Physics I – Mechanics (UNT-FACET) Comprehensive Notes

Introduction to Science and Physics

  • Science (from Latin scientia) seeks ordered explanations of natural phenomena using observation, experimentation and logical reasoning (Scientific Method).
  • Branches of Natural Science:
    • Life Sciences: Biology (Zoology, Botany).
    • Physical Sciences: Geology, Astronomy, Chemistry, Physics.
  • Physics (from Greek physis – nature) is the foundational physical science: studies matter, energy, motion, forces, light, electricity, atomic and nuclear structure.
  • Engineering applies principles of Physics & Chemistry to solve real-world problems.

Scientific Method & Models

  • No rigid recipe; includes observation, hypothesis, experimentation, prediction, communication.
  • Physicists build idealised mathematical models to simplify complex reality.
    • Example: ignoring air resistance when modelling a football’s flight.
  • Models are incomplete ➔ must match experiment within accepted uncertainty.

Measurement, SI Units & Dimensions

  • Measurement = interaction of object, apparatus and standard unit.
  • Physical quantity = number + unit.
  • Dimension exists when numerical value depends on chosen unit.
  • International System (SI) base units used in Mechanics
    • Length: metre (\text m) defined via speed of light.
    • Mass: kilogram (\text{kg}) defined via Planck constant h.
    • Time: second (\text s) via Cs-133 hyperfine frequency \Delta\nu_{\text{Cs}}.
  • Derived units: \text m\,\text s^{-1} (velocity), \text N = \text{kg}\,\text m\,\text s^{-2} (force), \text{Pa} = \text N\,\text m^{-2} (pressure), \text J = \text N\,\text m (energy).
  • Argentine SIMELA identical to SI but allows ^{\circ}\text C.

Scientific Notation & Significant Figures

  • Numbers expressed as a \times 10^{n} with one non-zero integer digit (1 \le a < 10).
  • Significant figures: all certain digits + first uncertain digit.
  • Absolute error e{\text{abs}} = |m{\text{real}}-m_{\text{med}}|.
  • Relative error \varepsilon =\dfrac{e{\text{abs}}}{m{\text{med}}} (often %).

Error Propagation

  • For product/quotient, relative errors add.
  • For sum/difference, absolute errors add (rounded to least precise decimal place).

Dimensional Homogeneity

  • Equation valid only if both sides have identical dimensions.
  • Convert all units to SI before computation.

Kinematics in One Dimension

Position, Displacement & Distance

  • Vector position \vec r(t)=x\hat i + y\hat j + z\hat k.
  • Displacement \Delta \vec r = \vec r2-\vec r1 (vector) vs. distance (scalar path length).

Average & Instantaneous Velocity

  • \vec v_m = \dfrac{\Delta \vec r}{\Delta t}.
  • Instantaneous \vec v=\dfrac{d\vec r}{dt}, tangent to trajectory.

Average & Instantaneous Acceleration

  • \vec a_m = \dfrac{\Delta \vec v}{\Delta t}, \vec a=\dfrac{d\vec v}{dt}.

Special Motions

  1. Uniform Rectilinear Motion (MRU)
    • v=\text{cte}, x = x0 + v(t-t0).
  2. Uniformly Accelerated Rectilinear Motion (MRUV)
    • a=\text{cte}, v = v0 + a(t-t0),
    • x = x0 + v0 t + \tfrac12 a t^{2},
    • v^{2}=v_0^{2}+2a\Delta x.

Dynamics: Newton’s Laws

  1. Inertia: \vec F_{\text{res}}=0 \Rightarrow \vec v=\text{cte} (defines inertial frames).
  2. \vec F_{\text{res}} = m\vec a. Force measured in \text N.
  3. Action–reaction: \vec F{AB}=-\vec F{BA} (acts on different bodies).

Weight

  • Near Earth \vec P = m\vec g,\; g\approx 9.81\,\text{m s}^{-2}.

Contact Forces & Friction

  • Normal \vec N perpendicular to surface.
  • Static friction f{\text s}\le \mu{\text s}N; maximal f{\text s,max}=\mu{\text s}N.
  • Kinetic friction f{\text k}=\mu{\text k}N (\mu{\text k}

Systems with Strings & Pulleys

  • Ideal rope: massless, inextensible, same tension T throughout.
  • Ideal pulley: massless, frictionless; only changes direction of T.

Work, Energy & Power

  • Work by constant force: W = \vec F\cdot\Delta\vec r = F\Delta r\cos\alpha.
  • Variable force: W = \displaystyle\int{r1}^{r_2} \vec F\cdot d\vec r.
  • Kinetic energy: E_k = \tfrac12 mv^{2}.
  • Work–Energy theorem: W{\text{net}} = \Delta Ek.
  • Potential energies:
    • Gravitational near Earth: E_{pg}=mgz.
    • Elastic (Hooke): E_{pe}=\tfrac12 kx^{2}.
  • Conservative force: \vec F = -\nabla E_p, path-independent work.
  • Mechanical energy Em=Ek+\sum E_p; conserved if only conservative forces act.
  • Power: P = \dfrac{dW}{dt}=\vec F\cdot\vec v (unit \text W = \text{J s}^{-1}).

