FSc Pre-Engineering Physics Syllabus

State the dot and cross products of the unit vectors $\hat{i},\hat{j},\hat{k}$.

Dot: $\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1$ and $\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0$. Cross: $\hat{i}\times\hat{j}=\hat{k},\ \hat{j}\times\hat{k}=\hat{i},\ \hat{k}\times\hat{i}=\hat{j}$, and $\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=\hat{k}\times\hat{k}=0$.

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

Torque is the turning effect of a force about a pivot, equal to the product of the force and the perpendicular moment arm: $\tau = \vec{r}\times\vec{F}$, with magnitude $\tau = rF\sin\theta$. SI unit: $N\,m$.

State the two conditions of equilibrium.

First condition: the vector sum of all forces is zero, $\sum \vec{F} = 0$ (so $\sum F_x = 0$ and $\sum F_y = 0$). Second condition: the sum of all torques about any point is zero, $\sum \tau = 0$.

Distinguish between translational and rotational equilibrium.

Translational equilibrium means no net force acts, so the body has no linear acceleration ($\sum\vec{F}=0$). Rotational equilibrium means no net torque acts, so the body has no angular acceleration ($\sum\tau=0$).

Differentiate between distance and displacement, and between speed and velocity.

Distance is the total path length (scalar); displacement is the straight-line change in position with direction (vector). Speed is the rate of change of distance (scalar); velocity is the rate of change of displacement (vector).

Define acceleration and write its defining equation.

Acceleration is the rate of change of velocity with time: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$. SI unit: $m\,s^{-2}$.

State the three equations of uniformly accelerated motion.

$v_f = v_i + at$; $\quad S = v_i t + \frac{1}{2}at^{2}$; $\quad 2aS = v_f^{2} - v_i^{2}$.

State Newton's three laws of motion.

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

Define linear momentum and state its SI unit.

Linear momentum is the product of mass and velocity, $\vec{p} = m\vec{v}$. It is a vector with SI unit $kg\,m\,s^{-1}$ (equivalently $N\,s$).

Express Newton's second law in terms of momentum.

The net force equals the time rate of change of momentum: $\vec{F} = \frac{\Delta \vec{p}}{\Delta t} = \frac{m(v_f - v_i)}{t}$.

Define impulse and state the impulse-momentum theorem.

Impulse is the product of force and the time for which it acts, $\vec{J} = \vec{F}\,\Delta t$. The impulse-momentum theorem states that impulse equals the change in momentum: $\vec{F}\,\Delta t = \Delta\vec{p} = m v_f - m v_i$. SI unit: $N\,s$.

State the law of conservation of momentum.

In the absence of an external force, the total momentum of an isolated system remains constant: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$.

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 converts to heat/sound); in a perfectly inelastic collision the bodies stick together after impact.

For a one-dimensional elastic collision, give the final velocity of body 1.

$v_1 = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)u_1 + \left(\frac{2m_2}{m_1 + m_2}\right)u_2$, where $u_1,u_2$ are the initial velocities.

Define projectile motion and state the nature of its horizontal and vertical motions.

Projectile motion is two-dimensional motion under gravity alone. The horizontal component has constant velocity (zero acceleration); the vertical component is uniformly accelerated motion with acceleration $g$ downward. The two are independent.

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

Time of flight: $T = \frac{2v_i\sin\theta}{g}$. Maximum height: $H = \frac{v_i^{2}\sin^{2}\theta}{2g}$. Range: $R = \frac{v_i^{2}\sin 2\theta}{g}$.

At what angle is the horizontal range of a projectile maximum, and what is that maximum range?

The range is maximum at $\theta = 45^{\circ}$ (since $\sin 2\theta = 1$), giving $R_{max} = \frac{v_i^{2}}{g}$.

Define work and state when it is positive, negative or zero.

Work is the product of force and displacement in the direction of the force: $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$. It is positive for $\theta < 90^{\circ}$, negative for $\theta > 90^{\circ}$, and zero when $\theta = 90^{\circ}$. SI unit: joule ($J$).

State the work-energy theorem.

The net work done on a body equals the change in its kinetic energy: $W_{net} = \Delta KE = \frac{1}{2}mv_f^{2} - \frac{1}{2}mv_i^{2}$.

Write the formulas for kinetic energy and gravitational potential energy.

