PAF Aeronautical Engineering Physics Syllabus

Write the formulas for time of flight, maximum height, and range of a projectile launched at speed $v_0$ and angle $\theta$.

$$T = \frac{2 v_0 \sin\theta}{g},\quad H = \frac{v_0^{2}\sin^{2}\theta}{2g},\quad R = \frac{v_0^{2}\sin 2\theta}{g}.$$

At what launch angle is the range of a projectile maximum (on level ground), and why?

At $\theta = 45^{\circ}$, because $R = \frac{v_0^{2}\sin 2\theta}{g}$ is maximum when $\sin 2\theta = 1$, i.e. $2\theta = 90^{\circ}$.

What is the shape of a projectile's trajectory, and what are its velocity components at the highest point?

The trajectory is a parabola. At the highest point the vertical velocity is zero and the horizontal velocity remains $v_0\cos\theta$.

State Newton's three laws of motion.

First law (inertia): a body stays at rest or in uniform motion unless acted on by a net external force. Second law: $\vec{F} = m\vec{a}$. Third law: for every action there is an equal and opposite reaction.

Define inertia and state how it depends on mass.

Inertia is the tendency of a body to resist changes in its state of rest or uniform motion. It is directly proportional to the body's mass; greater mass means greater inertia.

State Newton's second law in terms of momentum.

The net force equals the rate of change of momentum: $$\vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}.$$

What is the difference between mass and weight? Give the formula relating them.

Mass is the amount of matter (scalar, constant, in $\text{kg}$). Weight is the gravitational force on the body (vector, varies with $g$, in $\text{N}$): $W = mg$.

Define the work done by a constant force, including the role of angle.

Work $W = F s \cos\theta$, where $F$ is the force, $s$ the displacement, and $\theta$ the angle between them. Work is a scalar measured in joules ($\text{J}$).

Write the formulas for kinetic energy and gravitational potential energy.

Kinetic energy: $E_k = \tfrac{1}{2}m v^{2}$. Gravitational potential energy (near Earth): $E_p = m g h$.

State the work-energy theorem.

The net work done on a body equals the change in its kinetic energy: $$W_{net} = \Delta E_k = \tfrac{1}{2}m v^{2} - \tfrac{1}{2}m u^{2}.$$

Define power and give its formula in terms of force and velocity.

Power is the rate of doing work: $P = \frac{W}{t}$. For a force on a body moving at velocity $v$: $P = F v\cos\theta$. SI unit: watt ($\text{W}$).

State the principle of conservation of energy.

Energy can neither be created nor destroyed, only transformed from one form to another; the total energy of an isolated system remains constant.

For an object in uniform circular motion, write the formulas for centripetal acceleration and centripetal force.

Centripetal acceleration: $a_c = \frac{v^{2}}{r} = \omega^{2} r$. Centripetal force: $F_c = \frac{m v^{2}}{r} = m\omega^{2} r$, directed toward the centre.

Relate linear speed $v$ to angular velocity $\omega$, and define the period of circular motion.

$v = r\omega$, where $r$ is the radius. The period $T = \frac{2\pi}{\omega} = \frac{2\pi r}{v}$ is the time for one complete revolution.

Define linear momentum and state the law of conservation of momentum.

Linear momentum $\vec{p} = m\vec{v}$. In an isolated system (no external force), total momentum is conserved: $\sum \vec{p}_{before} = \sum \vec{p}_{after}$.

Define impulse and state the impulse-momentum theorem.

Impulse $= \vec{F}\,\Delta t$ (force times time). The impulse-momentum theorem: impulse equals change in momentum, $\vec{F}\,\Delta t = \Delta \vec{p} = m\vec{v} - m\vec{u}$.

Compare elastic and inelastic collisions.

In an elastic collision both momentum and kinetic energy are conserved. In an inelastic collision momentum is conserved but kinetic energy is not (some is converted to heat/deformation); in a perfectly inelastic collision the bodies stick together.

Define torque (moment of a force) and give its formula.

Torque is the turning effect of a force about a pivot: $\tau = F\, d = r F\sin\theta$, where $d$ is the perpendicular distance (moment arm). SI unit: $\text{N m}$.

State the principle of moments for a body in rotational equilibrium.

