1:56 PM Thu Jan 25 Problem 1.4. Does it ever make sense to say that one object is "twice as hot" as\ another? D\ Explain.\ Problem 1.5. When you're sick with a fever and you take your temperature with\ a thermometer, approximately what is the relaxation time?\ 6\ Chapter 1 Energy in Thermal Physics\ Problem 1.6. Give an example to illustrate why you cannot accurately judge the temperature of an object by how hot or cold it feels to the touch.\ Problem 1.7. When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 5500 . The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient,

\\\\beta

:\

\\\\beta -=(\\\\Delta (V)/(V))/(\\\\Delta T)

\ (where

V

is volume,

T

is temperature, and

\\\\Delta

signifies a change, which in this case should really be infinitesimal if

\\\\beta

is to be well defined). So for mercury,

\\\\beta =(1)/(5500)K^(-1)=1.81\\\\times 10^(-4)K^(-1)

. (The exact value varies with temperature, but between

0\\\\deg C

and

200\\\\deg C

the variation is less than

1%

.)\ (a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.\ (b) The thermal expansion coefficient of water varies significantly with temperature: It is

7.5\\\\times 10^(-4)K^(-1)

at

100\\\\deg C

, but decreases as the temperature is lowered until it becomes zero at

4\\\\deg C

. Below

4\\\\deg C

it is slightly negative, reaching a value of

-0.68\\\\times 10^(-4)K^(-1)

at

0\\\\deg C

. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive.\ Problem 1.8. For a solid, we also define the linear thermal expansion coefficient,

\\\\alpha

, as the fractional increase in length per degree:\

\\\\alpha -=(\\\\Delta (L)/(L))/(\\\\Delta T)

\ (a) For steel,

\\\\alpha

is

1.1\\\\times 10^(-5)K^(-1)

. Estimate the total variation in length of a 1-km steel bridge between a cold winter night and a hot summer day.\ (b) The dial thermometer in Figure 1.2 uses a coiled metal strip made of two different metals laminated together. Explain how this works.\ (c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions:

\\\\beta =

\\\\alpha _(x)+\\\\alpha _(y)+\\\\alpha _(z)

. (So for an isotropic solid, which expands the same in all directions,

\\\\beta =3\\\\alpha

.)\ 1.2 The Ideal Gas\ Many of the properties of a low-density gas can be summarized in the famous ideal gas law,\

PV=nRT,

\ where

P=

pressure,

V=

volume,

n=

number of moles of gas,

R

is a universal constant, and

T

is the temperature in kelvins. (If you were to plug a Celsius

Problem 1.6. Give an example to illustrate why you cannot accurately judge the temperature of an object by how hot or cold it feels to the touch. Problem 1.7. When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 5500 . The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, : TV/V (where V is volume, T is temperature, and signifies a change, which in this case should really be infinitesimal if is to be well defined). So for mercury, =1/5500K1=1.81104K1. (The exact value varies with temperature, but between 0C and 200C the variation is less than 1%.) (a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible. (b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5104K1 at 100C, but decreases as the temperature is lowered until it becomes zero at 4C. Below 4C it is slightly negative, reaching a value of 0.68104K1 at 0C. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive. Problem 1.8. For a solid, we also define the linear thermal expansion coefficient, , as the fractional increase in length per degree: TL/L. (a) For steel, is 1.1105K1. Estimate the total variation in length of a 1-km steel bridge between a cold winter night and a hot summer day. (b) The dial thermometer in Figure 1.2 uses a coiled metal strip made of two different metals laminated together. Explain how this works. (c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions: = x+y+z. (So for an isotropic solid, which expands the same in all directions, =3.) 1.2 The Ideal Gas Many of the properties of a low-density gas can be summarized in the famous ideal gas law, PV=nRT, where P= pressure, V= volume, n= number of moles of gas, R is a universal constant, and T is the temperature in kelvins. (If you were to plug a Celsius