MDCAT Physics Syllabus

For a car on a banked road (frictionless), what is the ideal banking angle?

$$\tan\theta = \frac{v^{2}}{rg},$$ so the ideal speed for a bank angle $\theta$ is $v = \sqrt{rg\tan\theta}$.

State Newton's law of universal gravitation.

Every two point masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: $$F = \frac{Gm_{1}m_{2}}{r^{2}}.$$

Give expressions for gravitational acceleration $g$ at Earth's surface and for orbital velocity.

$g = \frac{GM}{R^{2}}$ at the surface. Orbital velocity of a satellite at radius $r$: $v = \sqrt{\frac{GM}{r}}$.

What is escape velocity and its expression from a planet's surface?

The minimum speed needed to escape a planet's gravity without further propulsion: $$v_{e} = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}.$$ For Earth, $v_{e} \approx 11.2\ \text{km/s}$.

How does $g$ vary with altitude $h$ above Earth's surface (for $h \ll R$)?

$$g_{h} = g\left(\frac{R}{R+h}\right)^{2} \approx g\left(1 - \frac{2h}{R}\right).$$ Gravitational acceleration decreases with increasing altitude.

Define simple harmonic motion (SHM) and give its defining equation.

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

Give the time period of a simple pendulum and of a mass-spring system.

Simple pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$. Mass-spring system: $T = 2\pi\sqrt{\frac{m}{k}}$.

For SHM $x = A\sin(\omega t)$, write the velocity and its maximum value.

$v = A\omega\cos(\omega t) = \omega\sqrt{A^{2} - x^{2}}$. Maximum velocity $v_{\max} = A\omega$ occurs at the equilibrium position ($x = 0$).

In SHM, where are kinetic and potential energy maximum, and what is the total energy?

KE is maximum at equilibrium ($x = 0$); PE is maximum at extremes ($x = \pm A$). Total energy is constant: $$E = \tfrac{1}{2}kA^{2} = \tfrac{1}{2}m\omega^{2}A^{2}.$$

Distinguish between transverse and longitudinal waves with examples.

In transverse waves particles vibrate perpendicular to the direction of propagation (e.g. light, waves on a string). In longitudinal waves particles vibrate parallel to propagation, forming compressions and rarefactions (e.g. sound waves).

State the wave equation relating speed, frequency, and wavelength.

$$v = f\lambda,$$ where $v$ is wave speed, $f$ is frequency, and $\lambda$ is wavelength. Also $v = \frac{\lambda}{T}$.

State the principle of superposition and define constructive/destructive interference.

When waves overlap, the resultant displacement equals the vector sum of individual displacements. Constructive interference occurs at path difference $n\lambda$; destructive at $\left(n + \tfrac{1}{2}\right)\lambda$.

What is the condition for stationary (standing) waves and the distance between adjacent nodes?

Standing waves form when two identical waves travel in opposite directions and superpose. Adjacent nodes (or antinodes) are separated by $\frac{\lambda}{2}$; a node and its nearest antinode by $\frac{\lambda}{4}$.

Give the speed of sound in air and how it depends on temperature.

In air at $0^{\circ}\text{C}$, sound travels at about $331\ \text{m/s}$ (~$343\ \text{m/s}$ at room temperature). Speed increases with temperature: $v \propto \sqrt{T}$ (absolute temperature).

State the Doppler effect and the observed frequency formula for a moving source/observer.

The Doppler effect is the apparent change in frequency due to relative motion between source and observer. $$f' = f\left(\frac{v \pm v_{o}}{v \mp v_{s}}\right),$$ frequency rises on approach and falls on recession.

What are beats and the beat frequency?

Beats are periodic variations in loudness from superposition of two waves of slightly different frequencies. Beat frequency $f_{\text{beat}} = |f_{1} - f_{2}|$.

Distinguish between heat and temperature.

Temperature is a measure of the average kinetic energy of the particles of a body (a state property). Heat is the energy transferred between bodies due to a temperature difference (energy in transit).

