Energy, fluids and ideal gases

Follow the energy transfer

When you pull a crate across a floor, the force transfers energy if the point where it acts moves. For a constant force on an object moving along a straight path, work (lavoro) is the force component along the displacement multiplied by the displacement. The unit is the joule, one newton metre. A pull in the direction of travel does positive work. Sliding friction on a crate moving across a stationary floor does negative work. The normal force does zero work during horizontal motion because it acts perpendicular to the displacement.

The sum of the work done by all forces on a particle equals the change in its kinetic energy, $K=mv^2/2$. Doubling the speed of the same mass quadruples its kinetic energy. A useful hand method is to calculate the two kinetic energies separately and subtract the starting value from the final value. If a problem asks for work done by the pulling force alone, decide which other forces contributed before using the energy change as your answer.

Near Earth's surface a vertical rise of height $h$ increases gravitational potential energy by $mgh$. Measure the height vertically even when the route is a long ramp. You can include gravity through its work or through a potential-energy change. Counting it both ways doubles its contribution. When friction removes mechanical energy, the missing mechanical energy can appear as thermal energy in the surfaces. Power describes how quickly energy transfers. The same lift completed in half the time requires twice the average power.

Pressure at a point and lift on an object

Pressure is force per unit area. In a stationary liquid of uniform density $\rho$, descending a vertical distance $h$ increases pressure by $\rho gh$. Picture a slender column of liquid above a sensor. Doubling the column's area doubles both the weight and the area supporting it, so the pressure increase stays the same. The shape of the surrounding vessel drops out of the calculation. To obtain absolute pressure, add the pressure at the surface. A gauge referenced to the surrounding atmosphere reads the increase alone in an open tank.

The pressure on an immersed body acts inward on every face. For a rectangular block with horizontal top and bottom, the pressure on the bottom exceeds that on the top. Multiplying the difference by the face area gives the weight of the fluid the block displaces. Archimedes' principle gives this upward buoyant force (spinta di Archimede) for any shape. Use the density of the fluid and the volume actually submerged. A freely floating object at rest displaces fluid of weight equal to its own. A completely submerged object held by a string may have an extra force in its balance.

Temperature in a gas law

For an ideal gas in equilibrium, pressure, volume, amount of gas and absolute temperature obey $pV=nRT$. Keep the same sealed sample and the amount stays fixed. Divide the state equation for one state by that for another. The gas constant and amount cancel, leaving $p_1V_1/T_1=p_2V_2/T_2$. A rigid vessel keeps volume fixed. A freely moving piston subject to a fixed external pressure can instead allow a change in volume at the same equilibrium pressure.

Use kelvin and absolute pressure when taking ratios. A rise from 20 to 40 degrees Celsius is a rise from about 293 to 313 kelvin. The absolute temperature increases modestly even though the Celsius reading doubles. When only the temperature difference enters a heat calculation, one kelvin of change equals one Celsius degree of change. For a body remaining in the same phase with constant specific heat capacity $c$, energy supplied as heat is $mc\Delta T$ if other energy exchanges are negligible. Equal heating of two samples need not produce equal temperature rises. The larger heat capacity spreads the same energy over a smaller temperature change.