Physics 1-T Final Exam Comprehensive Notes

Vector and Scalar Quantities

• Vector ≡ quantity with both magnitude and direction

  • Canonical examples: displacement $\vec d$, velocity $\vec v$, acceleration $\vec a$, force $\vec F$, momentum $\vec p$, impulse $\vec J$, electric field $\vec E$, magnetic field $\vec B$.
  • Significance: vectors must obey head-to-tail (graphical) or component (analytical) addition. Direction determines physical effect (e.g.
  • Opposite forces cancel.
  • Parallel vectors may be added/subtracted algebraically.)

• Scalar ≡ quantity with magnitude only

  • Common examples: distance $d$, speed $v_{\text{speed}}$, mass $m$, energy $E$, work $W$, power $P$, temperature $T$, electric potential $V$, charge magnitude $|q|$, time $t$.

• Practical tip: when vectors are parallel / antiparallel (share line of action) you may simply add or subtract their magnitudes with sign. At any other relative angle, draw a diagram and use components or the Law of Cosines.

Measurement, Significant Figures, & Units

• Magnitude → the size/amount of a physical quantity (always positive).
• Significant–figure (sig-fig) count examples:
  • $97\;\text{m}$ → 2 sig figs.
  • $97.0\;\text{m}$ → 3.
  • $0.97\;\text{m}$ → 2.
  • $0.0097\;\text{m}$ → 2 (leading zeros are placeholders).
  • $970\;\text{m}$ → 2 (trailing zero not significant without a decimal point).
• If unsure of units, dimensional analysis (treat units like algebra) will restore correct dimensions.
• Slope of a graph = $\dfrac{\Delta y}{\Delta x}$ ⇒ always interpret by reading the axis labels and units.

Vector & Free-Body Diagrams (FBD)

• Vector diagram: sketch of all vectors (displacements, velocities, etc.) tip-to-tail; resultant drawn from start to finish.
• Free-body diagram: isolate one object; draw and label every real force acting now (gravitational $\vec W$, normal $\vec N$, friction $\vec f$, tension $\vec T$, applied $\vec F_{\text{app}}$, electric $q\vec E$, magnetic $q\vec v\times\vec B$, etc.). Usually begin with weight $\vec W$ because it is almost always present and points toward Earth.
  • Velocities and accelerations are placed off to the side—they are not forces.
• Normal = “perpendicular” to a surface. Friction acts parallel to the surface, opposite the direction of (impending) motion.

Fundamental Conservation Laws

• A quantity is “conserved” when its total remains constant in an isolated system.
  • Throughout Physics 1-T we used conservation of momentum $\sum\vec p = \text{const}$ and energy $\sum E = \text{const}$.

Kinematics & Dynamics Essentials

• Average speed $v<em>{\text{avg}} = \dfrac{d</em>{\text{total}}}{t_{\text{total}}}$.
• “Rates” hierarchy (all take a time derivative):
  1. Speed: $v = \dfrac{\mathrm d d}{\mathrm d t}$.
  2. Acceleration: $a = \dfrac{\mathrm d v}{\mathrm d t}$ (rate of velocity change).
  3. Power: $P = \dfrac{\mathrm d E}{\mathrm d t}$.
  4. Current: $I = \dfrac{\mathrm d Q}{\mathrm d t}$.
  5. Net force: $\vec F_{\text{net}} = \dfrac{\mathrm d \vec p}{\mathrm d t}$ (Newton’s Second Law in its most general, momentum form).
• Weight = gravitational force: $\vec W = m\vec g$.
• Newton’s 3rd Law: forces between two interacting bodies are equal in magnitude, opposite in direction.

Mass & Force Comparisons

• Same force on masses $M$ and $4M$ ⇒ accelerations are inversely proportional to mass; $a<em>{M} = 4 a</em>{4M}$.
• Same force on equal-mass charges +$Q$ and –$4Q$: acceleration magnitudes are equal (direction may differ if fields differ).

