ETEA Engineering Test Physics Syllabus

Write the dimensions of work/energy, power and pressure.

Work/Energy: $[M L^{2} T^{-2}]$; Power: $[M L^{2} T^{-3}]$; Pressure: $[M L^{-1} T^{-2}]$.

State the principle of homogeneity of dimensions used in dimensional analysis.

For any physically valid equation, the dimensions of every term on both sides must be identical. Only quantities with the same dimensions can be added, subtracted or equated.

Use dimensions to check whether $v = u + at$ is dimensionally correct.

$[v]=[L T^{-1}]$, $[u]=[L T^{-1}]$, $[at]=[L T^{-2}][T]=[L T^{-1}]$. All terms have dimension $[L T^{-1}]$, so the equation is dimensionally consistent (homogeneous).

Derive, using dimensional analysis, the form of the period $T$ of a simple pendulum in terms of length $l$ and $g$.

Assume $T \propto l^{a} g^{b}$. Dimensionally $[T] = [L]^{a}[L T^{-2}]^{b}$, giving $a+b=0$ and $-2b=1$, so $b=-\tfrac{1}{2}$, $a=\tfrac{1}{2}$. Thus $T \propto \sqrt{\dfrac{l}{g}}$.

What are the limitations of dimensional analysis?

It cannot determine dimensionless constants (like $2\pi$), cannot derive relations with more than one term (e.g. with $+$/$-$), cannot handle trigonometric, exponential or logarithmic functions, and cannot distinguish quantities with the same dimensions.

What is the difference between a scalar and a vector quantity? Give two examples of each.

A scalar has only magnitude (e.g. mass, temperature, speed, energy). A vector has both magnitude and direction (e.g. displacement, velocity, force, acceleration) and obeys vector addition rules.

State the head-to-tail (polygon) rule for adding vectors.

Draw the vectors to scale so the tail of each successive vector starts at the head of the previous one. The resultant is the vector drawn from the tail of the first vector to the head of the last vector.

For two vectors of magnitudes $A$ and $B$ with angle $\theta$ between them, give the magnitude and direction of the resultant.

Magnitude: $R = \sqrt{A^{2} + B^{2} + 2AB\cos\theta}$. Direction relative to $A$: $\tan\alpha = \dfrac{B\sin\theta}{A + B\cos\theta}$.

What are the rectangular components of a vector $\vec{A}$ making angle $\theta$ with the x-axis?

$A_{x} = A\cos\theta$ and $A_{y} = A\sin\theta$, where $A$ is the magnitude of $\vec{A}$.

Given the rectangular components $A_{x}$ and $A_{y}$, how do you find the magnitude and direction of the vector?

Magnitude: $A = \sqrt{A_{x}^{2} + A_{y}^{2}}$. Direction: $\theta = \tan^{-1}\!\left(\dfrac{A_{y}}{A_{x}}\right)$ measured from the x-axis.

What is a unit vector, and how is it obtained from a vector $\vec{A}$?

A unit vector has magnitude $1$ and specifies direction only. It is obtained by dividing a vector by its magnitude: $\hat{A} = \dfrac{\vec{A}}{|\vec{A}|}$.

What do the unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$ represent?

They are mutually perpendicular unit vectors of magnitude $1$ directed along the positive $x$-, $y$- and $z$-axes respectively in a right-handed Cartesian coordinate system.

Write a vector $\vec{A}$ in terms of its rectangular components and unit vectors, and give its magnitude in 3D.

$\vec{A} = A_{x}\hat{i} + A_{y}\hat{j} + A_{z}\hat{k}$, with magnitude $|\vec{A}| = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}$.

Define the scalar (dot) product of two vectors and state whether the result is scalar or vector.

$\vec{A}\cdot\vec{B} = AB\cos\theta$, where $\theta$ is the angle between them. The result is a scalar quantity.

Write the scalar product of $\vec{A}$ and $\vec{B}$ in terms of their components.

$\vec{A}\cdot\vec{B} = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}$.

Evaluate the dot products $\hat{i}\cdot\hat{i}$, $\hat{j}\cdot\hat{j}$, $\hat{k}\cdot\hat{k}$ and $\hat{i}\cdot\hat{j}$.

