Physics Study Notes: Momentum, Rotational Dynamics, Fluids, and Oscillations

Linear Momentum and Collisions

Linear Momentum

Linear momentum is a measure of how fast and heavy an object is, defined as the product of mass and velocity. It is a vector quantity and plays a central role in analyzing motion and collisions.

• Definition: $\vec{p} = m \vec{v}$

• Newton's Second Law (Momentum Form): $\frac{d\vec{p}}{dt} = \vec{F}_{\text{net}}$

• Momentum Conservation: In an isolated system (no net external force), total momentum is conserved: $\vec{p}_{\text{tot}} = \text{const.}$

• Applications: Conservation of momentum is especially useful for analyzing collisions and explosions, where external forces are negligible compared to internal forces.

Types of Collisions

• Totally Inelastic Events: Objects stick together after collision; only momentum is conserved, not kinetic energy. Examples: Ballistic pendulum, fusion reaction, radioactive decay, explosions.

• Partially Inelastic Events: Objects deform but do not stick; only momentum is conserved, but kinetic energy changes less than in totally inelastic events.

• Elastic Events: Hard objects collide and separate; both momentum and kinetic energy are conserved. Examples: Billiard balls, hockey pucks. $K_i = K_f$

Impulse and Center of Mass

Impulse

Impulse quantifies the effect of a force acting over a time interval, resulting in a change in momentum.

• Definition: $\vec{J} = \Delta \vec{p} = \int \vec{F}_{\text{net}} dt$

• Impulse allows estimation of average internal force during brief events.

Center of Mass

The center of mass is the unique point in a system of particles where Newton's second law applies as if all mass were concentrated at that point.

• Newton's Second Law for Center of Mass: $\vec{F}_{\text{net, ext}} = M \vec{a}_{\text{cm}}$

• Position of Center of Mass:

  • For point masses: $\vec{r}_{\text{cm}} = (\sum m_i \vec{r}_i)/M$

  • For continuous mass distribution: $\vec{r}_{\text{cm}} = (\int dm \vec{r})/M$

Variable Mass Systems

• Analyzed like a collision between $m$ and $dm$.

• Newton's Second Law for Variable Mass: $\vec{F}_{\text{net, ext}} = m \frac{d\vec{v}}{dt} - \vec{v}_r \frac{dm}{dt}$

• Direction and rate of mass change can affect the net force.

Rotational Dynamics

Rotational Motion

Rigid objects rotating about a fixed axis or an axis through the center of mass exhibit rotational motion, described by angular variables.

• Angular Position: $\theta(t)$

• Angular Velocity: $\omega(t) = d\theta/dt$

• Angular Acceleration: $\alpha = d\omega/dt = d^2\theta/dt^2$

• If $\alpha$ is constant, kinematic equations apply.

• Linear-Angular Relationships:

  • Displacement: $s = R\theta$

  • Velocity: $v = R\omega$

  • Tangential Acceleration: $a_{\text{tan}} = R\alpha$

Torque and Rotational Inertia

• Torque: $\tau = r F \sin\theta$

• Lever Arm: Perpendicular distance from rotation axis to line of action of force.

• Rotational Inertia (Moment of Inertia):

  • For point masses: $I = \sum m r^2$

  • For continuous mass: $I = \int dm\, r^2$

• Newton's Second Law for Rotation: $\tau_{\text{net}} = I \alpha$

Parallel and Perpendicular Axis Theorems

• Parallel Axis Theorem: $I_A = I_{\text{cm}} + M d^2$

• Perpendicular Axis Theorem: $I_z = I_x + I_y$ (for planar objects)

Energy in Rotational Motion

• Rotational Kinetic Energy: $K_{\text{rot}} = I \omega^2 / 2$

• Work-Kinetic Energy Theorem: $\int \tau_{\text{net}} d\theta = (\omega_f^2 - \omega_i^2)/2$

Rolling Without Slipping

• For rigid objects, a single point of contact with zero net velocity ensures static friction and normal force do no work.

• Mechanical energy is conserved if $W_{\text{nc}} = 0$:

• $\Delta U_g + \Delta K_{\text{tr}} + \Delta K_{\text{rot}} = 0$

Angular Momentum and Equilibrium

Angular Momentum

• For a symmetric object: $\vec{L} = I \vec{\omega}$

• Time Rate of Change: $\frac{d\vec{L}}{dt} = \vec{\tau}_{\text{ext}}$

• For a point mass: $\vec{L} = \vec{r} \times \vec{p}$

• Projection along a direction: $L = r p \sin\theta$

Angular Momentum Conservation

• If $\vec{\tau}_{\text{ext}} = 0$, then $\vec{L}_{\text{tot}} = \text{const.}$

• Only forces perpendicular to the axis of rotation can supply torque.

