Physics 1B - Tutorial #1

1. What temperature has the same numerical value in Celsius (Centigrade) and Fahrenheit? Is there a temperature which has the same value on the Celsius and Kelvin (thermodynamic) scales?
   T_C = T_F at -40°

2. A circular iron plate has a coefficient of linear expansion of 10^-5K^-1. At 20°C it has a central circular hole of radius 2 inches. The plate is then heated to 40°C.
   • Does the size of the hole increase, decrease or remain the same?
     The radius of the hole increases.

   • Calculate the numerical value of the radius at 40C.
     r = 2.0004 inches.

3. The surface area of the earth is approximately 5 x 10¹⁴m². The scale height (thickness) of the earth's atmosphere is 10⁴m. A cubic meter of air at room temperature and a pressure, P = 1 atmosphere, has a mass of 1.3kg.
   • Estimate the mass of air on the earth.
     Volume of air = (5 x 10¹⁴m²) x (10⁴m) = 5 x 10¹⁸m³. Mass of air = (5 x 10¹⁸m³) x (1.3 kg/m³) = 6.5 x 10¹⁸kg.

   • A mole of air has a mass of 0.029kg (Why?). How many molecules does the earth's atmosphere contain?
     Air is 80% N₂ and 20% O₂ so the mass of 1 mole is between 28 and 32 grams (0.8 x 28g + 0.2 x 32g). Number of moles = 2.2 x 10²⁰ moles. Number of molecules = (2.2 x 10²⁰) x (6 x 10²³) = 1.3 x 10⁴⁴ molecules.

   • Assuming that the atmosphere has become completely mixed over the last 2000 years, how many molecules from Julius Ceasar's last breath are in your lungs at the current moment.
     Estimate the volume of the lungs as two cylinders 30cm in length, 5cm in radius.
     V = 2 lungs x 3(~pi) x (0.05m)² x 0.3m = 0.005 m³.
     Julius Caesar's last gasp had
     (1.3x10⁴⁴)x(0.005 m³)/(5x10¹⁸m³) = 10²³ molecules,
     of which
     (10²³)x(0.005 m³)/(5x10¹⁸m³)
     (about 100) are in your lungs right now.

4. You use superglue to attach a strip of copper to a strip of aluminum at 20C. Coefficients of linear expansion: Cu = 16.6 x 10^-6K^-1; Al = 25 x 10^-6K^-1.
   • What will happen if you heat the strips to 40C? Make a sketch.
     The strips will bend into an arc of circle; the more expansive material will be on the outer circumference of the arc.

   • What will happen if you cool the strips to 0C? Make a sketch.
     The strips will bend into an arc in the opposite direction.

   • How will the behaviour of the strips depend upon thickness: i.e. if they are very thick (like metal bars) or very thin (like paper strips).

     Thin strips will bend more than thick bars.

   • Can you think of practical uses of devices employing this behaviour?
     Thermostats, of course; any mechanism in which you want to regulate temperature or turn a device on or off if it gets too hot or too cold.

   • For a device made of two metal strips such as this one, what parameters are important in determining how the strips bend? Which would make bending larger? Which would make bending smaller? (Consider the coefficients of linear expansion, initial & final temperatures, strip length & thickness, anything else that you think might be important.)
     
     • Strip length: longer strips will bend more.
     • Temperature difference from "normal": greater T difference, greater bending.
     • Difference in coefficients of expansion between materials; the greater the difference in coefficients, the greater the bending.

   • (OPTIONAL) Suggest a formula to describe the bending using the parameters discussed above in (e).
     Let the strip consist of two metals, with expansion coefficients ₁ and ₂, each with thickness t/2 and original length, l₀. Suppose that metal #1 expands more (or contracts less) than metal #2 so that, at temperature T = T₀ + T, the length of metal #1 is l₁ = R, where R is the (unknown) radius of the arc formed by the bimetallic strip and is the angle subtended by the arc. Metal #2 has length l₂ = (R + t/2). The difference in these lengths is (t/2) which must equal (₂ - ₁) T l₀. Solving for = 2(₂ - ₁) T l₀/t.

5. As you learned in Physics 1A, rocket propulsion employs the principle of Conservation of Linear Momentum. Ideally, the burning (oxidation) of hydrocarbon fuels produces water (H₂O) and carbon dioxide (CO₂) which are expelled at high velocity. Sketch a diagram of how the burning of rocket fuel propels a rocket forward.
   • A rocket of mass 1 metric ton (1000 kg) expels CO₂ with an exhaust speed of 1000m/s. How much CO₂ must be ejected to accelerate the rocket by 1000 m/s.
     Let m_r be the initial mass of the rocket including all its fuel and m_f be the mass of fuel expended. In the initial reference frame of the rocket, m_fv_f = (m_r-m_f)v_r. In this case v_f(backward) = v_r(forward) and m_f = m_r/2 = 500kg. (Note that the expended fuel is accelerating both the rocket and the unexpended fuel - the most effective way to accelerate the rocket is to expel the fuel at the highest possible velocity, hence at the highest possible temperature.)

   • Compare the rms speeds of water and carbon dioxide at a temperature of 1000 K.
     m(H₂ O) = 18 g/mole; m(CO₂) = 76 g/mole. The speed of (lighter) water molecules will always be a factor of sqrt(76/18) = 2.06 higher than carbon dioxide. At 1000K water molecules will have v_rms = 1175 m/s and corbon dioxide, v_rms = 570 m/s.

     Does the rms speed depend upon the pressure?
     Nope; then what role does the pressure in the balloon play in propelling it forward in the next question?

   • Use a balloon to demonstrate rocket propulsion. What is the typical speed of the escaping molecules? Do they carry away momentum?
     The molecules escaping from the balloon are moving at speeds typical of room temperature, v_rms = 500 m/s. Remember that momentum is a vector; the escaping molecules carry away (negative) momentum leaving the balloon with a net positive momentum.

6. The next time you are home, take apart your thermostat and demonstrate to your parents how a bi-metallic strip works. Four brownie points if you can reassemble the thermostat and your home heating system still works,

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