Physics 211: Gravity, Torque, Rotational Dynamics, and Angular Momentum

Comprehensive practice vocabulary flashcards covering Newton's law of gravitation, planetary motion, torque, static equilibrium, center of mass, moment of inertia, rolling mechanics, and angular momentum conservation from Physics 211 lectures.

Newton's Law of Universal Gravitation

The law stating that any two objects attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: $F = G \frac{m_1 m_2}{r^2}$, where $G = 6.6743 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$.

Inverse-Square Law

A geometric relationship where the intensity of a field or force radiating uniformly in three-dimensional space decreases in proportion to the surface area of an expanding sphere, scaling as $\frac{1}{r^2}$.

Gravitational Field Strength ($g$)

The free-fall acceleration produced by a body of mass $M$ at radial distance $r$, calculated as $g = \frac{GM}{r^2}$, which evaluates to approximately $9.82\,\text{m/s}^2$ at Earth's average surface radius ($r = 6{,}371\,\text{km}$).

Shell Theorem

A principle derived from Gauss's law stating that inside a spherical body, the net gravitational force exerted by all mass at radii greater than the observer's position is zero, meaning only the enclosed mass below contributes: $g = \frac{G M_{\text{below}}}{r_{\text{below}}^2}$.

Preliminary Reference Earth Model (PREM)

A radial density model of Earth demonstrating that gravitational acceleration does not decrease linearly below the crust, but instead peaks at $10.7\,\text{m/s}^2$ near the core-mantle boundary before dropping to zero at the center.

Apparent Weightlessness in Orbit

The condition of zero perceived weight experienced in orbit (e.g., at the ISS, $h = 400\,\text{km}$, where $g \approx 8.7\,\text{m/s}^2$) because gravity provides the exact centripetal acceleration required for both the station and astronauts, causing them to fall together with zero contact normal force.

Gravitational Potential Energy ($U(r)$)

The potential energy of a two-body system interacting via gravity across radial distance $r$, defined as $U(r) = -\frac{GMm}{r}$, which is negative everywhere and approaches zero as separation distance $r \rightarrow \infty$.

Escape Velocity ($v_{\text{escape}}$)

The minimum speed required at launch to escape a celestial body's gravitational field with zero residual kinetic energy at infinity, given by $v_{\text{escape}} = \sqrt{\frac{2GM}{r_E}}$ ($\approx 11{,}200\,\text{m/s}$ for Earth).

Space Elevator

A conceptual transport mechanism anchored to the equator and extending past geostationary orbit ($35{,}786\,\text{km}$) that delivers payloads using ground-supplied power, requiring $\Delta U = 53\,\text{MJ/kg}$ ($85\%$ of Earth's total escape energy).

Orbital Speed (Circular Orbit)

The tangential velocity necessary to sustain a circular orbit around a central body of mass $M$ at orbital radius $r$, given by $v = \sqrt{\frac{GM}{r}}$ ($\approx 7{,}700\,\text{m/s}$ for the ISS at $h = 400\,\text{km}$).

Orbital Period ($T$)

The elapsed time for a satellite to execute one full orbit, derived from Kepler's third law as $T = \frac{2\pi}{\sqrt{GM}} r^{3/2}$ ($\approx 5{,}500\,\text{s} \approx 1.5\,\text{hours}$ for the ISS).

Torque ($\vec{\tau}$)

The rotational equivalent of force that produces an angular acceleration $\vec{\alpha}$, defined mathematically by the vector cross product $\vec{\tau} = \vec{r} \times \vec{F}$, with scalar magnitude $\tau = rF\sin(\phi)$.

Right-Hand Rule for Vector Cross Products

A geometric rule to determine the direction of $\vec{a} \times \vec{b}$, where the fingers of the right hand curl from vector $\vec{a}$ toward vector $\vec{b}$ and the outstretched thumb points in the resulting perpendicular direction.

Aircraft Rotational Degrees of Freedom

The three independent rotational modes of an aircraft about orthogonal axes: roll (rotation about the longitudinal fuselage axis), pitch (nose up/down rotation), and yaw (nose left/right rotation).