Friction & Applied-Force Scenarios (55 kg block)

1. No friction: push with $245\,\text N$ → block accelerates $a = \dfrac{F}{m} = \dfrac{245}{55.0} \approx 4.45\,\text{m/s}^2$.
2. Encounter rough patch: friction $225\,\text N$ opposite motion.
   • Continue to push 245 N ⇒ net $F = 245-225 = 20\,\text N$ ⇒ smaller acceleration.
   • Push 225 N ⇒ net $F = 0$ ⇒ constant velocity.
   • Push 175 N ⇒ net $F = -(225-175) = -50\,\text N$ ⇒ deceleration.
   • Let go ⇒ only friction acts; object slows over time, not instantaneously.

Spring–Piston Two-Mass System (M & 1.5 M)

• Work done compressing spring: $W=25\,\text J$ ⇒ potential energy stored $U_s=25\,\text J$.
• When lock is released on frictionless surface:
  • Internal spring force equal & opposite on each mass (Newton III).
  • Smaller mass $M$ gains larger acceleration.
  • Final speeds differ (inverse mass ratio); directions opposite.
• Conserved: total mechanical energy (25 J) & total momentum of the system.
• Post-interaction energy form: pure kinetic $E_k = 25\,\text J$—converted from spring potential.

Momentum, Impulse & Work Example (3.2 kg block)

• $\Delta v = 2.4-1.9 = 0.5\,\text{m/s West}$, so
  • \Delta \vec p = m\Delta \vec v = 3.2\times 0.5 = 1.6\,\text{kg·m/s West}.
  • \Delta KE = \tfrac12 m (vf^2-vi^2) = \tfrac12\times 3.2(2.4^2-1.9^2) \approx 0.4\,\text J}.
  • Impulse $\vec J = \Delta\vec p$; Net work $W_{\text{net}} = \Delta KE$.

Energy & Momentum Scaling Relations

• Reduce speed to $\tfrac15 v_0$:
  • $p \propto v$ ⇒ $p<em>{\text{new}} = \tfrac15 p</em>0$.
  • $KE \propto v^2$ ⇒ $KE<em>{\text{new}} = (\tfrac15)^2 KE</em>0 = \tfrac1{25} KE_0$.
• Halve mass and triple speed: $KE \propto m v^2$ ⇒ $KE<em>{\text{new}} = (\tfrac12)(3^2) KE</em>0 = 4.5 KE_0$.
• Uniform circular motion: $a<em>c = \dfrac{v^2}{r}$. If $v\to\tfrac14 v</em>0$ and $r\to 2r<em>0$ ⇒ $a</em>c\to\dfrac{(\tfrac14 v<em>0)^2}{2r</em>0}=\tfrac1{32} a_{c0}$.

Gravitation & Electrostatics Scaling

• Newton: $F<em>g = G\dfrac{m</em>1 m<em>2}{r^2}$. Coulomb: $F</em>e = k\dfrac{|q<em>1 q</em>2|}{r^2}$.
• Separation $r\to\tfrac32 r$ ⇒ $F\to (\tfrac23)^2 F<em>0 = \tfrac49 F</em>0$.
• Separation $r\to 6r$, mass $m<em>1\to\tfrac12 m</em>1$ ⇒ $F\to (\tfrac12)(\tfrac1{36}) F<em>0 = \tfrac1{72} F</em>0$.
• Tripling both charges and halving separation: $F\to (3\cdot3)(\tfrac12)^{-2}F<em>0 = 36F</em>0$.
• Electric field between parallel plates is uniform, magnitude $E = \dfrac{V}{d}$, direction from + to –.
  • Replacing a test charge does not change the field. Force scales with charge: $\vec F = q \vec E$. Acceleration $a = \dfrac{F}{m} = \dfrac{qE}{m}$ ⇒ halving mass & dividing charge by 3 doubles acceleration.