Convert between Celsius, Kelvin, and Fahrenheit temperature scales.

$T_{K} = T_{C} + 273.15$; $\ T_{F} = \frac{9}{5}T_{C} + 32$. Absolute zero is $0\ \text{K} = -273.15^{\circ}\text{C}$.

Give the equations for linear thermal expansion and specific heat.

Linear expansion: $\Delta L = \alpha L_{0}\Delta T$. Heat for temperature change: $Q = mc\Delta T$, where $c$ is the specific heat capacity.

What is latent heat, and write the equation for a phase change.

Latent heat is the heat absorbed or released during a phase change at constant temperature: $Q = mL$, where $L$ is the specific latent heat (of fusion or vaporization).

State the first law of thermodynamics.

Energy is conserved: the heat added to a system equals the increase in internal energy plus the work done by the system. $$\Delta Q = \Delta U + \Delta W.$$

Describe isothermal and adiabatic processes.

Isothermal: temperature constant ($\Delta U = 0$, so $\Delta Q = \Delta W$), obeying $PV = \text{constant}$. Adiabatic: no heat exchange ($\Delta Q = 0$, so $\Delta U = -\Delta W$), obeying $PV^{\gamma} = \text{constant}$.

Define isobaric and isochoric processes.

Isobaric: constant pressure; work done $W = P\Delta V$. Isochoric (isovolumetric): constant volume; no work is done ($W = 0$), so $\Delta Q = \Delta U$.

State the second law of thermodynamics and Carnot engine efficiency.

Heat cannot spontaneously flow from a colder to a hotter body; no engine can be 100% efficient. Carnot efficiency: $$\eta = 1 - \frac{T_{C}}{T_{H}},$$ with temperatures in kelvin.

State the postulates of the kinetic theory of gases.

Gas consists of many tiny molecules in random motion; molecular volume is negligible; collisions are perfectly elastic; there are no intermolecular forces except during collisions; and average KE is proportional to absolute temperature.

Give the ideal gas law and the kinetic theory expression for pressure.

Ideal gas law: $PV = nRT$. Kinetic pressure: $$P = \frac{1}{3}\frac{Nm\overline{v^{2}}}{V} = \frac{1}{3}\rho\overline{v^{2}}.$$

Relate average kinetic energy of gas molecules and rms speed to temperature.

Average translational KE per molecule: $\overline{KE} = \frac{3}{2}k_{B}T$. Root-mean-square speed: $v_{\text{rms}} = \sqrt{\frac{3k_{B}T}{m}} = \sqrt{\frac{3RT}{M}}$.

State Coulomb's law and the value of the Coulomb constant.

The electrostatic force between two point charges is $$F = k\frac{q_{1}q_{2}}{r^{2}}, \quad k = \frac{1}{4\pi\varepsilon_{0}} \approx 9 \times 10^{9}\ \text{N·m}^{2}/\text{C}^{2}.$$

Define electric field intensity and give the field of a point charge.

Electric field is force per unit positive test charge: $\vec{E} = \frac{\vec{F}}{q_{0}}$, in $\text{N/C}$ or $\text{V/m}$. For a point charge: $E = k\frac{q}{r^{2}}$, directed radially.

State Gauss's law.

The total electric flux through a closed surface equals the enclosed charge divided by $\varepsilon_{0}$: $$\Phi_{E} = \oint \vec{E}\cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_{0}}.$$

Define capacitance and give the capacitance of a parallel-plate capacitor.

Capacitance is charge stored per unit potential: $C = \frac{Q}{V}$, in farads. Parallel-plate capacitor: $$C = \frac{\varepsilon_{0}\varepsilon_{r}A}{d},$$ where $A$ is plate area and $d$ their separation.

Give the equivalent capacitance for capacitors in series and in parallel, and the energy stored.

Series: $\frac{1}{C_{\text{eq}}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \cdots$. Parallel: $C_{\text{eq}} = C_{1} + C_{2} + \cdots$. Energy stored: $U = \frac{1}{2}CV^{2} = \frac{Q^{2}}{2C}$.