Horizontal Stabilizer (Airplane Tail)

A tail surface designed to generate an aerodynamic downward force ($\vec{F}_{\text{tail}}$), producing a torque that balances the gravitational torque around the wing lift axis to maintain steady pitch equilibrium.

Center of Mass ($\vec{r}_{\text{cm}}$)

The mean location of a body's total mass distribution, determined by the first moment of mass: $x_{\text{cm}} = \frac{1}{M}\int x\rho(x)\,dx$ for continuous bodies, or $\vec{r}_{\text{cm}} = \frac{\sum m_i \vec{r}_i}{M}$ for discrete systems.

Static Equilibrium

The state of a stationary extended object remaining completely at rest without linear or rotational acceleration, which requires both $\vec{F}_{\text{net}} = 0$ and $\vec{\tau}_{\text{net}} = 0$.

Stable Equilibrium

An equilibrium state where a slight displacement generates a restorative force returning the body to its original position, coinciding with a local minimum in the potential energy curve ($\frac{dU}{dx} = 0$).

Moment of Inertia ($I$)

The rotational analogue of mass quantifying an object's resistance to angular acceleration about an axis, calculated for discrete masses as $I = \sum m_i r_i^2$ and for continuous bodies as $I = \int r^2\,dm$.

Shape Factor ($\beta$)

The dimensionless geometric multiplier in the moment of inertia expression $I = \beta MR^2$, equal to $1$ for a cylindrical shell/hoop, $\frac{1}{2}$ for a solid cylinder/disk, and $\frac{2}{5}$ for a solid sphere.

Parallel-Axis Theorem

A theorem stating that the moment of inertia $I$ around any axis parallel to an axis passing through the center of mass is equal to $I = I_{\text{cm}} + Md^2$, where $d$ is the perpendicular distance between the two axes.

Rotational Kinetic Energy ($K_r$)

The kinetic energy associated with the rotation of an object about an axis, defined by $K_r = \frac{1}{2}I\omega^2$.

Rolling Without Slipping

A combined motion where a rolling body rotates while translating such that the contact point with the surface is instantaneously at rest ($v = 0$), enforcing the linear-angular constraint $v = r\omega$.

Rotational-to-Translational Energy Ratio

The constant proportion between rotational and center-of-mass translational kinetic energies for an object rolling without slipping, which identically matches the shape factor: $\frac{K_r}{K_t} = \beta$.

Acceleration of a Rolling Object on an Incline

The linear acceleration of an object of mass $m$ and radius $r$ rolling without slipping down an incline of angle $\theta$, given by $a = \frac{g\sin(\theta)}{1+\beta}$.

Incline Slipping Condition

The inequality defining the tilt threshold where required static friction exceeds maximum static friction ($f_s > \mu_s N$), causing a rolling object to slip: $\tan(\theta) > \left(\frac{1+\beta}{\beta}\right)\mu_s$.

Loop-the-Loop Minimum Release Height ($h$)

The minimum release height required above the base for an object to roll without slipping around a vertical loop of radius $R$ without falling off at the peak ($N_{\text{top}} = 0$): $h = \left(\frac{5+\beta}{2}\right)R$.

Angular Momentum ($\vec{L}$)

The rotational counterpart of linear momentum, defined for a rigid body rotating about a symmetry axis as $\vec{L} = I\vec{\omega}$ and for a particle as the cross product $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$.

Law of Conservation of Angular Momentum

The fundamental conservation law stating that if the net external torque acting on a system is zero ($\vec{\tau}_{\text{net}} = 0$), the total angular momentum vector is constant: $\frac{d\vec{L}}{dt} = 0$.

Helicopter Tail Rotor

An auxiliary vertical rotor at the tail of a helicopter providing horizontal thrust to create a torque opposite to that produced on the fuselage by the main rotor, thereby preserving $\vec{\tau}_{\text{net}} = 0$ and preventing fuselage rotation.

Precession

The continuous rotation of the spin axis of a spinning body around an orthogonal axis, occurring when a net torque is exerted perpendicular to the existing spin angular momentum vector ($\vec{\tau} = \frac{d\vec{L}}{dt} \perp \vec{L}$).