stay within the elastic limit. Okay? Substitute: two times ten to the power eleven, times ten to the power minus four, times one point two times ten to the power minus five, times forty. Right? Powers of ten first: eleven minus four minus five is two, so ten squared, one hundred. And two times one point two times forty is ninety six. So F is nine thousand six hundred newtons: nine point six kilonewtons. Close to the weight of a one tonne mass, from one small rod! That is why rails need their gaps. Clear?

8:08Quick revision: alpha, beta, gamma, liquids, thermal stress

So, quick revision. Solids: delta L equals L alpha delta T, beta is two alpha, gamma is three alpha, and a hole expands like the metal. Liquids: gamma apparent is gamma real minus gamma of the vessel, and water is densest at four degrees Celsius. Clamped rod: stress Y alpha delta T, force Y A alpha delta T, whatever the length. Now go and try the practice problems on this page. All the best!