For a body in rotational equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments: $\sum \tau_{clockwise} = \sum \tau_{anticlockwise}$, i.e. net torque $= 0$.

Define the centre of gravity and the centre of mass.

Centre of gravity is the point where the whole weight of a body appears to act. Centre of mass is the point where the total mass may be considered concentrated. In a uniform gravitational field they coincide.

Define angular momentum and state its conservation law.

Angular momentum $L = I\omega$ (moment of inertia times angular velocity); for a particle $L = m v r$. When no net external torque acts, angular momentum is conserved: $I_1\omega_1 = I_2\omega_2$.

Define moment of inertia and write the rotational analogue of Newton's second law.

Moment of inertia $I = \sum m_i r_i^{2}$ measures rotational inertia about an axis. The rotational form of Newton's second law is $\tau = I\alpha$, where $\alpha$ is angular acceleration.

Define pressure in a fluid and write its dependence on depth.

Pressure $P = \frac{F}{A}$ (force per unit area). At depth $h$ in a fluid of density $\rho$: $P = \rho g h$ (gauge pressure); absolute pressure $= P_{atm} + \rho g h$. SI unit: pascal ($\text{Pa}$).

State Pascal's principle and Archimedes' principle.

Pascal's principle: pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. Archimedes' principle: the upthrust (buoyant force) on a body equals the weight of the fluid it displaces, $F_B = \rho_{fluid}\, V g$.

State the equation of continuity for an incompressible fluid and what it expresses.

$A_1 v_1 = A_2 v_2$ (or $A v = \text{constant}$). It expresses conservation of mass/volume flow rate: where the pipe is narrower the fluid flows faster.

State Bernoulli's equation for an ideal fluid.

$$P + \tfrac{1}{2}\rho v^{2} + \rho g h = \text{constant},$$ relating pressure, kinetic energy per unit volume, and potential energy per unit volume along a streamline.

Explain the Bernoulli principle in words and give one everyday application.

Where a fluid's speed is high, its pressure is low, and vice versa. Applications include aircraft lift (faster air over the wing gives lower pressure above), the carburettor, and the spin of a ball (swing/curve).

State Newton's law of universal gravitation with its formula.

Every two masses attract with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: $$F = G\frac{m_1 m_2}{r^{2}},\quad G = 6.67\times10^{-11}\ \text{N m}^2\text{kg}^{-2}.$$

Derive the relation between $g$ at Earth's surface and $G$, the Earth's mass $M$ and radius $R$.

Equating $mg = G\frac{Mm}{R^{2}}$ gives $$g = \frac{GM}{R^{2}}.$$ This shows $g$ is independent of the falling object's mass.

Derive the orbital speed of a satellite in a circular orbit of radius $r$ around a planet of mass $M$.

Setting gravity as the centripetal force, $\frac{GMm}{r^{2}} = \frac{m v^{2}}{r}$, gives $$v = \sqrt{\frac{GM}{r}}.$$

What is a geostationary satellite and what is its orbital period?

A geostationary satellite orbits in the equatorial plane in the same direction as Earth's rotation with a period of $24\ \text{h}$ ($\approx 86400\ \text{s}$), so it stays fixed above one point on Earth, at an altitude of about $36000\ \text{km}$.

Define escape velocity and give its formula for Earth.

Escape velocity is the minimum speed needed to escape a planet's gravity without further propulsion: $$v_{esc} = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \approx 11.2\ \text{km s}^{-1}\ \text{for Earth}.$$

Define Simple Harmonic Motion (SHM) and write its defining equation.

SHM is oscillatory motion in which the restoring force (and acceleration) is directly proportional to the displacement and directed toward the equilibrium position: $$a = -\omega^{2} x.$$

Write the formulas for the period of a simple pendulum and a mass-spring system in SHM.

Simple pendulum: $T = 2\pi\sqrt{\dfrac{l}{g}}$. Mass-spring system: $T = 2\pi\sqrt{\dfrac{m}{k}}$, where $k$ is the spring constant.

For SHM with amplitude $x_0$ and angular frequency $\omega$, write the expressions for velocity and maximum velocity.

Velocity at displacement $x$: $v = \omega\sqrt{x_0^{2} - x^{2}}$. Maximum velocity (at equilibrium, $x = 0$): $v_{max} = \omega x_0$.