Thermodynamic finals revision

1. Thermal Physics (Heat, Expansion, Work)

This is about how stuff gets hot or cold, changes size, and what happens when it does.

• Heat (Q = mcΔT and Q = mL)

  • Think: Heating up a cup of tea vs. melting ice.

  • Q = mcΔT (Heating/Cooling without changing state): This recipe is for when you're just making something hotter or colder without it melting or boiling.

    • Q: How much heat energy you add (like how much gas you use on the stove).

    • m: How much stuff you have (mass, like how much water is in your cup).

    • c: "Specific Heat Capacity." This is like a material's "stubbornness" to change temperature. Water is very stubborn (high 'c'), so it takes a lot of heat to warm up. Metal isn't as stubborn (lower 'c'), so it heats up fast.

    • ΔT: How much the temperature changes (your tea went from warm to hot).

  • Q = mL (Changing State - Melting/Boiling): This recipe is for when your stuff is melting or boiling (or freezing/condensing). The temperature doesn't change during this process! All the heat goes into breaking/forming bonds.

    • Q: Heat energy.

    • m: Mass.

    • L: "Latent Heat." This is the special amount of heat needed to melt/freeze (Latent Heat of Fusion, Lf) or boil/condense (Latent Heat of Vaporization, Lv) one kilogram of a substance. It's like the extra energy needed to turn ice into water, even though both are 0°C.

• Linear Expansion (ΔL = αLΔT)

  • Think: A metal bridge getting longer on a hot day.

  • ΔL: How much longer (or shorter) the object gets.

  • α (alpha): "Coefficient of Linear Expansion." This is how much a material expands per degree of temperature change. Different materials expand different amounts.

  • L: The original length of the object.

  • ΔT: The change in temperature.

  • Important for Volumetric Expansion (Like the Aluminium Cube!): If a material expands in length, it also expands in width and height. For a simple cube, the volume expansion coefficient (β) is roughly 3 times the linear one (β≈3α). So, if you need volume change, you'd use ΔV=V0​βΔT=V0​(3α)ΔT.

• Thermal Conductivity (P = kA(dT/dx))

  • Think: How fast heat flows through a pan handle.

  • P: Power, or the rate of heat flow (how many Joules per second).

  • k: "Thermal Conductivity." This is how good a material is at letting heat pass through it. Metal has a high 'k' (good conductor), air has a low 'k' (good insulator).

  • A: The area that the heat is flowing through (like the cross-section of the rod).

  • dT/dx: This is like the "steepness" of the temperature difference. It's (Temperature difference) / (Length the heat flows over). Basically, a bigger temperature difference or a shorter distance means faster heat flow.

  • In the Tank Problem: Heat flows from hot to cold through the copper rods. At "steady state," the heat flowing into the ice from one side equals the heat flowing out of the ice to the other side. Think of it like a steady river flow.

• Work (dW = -PdV or W = PΔV for work by system)

  • Think: A balloon expanding and pushing on the air around it.

  • W: Work done (energy transferred because of a force moving a distance).

  • P: Pressure (the force per unit area that the system is pushing against, like atmospheric pressure).

  • ΔV: Change in volume. If the volume gets bigger (ΔV is positive), the system does positive work on the surroundings. If the system shrinks (ΔV is negative), the surroundings do work on the system.

  • For the Aluminium Cube: The heated cube expands, pushing against the surrounding air (atmospheric pressure). So it does work.

2. Rotational Motion (Angular Momentum)

This is about spinning things.

• Angular Momentum (L = Iω)

  • Think: A figure skater pulling their arms in to spin faster.

  • L: "Angular Momentum." How much "spin" an object has.

  • I (Moment of Inertia): This is like "rotational mass." It's not just how heavy something is, but where that weight is. If the mass is far from the spinning center (like a skater's arms out), 'I' is big. If the mass is close to the center (arms pulled in), 'I' is small.

    • For a solid disk: I=21​MR2 (The mass is spread out).

    • For a point mass: I=mr2 (All the mass is at one distance 'r').

  • ω (omega): "Angular Velocity." How fast it's spinning (like RPM, but in radians per second).

• Conservation of Angular Momentum:

  • The Big Rule: If nothing from outside the system is trying to twist it (no external torque), then the total amount of spin (total L) stays the same.

  • The Skater Analogy: When a skater pulls their arms in, their 'I' (rotational mass) gets smaller. Since 'L' must stay the same, their 'ω' (spin speed) must increase.

  • In the Disk Problem: When the clay drops on the disk, it's like the disk suddenly gaining more "rotational mass" (its 'I' increases) at a certain distance. Since no outside force is twisting the whole disk-clay system, the total spin (L) must stay the same. To compensate for the bigger 'I', the 'ω' (spin speed) has to get slower.

3. Forces and Motion (Terminal Velocity)

• Terminal Velocity:

  • Think: A sky diver falling.

  • What's Happening: As a sky diver falls, gravity pulls them down (their weight, Fg​=mg). But the air pushes up on them (air resistance). The faster they go, the stronger the air resistance.

  • The "Terminal" Part: Eventually, the upward air resistance force becomes exactly equal to the downward gravitational force. When these two forces balance, the net force is zero.

  • Newton's First Law: If the net force is zero, there's no acceleration. This means the sky diver stops speeding up and falls at a constant, maximum speed, which is their "terminal velocity."

  • The Key for the Problem: At terminal velocity, the resistive force is equal to the weight of the object (Fresistive​=mg). So, if you know the resistive force, you know the weight, and you can find the mass using m=Weight/g.