Kinetic energy: $KE = \frac{1}{2}mv^{2}$. Gravitational potential energy (near Earth's surface): $PE = mgh$.

State the law of conservation of energy.

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

For a freely falling body, state the principle of conservation of mechanical energy.

In the absence of friction, the sum of kinetic and potential energy is constant: $\frac{1}{2}mv^{2} + mgh = \text{constant}$. Energy continuously interchanges between PE and KE during the fall.

Define power and give its formula and SI unit.

Power is the rate of doing work (or rate of energy transfer): $P = \frac{W}{t}$. It can also be written $P = \vec{F}\cdot\vec{v}$. SI unit: watt ($W = J\,s^{-1}$).

Define efficiency and write its formula.

Efficiency is the ratio of useful output to total input, $\eta = \frac{\text{output}}{\text{input}}\times 100\%$ (energy or power). It is always less than $100\%$ because of energy losses such as friction and heat.

Define angular displacement and state the relation between linear distance and angular displacement.

Angular displacement $\theta$ is the angle swept by a body in circular motion, measured in radians. The arc length is $S = r\theta$, where $r$ is the radius.

Define angular velocity and angular acceleration, and relate angular velocity to linear (tangential) velocity.

Angular velocity $\omega = \frac{\Delta\theta}{\Delta t}$ (rad $s^{-1}$); angular acceleration $\alpha = \frac{\Delta\omega}{\Delta t}$ (rad $s^{-2}$). Linear velocity is related by $v = r\omega$, and tangential acceleration by $a = r\alpha$.

Define centripetal acceleration and give its formula.

Centripetal acceleration is the acceleration directed toward the centre of a circular path that changes the direction of velocity: $a_c = \frac{v^{2}}{r} = r\omega^{2}$.

Define centripetal force and write its two equivalent formulas.

Centripetal force is the net force directed toward the centre that keeps a body moving in a circle: $F_c = \frac{mv^{2}}{r} = mr\omega^{2}$.

State Newton's law of universal gravitation and write its equation.

Every two point masses attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them: $F = G\dfrac{m_1 m_2}{r^{2}}$, where $G = 6.67\times 10^{-11}\ N\,m^{2}\,kg^{-2}$.

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

The gravitational force provides the centripetal force: $\frac{GMm}{r^{2}} = \frac{mv^{2}}{r}$, giving the orbital speed $v = \sqrt{\dfrac{GM}{r}}$.

What is the minimum (critical) orbital speed for a satellite close to Earth's surface, and its approximate value?

For a low orbit $r \approx R$ (Earth's radius), so $v = \sqrt{gR} \approx \sqrt{9.8 \times 6.4\times 10^{6}} \approx 7.9\ km\,s^{-1}$ (about $7900\ m\,s^{-1}$).

What is a geostationary (geosynchronous) satellite?

A satellite that orbits the Earth in the equatorial plane with a period of $24$ hours, in the same direction as Earth's rotation, so it appears stationary relative to a point on Earth. Its orbital radius is about $4.23\times 10^{7}\ m$ ($\approx 36000\ km$ above the surface).

Define viscosity and state the SI unit of the coefficient of viscosity.

Viscosity is the property of a fluid by which it opposes the relative motion between its layers (internal friction). The coefficient of viscosity $\eta$ has SI unit $N\,s\,m^{-2}$ (pascal-second, $Pa\,s$).

State Stokes's law for the drag force on a sphere moving through a viscous fluid.

$F = 6\pi\eta r v$, where $\eta$ is the coefficient of viscosity, $r$ the radius of the sphere and $v$ its speed relative to the fluid.

Define terminal velocity and write its expression for a sphere falling through a viscous fluid.

Terminal velocity is the constant maximum velocity attained when the drag and buoyant forces balance the weight, so net force (and acceleration) is zero: $v_t = \dfrac{2 r^{2} (\rho - \rho_0) g}{9\eta}$, where $\rho$ is the density of the sphere and $\rho_0$ that of the fluid.

State the equation of continuity and the principle it expresses.

For an incompressible fluid in steady flow, $A_1 v_1 = A_2 v_2$ (i.e. $Av = \text{constant}$). It expresses conservation of mass: the volume flow rate is constant, so fluid speeds up where the cross-sectional area is smaller.