(a) Define stable equilibrium as applied to a rigid body. (b) Sketch a block and tackle system of pulleys with a velocity ratio of 3. (c) At the beginning o...

(a) Stable equilibrium

A rigid body is in stable equilibrium if, when it is slightly displaced and released, it tends to return to its original position. Its centre of gravity rises on displacement, so that its potential energy is a minimum at the equilibrium position.

(b) Block and tackle of velocity ratio 3

The rope is fixed to the movable block. It then passes over the left fixed pulley, round the movable pulley, and over the right fixed pulley. Thus, the movable block is supported by three tensioned sections of the rope, giving:

\[\text{V.R.}=3\]

(c) Temperature of the gas at the end of the race

Since the volume of the tyre is constant,

\[\frac{P_1}{T_1}=\frac{P_2}{T_2}\]

\[P_1=4.5\times10^5\ \text{Pa},\qquad P_2=4.6\times10^5\ \text{Pa}\]

\[T_1=20^\circ\text{C}+273=293\ \text{K}\]

\[T_2=\frac{P_2T_1}{P_1}=\frac{(4.6\times10^5)(293)}{4.5\times10^5}=299.5\ \text{K}\]

\[\boxed{T_2=299.5\ \text{K}\approx 26.5^\circ\text{C}}\]

(d)(i) Temperature scales

The calibration readings are:

Let \(R_\theta\) be the resistance of the platinum thermometer in the liquid bath and \(P_\theta\) be the pressure of the gas thermometer in the same bath.

(α) Platinum resistance thermometer

\[\frac{R_\theta-5.67}{7.75-5.67}=\frac{T_R-273}{373-273}\]

\[\boxed{T_R=273+\frac{100(R_\theta-5.67)}{2.08}\ \text{K}}\]

(β) Constant-volume gas thermometer

\[\frac{P_\theta-7.13\times10^4}{(9.74-7.13)\times10^4}=\frac{T_G-273}{373-273}\]

\[\boxed{T_G=273+\frac{100(P_\theta-7.13\times10^4)}{2.61\times10^4}\ \text{K}}\]

(d)(ii) Percentage error of the platinum resistance thermometer

Taking the constant-volume gas thermometer temperature as the reference temperature,

\[\boxed{\%\,\text{error}=\frac{T_R-T_G}{T_G}\times100\%}\]

where \(T_R\) and \(T_G\) are obtained from the two boxed calibration equations above.