$\hat{i}\cdot\hat{i} = \hat{j}\cdot\hat{j} = \hat{k}\cdot\hat{k} = 1$ and $\hat{i}\cdot\hat{j} = \hat{j}\cdot\hat{k} = \hat{k}\cdot\hat{i} = 0$.

State two key properties of the scalar product and give one physical example.

It is commutative: $\vec{A}\cdot\vec{B} = \vec{B}\cdot\vec{A}$; it is zero for perpendicular vectors ($\theta = 90^{\circ}$). Physical example: work done $W = \vec{F}\cdot\vec{d}$.

Define the vector (cross) product of two vectors, including its magnitude and direction.

$\vec{A}\times\vec{B} = AB\sin\theta\,\hat{n}$, where $\hat{n}$ is the unit vector perpendicular to the plane of $\vec{A}$ and $\vec{B}$ given by the right-hand rule. The result is a vector.

Write the vector product $\vec{A}\times\vec{B}$ as a determinant of components.

$\vec{A}\times\vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{vmatrix}$.

Evaluate $\hat{i}\times\hat{j}$, $\hat{j}\times\hat{k}$, $\hat{k}\times\hat{i}$ and $\hat{i}\times\hat{i}$.

$\hat{i}\times\hat{j} = \hat{k}$, $\hat{j}\times\hat{k} = \hat{i}$, $\hat{k}\times\hat{i} = \hat{j}$, and $\hat{i}\times\hat{i} = \vec{0}$.

Compare the scalar product and the vector product (result type, maximum value condition, commutativity).

Dot product gives a scalar, is maximum when vectors are parallel ($\theta=0^{\circ}$), and is commutative. Cross product gives a vector, is maximum when vectors are perpendicular ($\theta=90^{\circ}$), and is anti-commutative: $\vec{A}\times\vec{B} = -\,\vec{B}\times\vec{A}$.

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

Torque is the turning effect of a force about a pivot: $\vec{\tau} = \vec{r}\times\vec{F}$, with magnitude $\tau = rF\sin\theta$. Its SI unit is the newton metre ($\text{N}\,\text{m}$).

What is the moment arm of a force, and how does it affect torque?

The moment arm is the perpendicular distance from the axis of rotation to the line of action of the force. Torque equals force times moment arm; a larger moment arm produces a larger turning effect.

State the first condition of equilibrium.

The vector sum of all external forces acting on a body must be zero: $\sum \vec{F} = 0$, i.e. $\sum F_{x} = 0$ and $\sum F_{y} = 0$. This ensures no linear (translational) acceleration.

State the second condition of equilibrium.

The sum of all torques about any point must be zero: $\sum \vec{\tau} = 0$. This ensures no angular (rotational) acceleration.

What are the requirements for a body to be in complete (static) equilibrium?

Both conditions must hold simultaneously: $\sum \vec{F} = 0$ (no translational acceleration) and $\sum \vec{\tau} = 0$ (no rotational acceleration). The body is then in translational and rotational equilibrium.

Distinguish between distance and displacement.

Distance is the total path length travelled; it is a scalar and is always positive. Displacement is the straight-line change in position from start to end with direction; it is a vector and can be zero or negative.

Define average velocity and instantaneous velocity.

Average velocity is total displacement over total time, $\vec{v}_{avg} = \dfrac{\Delta \vec{x}}{\Delta t}$. Instantaneous velocity is the velocity at a single instant, $\vec{v} = \dfrac{d\vec{x}}{dt}$, the limit as $\Delta t \to 0$.

Define acceleration and write it in terms of velocity.

Acceleration is the rate of change of velocity with time, a vector: $\vec{a} = \dfrac{\Delta \vec{v}}{\Delta t}$, or instantaneously $\vec{a} = \dfrac{d\vec{v}}{dt}$. Its SI unit is $\text{m}\,\text{s}^{-2}$.

State the three equations of motion for uniform acceleration.

$v = u + at$, $\;\; s = ut + \tfrac{1}{2}at^{2}$, $\;\; v^{2} = u^{2} + 2as$, where $u$ is initial velocity, $v$ final velocity, $a$ acceleration, $s$ displacement and $t$ time.

How are velocity and acceleration obtained from a displacement-time graph and a velocity-time graph respectively?

On a displacement-time graph, velocity is the slope (gradient). On a velocity-time graph, acceleration is the slope and the displacement equals the area under the graph.