Static Equilibrium

• For rest, both translational and rotational motion must be zero:

• $\vec{F}_{\text{net}} = 0$

• $\vec{\tau}_{\text{net}} = 0$

• Pivot point can be chosen arbitrarily for torque calculations.

Equilibrium Stability

• Stable Equilibrium: System returns to original position after disturbance (e.g., simple pendulum).

• Unstable Equilibrium: System moves away from original position (e.g., ball at top of hill).

• Neutral Equilibrium: System stays in new position (e.g., ball on flat table).

Fluids and Fluid Dynamics

Fluids, Density, and Pressure

• Fluids: Include liquids and gases; intermolecular forces are weaker than in solids.

• Density: $\rho = m/V$

• Liquids are considered incompressible ($\rho = \text{const}$).

• Pressure: $P = F/A$ (scalar, independent of direction)

• Gauge Pressure: $P_{\text{gauge}} = P_{\text{abs}} - P_{\text{atm}}$

Equation of Fluid Statics

• For constant density liquids, pressure varies with depth:

• $P = P_0 + \rho g h$

Pascal's Principle

• A change in pressure applied to a confined incompressible fluid is transmitted undiminished throughout the fluid.

• Applications: Hydraulic jacks, braking systems.

Buoyant Force and Archimedes' Principle

• Buoyant Force: Upward force on objects in contact with a fluid, due to pressure variation with depth.

• Archimedes' Principle: Buoyant force equals the weight of fluid displaced by the object.

Fluid Dynamics

• Assume no viscosity and no turbulence for ideal fluids.

• Flow velocity described by streamlines (continuous lines tangent to velocity vectors).

• Volume Flow Rate: $Q_v = \Delta V/\Delta t = Av$

• Mass Flow Rate: $Q_m = \Delta m/\Delta t = \rho Av$

Continuity Equation

• Mass conservation along a flow tube: $\rho A v = \text{const}$

Bernoulli's Equation

• Energy conservation for incompressible fluids (y-axis upward):

• $P + \rho g y + \rho v^2/2 = \text{const.}$

Oscillatory Motion and Waves

Simple Harmonic Motion (SHM)

• Restoring force proportional to displacement: $\frac{d^2 x}{dt^2} + \omega^2 x = 0$

• General solution: $x(t) = A \cos(\omega t + \phi)$

• Parameters:

  • $A$ = amplitude

  • $\omega = 2\pi f$ = angular frequency

  • $f = 1/T$ = frequency

  • $T$ = period

  • $\phi$ = phase shift

• For mass-spring system: $\omega^2 = k/m$

Energy in SHM

• Kinetic energy: $K = m v^2 / 2$ (oscillates as $\sin^2$)

• Elastic potential energy: $U_{el} = k x^2 / 2$ (oscillates as $\cos^2$)

• Mechanical energy: $E = K + U_{el} = kA^2/2 = \text{const.}$

Other Types of SHO

• Torsional Pendulum: $\frac{d^2 \theta}{dt^2} + (\kappa/I) \theta = 0$, $\omega^2 = \kappa/I$

• Simple Pendulum (small angle): $\frac{d^2 \theta}{dt^2} + (g/l) \theta = 0$, $\omega^2 = g/l$

• Physical Pendulum (small angle): $\frac{d^2 \theta}{dt^2} + (mgL/I) \theta = 0$, $\omega^2 = mgL/I$

Damped Harmonic Oscillator

• General equation: $\frac{d^2 x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega^2 x = 0$

• Underdamped: $x(t) = A e^{-\gamma t} \cos(\omega' t + \phi)$, $\omega' = \sqrt{\omega^2 - \gamma^2}$

• Critically damped: System returns to equilibrium fastest without oscillating.

• Overdamped: System returns to equilibrium slowly, no oscillation.

Harmonic Waves

• Mechanical waves require a medium; described by periodic functions in time and space.

• Wave speed: $v = \lambda f$

• Waves on a Rope: $v = \sqrt{F_T/\mu}$

• Wave function (x-direction): $D(x,t) = A \sin(kx - \omega t + \phi)$

Standing Waves

• Result from superposition of two waves of equal amplitude and frequency traveling in opposite directions.

• Standing wave equation: $D_{\text{tot}}(x,t) = 2A \sin(kx) \cos(\omega t)$

• For rope of length $L$ clamped at both ends: $\lambda_n = 2L/n$, $n$ = positive integer

• For rope clamped at one end: $\lambda_{2n+1} = 4L/(2n+1)$, $n$ = positive integer

Additional info: These notes cover core topics from chapters on momentum, rotational dynamics, fluids, and oscillatory motion, providing definitions, equations, and examples relevant for college-level physics.