Circuit Principles

• Series resistors: $R_{\text{eq}} = \sum R$, same current. Adding two identical resistors in series triples total resistance ⇒ current drops to $\tfrac13$.
• Parallel resistors: equal voltage across each branch. Each path must drop the full source voltage before charges return to the battery.
• Power: $P = IV = \dfrac{V^2}{R} = I^2 R$. Quadrupling $V$ increases power by factor $4^2 = 16$ (since $P\propto V^2$).
• Junction rule (Kirchhoff’s): current entering = current leaving (charge conservation).

Waves & Optics

• Wave speed (no dispersion): $v = f\lambda$ determined solely by medium properties (e.g.
  string tension & mass per unit length: $v = \sqrt{\tfrac{T}{\mu}}$).
• Frequency is fixed by the source (oscillating object).
• If $f\to \tfrac54 f<em>0$ with same medium, $\lambda\to \tfrac{4}{5}\lambda</em>0$ (inverse proportionality).
• Defining wave behaviors: reflection, refraction, diffraction, interference.
• All incident, reflected, and refracted angles are measured with respect to the normal.
• Reality checks: speeds > $c$ or negative discriminants under radicals signal algebraic mistakes—re-check.

Lenses & Refraction Quick Reference

• Converging lens: thick in center, positive focal length, can form real inverted images (object outside $f$) or virtual upright images (object inside $f$), used in magnifiers, cameras.
• Diverging lens: thin center, negative focal length, always forms smaller virtual upright images, used in peepholes, laser beam expanders.
• Sense-making questions:
  • Is the second index of refraction higher or lower? Light slows in higher index (bends toward normal) and speeds up in lower index.
  • Do computed angles obey Snell’s Law $n<em>1\sin\theta</em>1 = n<em>2\sin\theta</em>2$ and lie between $0^\circ$ and $90^\circ$?

Magnetic Phenomena

• Magnetic forces act on moving charges, current-carrying wires, and magnetic dipoles.
• All magnetic fields originate from moving charges (currents) or intrinsic spin of electrons (microscopic currents).

Strategy & Heuristics

1. Always start physics problems with a clear diagram (vectors, FBD, circuit, rays, waves). It prevents sign errors.
2. Check limiting cases (e.g. friction = 0, mass → ∞) to test formulas.
3. Keep units throughout—catches algebra errors early.
4. Conservation laws often replace kinematics/dynamics when forces are unknown.
5. Never leave a calculation with more significant figures than the given data justify.

Common Formula Sheet (Physics 1-T)

• $\vec v_{\text{avg}} = \dfrac{\Delta \vec r}{\Delta t}$, $\vec a = \dfrac{\Delta \vec v}{\Delta t}$.
• $\vec F_{\text{net}} = m\vec a$.
• Work: $W = \vec F \cdot \vec d = Fd\cos\theta$.
• Kinetic energy: $KE = \tfrac12 m v^2$.
• Potential (spring): $U_s = \tfrac12 k x^2$.
• Potential (gravity near Earth): $U<em>g = mgh$; universal gravity: $U</em>g = -G\dfrac{m<em>1 m</em>2}{r}$.
• Power: $P = \dfrac{W}{t} = IV = I^2R = \dfrac{V^2}{R}$.
• Impulse–momentum: $\vec J = \int \vec F\,dt = \Delta \vec p$.
• Centripetal acceleration: $a_c = \dfrac{v^2}{r}$.
• Coulomb’s Law: $F = k\dfrac{|q<em>1 q</em>2|}{r^2}$.
• Ohm’s Law: $V = IR$.
• Snell’s Law: $n<em>1\sin\theta</em>1 = n<em>2\sin\theta</em>2$.
• Lens/Mirror equation: $\dfrac1f = \dfrac1d<em>o + \dfrac1d</em>i$.

Final Reminders

• Weight is just another name for gravitational force (appears three times on the review!).
• Three ways to accelerate: speed up, slow down, change direction.
• Uniform electric field exists only between parallel plates with opposite charges.
• Direction of electrostatic force: like charges repel, unlike attract.
• Series circuits share current; parallel circuits share voltage.
• Any speed claim > $c$ or negative under a square root is unphysical → re-evaluate.