Physics for Life Sciences, Part 2 - Rotational Motion (Study Guide 11)

Vocabulary practice flashcards covering rotational dynamics, moments of inertia, rotational kinetic energy, rolling motion, and angular momentum from Physics for Life Sciences (PHYS*1080).

Rotational Equilibrium

A condition in statics where the net torque acting on a body is zero ($\sum \tau = 0$), which causes the angular acceleration to be zero ($\alpha = 0$).

Newton's Second Law for Rotation

The rotational analog of linear dynamics ($F = ma$), stating that net torque equals moment of inertia multiplied by angular acceleration: $\tau_{\text{net}} = \sum \tau = I\alpha$.

Moment of Inertia ($I$)

A measure of an object's resistance to rotational acceleration depending on mass distribution relative to the rotation axis, defined for point masses as $I = \sum_i m_i r_i^2$ with units of $kg\,m^2$.

Moment of Inertia of a Solid Cylinder or Disk

The rotational inertia of a uniform solid cylinder or disk of mass $M$ and radius $R$ rotated about its central axis: $I = \frac{1}{2} M R^2$.

Moment of Inertia of a Thin-Walled Hoop or Hollow Cylinder

The rotational inertia of a hollow cylinder with inner radius $R_1$ and outer radius $R_2$, given by $I = \frac{1}{2} M (R_1^2 + R_2^2)$, which simplifies to $I = M R^2$ when the wall is very thin ($R_1 \approx R_2 = R$).

Moment of Inertia of a Solid Sphere

The rotational inertia of a uniform solid sphere of mass $M$ and radius $R$ about an axis through its center: $I = \frac{2}{5} M R^2$.

Moment of Inertia of a Thin-Walled Hollow Sphere

The rotational inertia of a spherical shell of mass $M$ and radius $R$ about an axis passing through its center: $I = \frac{2}{3} M R^2$.

Moment of Inertia of a Long Thin Rod

The rotational inertia of a rod of mass $M$ and length $L$, equal to $I = \frac{1}{12} M L^2$ when rotated through its center, and $I = \frac{1}{3} M L^2$ when rotated through one end.

Insect Wing Geometry and Moment of Inertia

The principle that broader objects have greater moment of inertia; wing (ii) possesses a larger $I$ about axis AB than wings (i) and (iii) because its mass is distributed farther from the rotation axis.

Coin Rotation Comparison ($I_A$ vs. $I_B$)

A demonstration showing that $I_A < I_B$ because spinning a coin on its edge (A) distributes mass closer to the rotation axis than spinning it flat like a disk (B).

Rotational Kinetic Energy ($K_{\text{rot}}$)

The kinetic energy associated with rotational motion, given by $K = \frac{1}{2} I \omega^2$, derived from $\frac{1}{2} \sum m_i v_i^2$ substituting $v = r\omega$.

Kinetic Energy of Rolling Objects

The combined energy of an object rolling without slipping, containing translational and rotational parts: $K = \frac{1}{2} M v_{\text{cm}}^2 + \frac{1}{2} I_{\text{cm}} \omega^2 = \frac{1}{2} M v_{\text{cm}}^2 \left(1 + \frac{I_{\text{cm}}}{M R^2}\right)$.

Rolling Without Slipping Condition

Motion requiring friction where the center-of-mass linear speed and angular speed are strictly related by $v_{\text{cm}} = \omega R$ (or $\omega = \frac{v_{\text{cm}}}{R}$).

Rolling Disc vs. Hoop Demonstration

When released down an incline, a solid disk wins against a hoop of equal mass because its lower moment of inertia ($I_{\text{disk}} = \frac{1}{2}MR^2 < I_{\text{hoop}} = MR^2$) channels less potential energy into rotational kinetic energy, yielding greater translational speed ($v_{\text{disk}} > v_{\text{hoop}}$).

Angular Momentum ($L$)

The rotational analog of linear momentum ($\vec{p} = m\vec{v}$), defined as the product of moment of inertia and angular velocity: $L = I\omega$, measured in units of $kg\,m^2/s$.

Conservation of Angular Momentum

The law stating that if no net external torque acts on a system ($\tau_{\text{net}} = 0$), its angular momentum remains constant: $I_1 \omega_1 = I_2 \omega_2$.

Figure Skater Effect

An illustration of conservation of angular momentum where pulling the arms inward decreases the moment of inertia ($I$), producing an inverse, proportional increase in angular speed ($\omega$).