3PHY MECHANICS

Flashcards using the notes for 3PHY Mechanics

Damped Motion

An oscillatory motion where friction or resistive forces are added to the system, causing energy to leave and the oscillations to decrease in amplitude over time.

Driven Motion

An oscillatory motion that occurs when a periodic external force is continuously applied to an oscillating Simple Harmonic Motion ($\text{SHM}$) system.

Resonance Frequency

The exact frequency and timing at which an external driving force must be applied to an oscillating system to maximize its amplitude.

Amplitude ($A$)

The maximum linear displacement measured from the equilibrium position to the point of furthest travel.

Simple Harmonic Motion (SHM)

A repeating back-and-forth motion where the net restoring force ($\Sigma F$) always points directly toward equilibrium and is proportional to displacement ($\Sigma F \propto y$).

Small Angle Approximation for Pendulums

The condition stating that for a pendulum swinging at small angles ($< 10^\circ$), the restoring force points toward equilibrium and is proportional to displacement, satisfying Simple Harmonic Motion.

Spring Constant ($k$)

A measure of spring stiffness in Hooke's Law ($F = -kx$), where the negative sign indicates that the restoring force acts opposite to the direction of displacement.

Time Period of a Spring ($T$)

The time in seconds required to complete one full cycle of oscillation on a spring, given by the formula $T = 2\pi\sqrt{\frac{m}{k}}$.

Time Period of a Pendulum ($T$)

The time in seconds for a pendulum of length $L$ to complete one full oscillation, calculated using $T = 2\pi\sqrt{\frac{L}{g}}$, where $g = 9.81\,\text{m/s}^2$.

Angular Velocity / Frequency ($\omega$)

The rate of rotation or phase change measured in radians per second ($\text{rad/s}$), calculated using $\omega = 2\pi f$ or $\omega = \frac{2\pi}{T}$.

Radian

A dimensionless measure of angular displacement where one full circle ($360^\circ$) is equal to $2\pi\,\text{radians}$.

Rotational Inertia ($I$)

A measure of an object's resistance to rotational acceleration, measured in $\text{kg}\,\text{m}^2$, which increases the further mass is distributed from the centre of rotation.

Rotational Kinetic Energy ($E\_{\text{k(rot)}}$)

The kinetic energy possessed by a body due to its rotational movement, calculated by the formula $E\_{\text{k(rot)}} = \frac{1}{2}I\omega^2$.

Torque ($\tau$)

A turning force measured in newton metres ($\text{N}\,\text{m}$), given by $\tau = F r$ when applied perpendicularly, or $\tau = F r\sin(\theta)$ when the force is applied at an angle.

Newton's Second Law for Rotation

The dynamical principle stating that an unbalanced net torque produces an angular acceleration on a body, expressed by $\tau = I\alpha$.

Angular Momentum ($L$)

The rotational momentum of a spinning body ($L = I\omega$) or linear particle ($L = mvr$), with standard units of $\text{kg}\,\text{m}^2/\text{s}$.

Conservation of Angular Momentum

The principle stating that if no external torque acts on a system, total angular momentum remains constant ($L\_{\text{before}} = L\_{\text{after}}$), so decreasing rotational inertia ($I$) increases angular velocity ($\omega$).

Circular Motion vs. Rotational Motion

Circular motion occurs when an object travels along an external path around an axis outside its own centre of mass, whereas rotational motion occurs when an object spins around an internal axis through its own centre of mass.

Centripetal Force ($\Sigma F$)

The net inward force directed toward the centre of a circular path that maintains circular motion, given by $\Sigma F = \frac{m v^2}{r}$.

Banked Corner

A curved road surface inclined at an angle such that the horizontal component of the normal force supplies the necessary inward centripetal force without relying solely on friction.

Geostationary Orbit

An equatorial orbit with an orbital period matching Earth's rotational period ($24\,\text{hours}$ or $86{,}400\,\text{s}$), keeping a satellite fixed over a single spot on Earth's surface.

Apparent Weightlessness (Microgravity)

The state experienced by astronauts in orbit due to being in continuous freefall around Earth, where the absence of a normal contact force eliminates the sensation of weight.

Newton's Universal Law of Gravitation

The fundamental law describing the mutual gravitational pull between two masses: $F\_g = \frac{G M m}{r^2}$, where $G = 6.67 \times 10^{-11}\,\text{N}\,\text{m}^2/\text{kg}^2$.

Centre of Mass (COM)

The average position of all the mass in an object or system, acting as the single point where the total gravitational force is considered to act.

Stability

The property of an object maintaining its equilibrium without tipping over, which holds as long as the vertical line from its centre of mass passes within its base of support.

Conservation of Linear Momentum

The principle stating that as long as no unbalanced external force acts on a system, total linear momentum ($p = mv$) remains equal before and after any interaction.

necessary for ball to be considered moving with SHM

Motion where the maximum acceleration occurs simultaneously with maximum negative displacement (mathematically, a ∝ -y)

restoring force acts towards equilibrium (F ∝ -y)

relationship with F = ma and Fc = mv²/r

both acquire centripetal force but uses different variables

Inertia when mass is distributed

Inertia in centre (pivot) is 0 as diameter is covered therefore r = 0 and I = mr² I would = 0 ONLY IN CENTRE but as mass is distributed further away from pivot I = >0

Inertia concentrated in edges / rims

Inertia is increased as radius is valued, according to I = mr²

bottom family in SHM

Used for when starts at max displacement

Top family

used for when starts at equlibrium

How to draw in referenced circle?

object oscillates ANTI clockwise (left)

you draw from x axis first how far the object oscillates (max y) (Q in image)

you then draw from the objects height across that y axis (blue dot in image)

then you draw the triangle (from centre to Q)

resonance

when current is at maximum

Current is maximum therefore impedance is at minimum