Pho.rs: Fuel tank and Gearbox

Условие

Edu

Русский

This problem concerns the physics of certain devices inside a car.

Part A. Fuel level sensor (3.2 points)

Let's consider a fuel tank in the shape of a cube with side length $a$. The tank contains a fuel up to a level $h$. There is a fixed vertical electrically conducting rod with a float inside the cube. The float is in electrical contact with the rod and is connected by a wire to an external circuit consisting of a resistor $R_0$ and a battery of voltage $\mathcal{E}$.

The resistance per unit length of the rod is $\lambda$, the resistance of the float may be neglected. The connecting wires are insulated.

A1 ^0.60 Write the expression that can be used to calculate the volume of fuel $V$ in the tank, given the voltmeter reading $U_1$ and the values of $R_0$, $\lambda$, $\mathcal{E}$ and $a$.

This expression is convenient, but if values of $R_0$ and $\lambda$ depend on temperature in different ways, the readings will vary with temperature. Therefore, a more complex circuit is proposed. It contains two resistors $R_1$ and $R_2$ made of the same material.

A2 ^1.00 Write the expression that can be used to calculate the volume of fuel $V$ in the tank, given the voltmeter reading $U_2$ and the values of $R_1/R_2$, $\mathcal{E}$ and $a$.

During operation of the car, the value of $R_1/R_2$ in the fuel gauge formula has drifted and became equal to one. The graph below shows the readings $V$ of the miscalibrated fuel gauge vs time as the tank is filled uniformly from empty to full.

A3 ^1.60 Determine the volume of the fuel tank $V_0$. Determine ratio $R_1/R_2$ that should be used in the fuel gauge for it to give correct readings.

Part B. Gearbox (6.8 points)

In this part we will analyze the operation of a gearbox, which serves as a mehcanical connection between the car's engine and its wheels. Let the engine axis be denoted by $E$, its angular velocity by $\omega$, and the wheel axis by $W$. For simplicity, let us assume that the car has only one wheel of radius $r$, which never slips.

Let us assume that the engine operates continuously and consumes fuel. As a result of fuel combustion, heat is released at a power $P_0$. The power efficiency $\eta$ of the engine depends on the angular velocity of its axis:
\[
\eta =
\begin{cases}
\alpha \dfrac{\omega}{\omega_0} \left( 1 - \dfrac{\omega}{\omega_0} \right) &,\omega < \omega_0 \\
0&,\omega \geq \omega_0
\end{cases}
\]

B1 ^0.60 Sketch a graph of $\eta$ vs $\omega$.

In all subsequent questions we consider only the case $\omega < \omega_0$

B2 ^0.50 Show that if the engine's axis rotates with the angular velocity $\omega$ and an external torque $M$ acts on it, then the mechanical power of the engine equals $\omega M$.

B3 ^1.00 Determine torque $M$ of the engine. Express the answer in terms of $P_0$, $\alpha$, $\omega$ and $\omega_0$.

The radius of the gear on the engine axis is $R_E$, the radius of the gear on the wheel axis is $R_W$. Let us denote $R_E/R_W=k$. Assume that the axes rotate without friction.

B4 ^1.20 Determine the angular velocity $\omega_W$ of the wheel and the torque $M_W$ acting on it from its axis $W$. Express the answer in terms of $k$, $\omega$ and $M$.

When the car moves with speed $v$, the enviroment excerts a drag force $-\beta v$ on it. With the engine running, the car gradually accelerates and reaches its terminal velocity.

B5 ^2.40 Determine the terminal velocity $v$ of the car with the engine running. Express the answer in terms of $P_0$, $\alpha$, $k$, $r$, $\omega_0$ and $\beta$. Sketch the graph of $v$ vs $k$.

B6 ^1.10 For what value of $k=k_\text{max}$ does the car velocity reach its maximum? Express the answer in terms of $\alpha$, $P_0$, $\beta$, $r$ and $\omega_0$.