Impulse & Linear Momentum

  • Momentum \vec p = m\vec v.
  • Impulse \vec J = \displaystyle\int{t1}^{t_2} \vec F dt = \Delta \vec p.
  • For constant \vec F, \vec J = \vec F\,\Delta t.

Systems of Particles

  • Center of mass: \vec r{\text{cm}} = \dfrac{\sum mi \vec ri}{\sum mi}.
  • \vec p{\text{sys}} = M\vec v{\text{cm}}, \displaystyle\frac{d\vec p{sys}}{dt}=\sum \vec F{\text{ext}}.
  • Internal forces cancel in pairs (Newton III), thus only external forces change \vec p_{\text{sys}}.
  • Linear momentum conserved in isolated system.

Collisions in 1-D

  • Perfectly elastic: Ek & p conserved. u1 = \frac{(m1-m2)v1+2m2v2}{m1+m2},\quad u2 = \frac{(m2-m1)v2+2m1v1}{m1+m_2}.
  • Perfectly inelastic: bodies stick; common velocity u=\dfrac{m1v1+m2v2}{m1+m2}.
  • Coefficient of restitution e=\dfrac{u2-u1}{v1-v2} (0\le e\le1).

Plane Motion: Projectile (Tiro Parabólico)

  • Decompose v0 into v{0x}=v0\cos\alpha, v{0y}=v_0\sin\alpha.
  • Trajectory y(x)=\tan\alpha\,x-\dfrac{g}{2v_0^{2}\cos^{2}\alpha}\,x^{2} (parabola).
  • Range on level ground x{\max}=\dfrac{v0^{2}\sin2\alpha}{g} (max at \alpha=45^{\circ}).
  • Time of flight t=\dfrac{2v_0\sin\alpha}{g}.

Circular Motion

Uniform Circular Motion (MCU)

  • |\vec v|=\text{cte},\; v=r\,\omega.
  • Centripetal acceleration a_c=\dfrac{v^{2}}{r}=\omega^{2}r to centre.
  • Force \vec Fc = m\vec ac.

Non-Uniform (MCUV)

  • Tangential acceleration a_t = \alpha r (\alpha=d\omega/dt).
  • Total \vec a = \vec ac + \vec at, magnitude a=\sqrt{ac^{2}+at^{2}}.

Angular Momentum & Torque

  • For particle: \vec L = \vec r\times\vec p.
  • Torque \vec M = \vec r\times\vec F = \dfrac{d\vec L}{dt}.
  • Isolated system (\sum \vec M_{\text{ext}}=0) ➔ \vec L=\text{cte}.

Rigid-Body Dynamics

Moment of Inertia I

  • I = \sum miri^{2}\;\; (\text{discrete}),\quad I=\int r^{2}\,dm (continuous).
  • Examples about CM axis
    • Thin ring I=MR^{2}.
    • Solid cylinder/disc I=\tfrac12 MR^{2}.
    • Solid sphere I=\tfrac25 MR^{2}.
    • Thin rod (axis through center) I=\tfrac1{12}ML^{2}.
  • Parallel-axis (Steiner): I = I_{\text{cm}}+Md^{2}.

Pure Rotation about Fixed Axis

  • \sum M_{\text{ext}} = I\alpha (rotational analogue of F=ma).
  • Rotational kinetic energy E_{k,rot}=\tfrac12 I\omega^{2}.
  • Work–Energy rot: W=\Delta E_{k,rot}.

General Plane Motion (translation + rotation)

  • Total Ek=\tfrac12 Mv{\text{cm}}^{2}+\tfrac12 I_{\text{cm}}\omega^{2}.
  • Total \vec L = \vec r{\text{cm}}\times M\vec v{\text{cm}} + I_{\text{cm}}\,\vec \omega (orbital + spin).

Rolling Without Slipping (velocity of contact point zero)

  • Condition v_{\text{cm}} = \omega R.
  • Down an incline (cylinder): a=\dfrac{g\sin\theta}{1+I_{\text{cm}}/(MR^{2})}.

Systems of Identical Masses (Machine of Atwood)

  • Two masses m1,m2 linked over pulley of inertia I, radius R:
    a = \dfrac{(m2-m1)g}{m1+m2+I/R^{2}}.

Key Take-aways

  • Always choose an inertial reference frame and suitable coordinates to simplify.
  • Check dimensional homogeneity and significant figures.
  • For particle systems isolate the body (DCL) and apply Newton II + kinematic links.
  • Use conservation laws (\vec p, \; E, \; \vec L) whenever external influences are zero or forces are conservative.
  • In rigid bodies distinguish translation ( analyses with F=ma) from rotation ( analyse with \tau=I\alpha).
  • Rolling objects combine both energies; slipping prevented by static friction.
  • Experimental accuracy hinges on unit consistency, error analysis and reporting correct significant figures.