Rotational Motion, Dynamics, and Periodic Motion: Mini-Study Guide

Rotational Motion and Dynamics

Momentum and Impulse

Momentum and impulse are fundamental concepts in both linear and rotational motion, describing how objects move and how their motion changes due to forces.

• Momentum (p): Defined as the product of mass and velocity. It is a vector quantity.

• Formula: $p = mv$

• Units: kg·m/s

• Impulse (J): The change in momentum, equal to the average force multiplied by the time interval.

• Formula: $J = F_{avg} \Delta t = \Delta p$

• Units: N·s or kg·m/s

• Both momentum and impulse are vectors and must be broken into components when solving problems.

• Example: A ball of mass 0.5 kg moving at 2 m/s has momentum $p = 0.5 \times 2 = 1$ kg·m/s.

Angular Kinematics

Angular kinematics describes the rotational analogs of linear motion, including angular position, velocity, and acceleration.

• Angular Position (\theta): Measured in radians.

• Angular Velocity (\omega): Rate of change of angular position, measured in rad/s.

• Angular Acceleration (\alpha): Rate of change of angular velocity, measured in rad/s2.

• Relationship: $\omega = \frac{\Delta \theta}{\Delta t}$, $\alpha = \frac{\Delta \omega}{\Delta t}$

• Arc Length (s): $s = r\theta$

• Conversion: $360^\circ = 2\pi$ radians, $180^\circ = \pi$ radians

• Example: If a wheel rotates 2 radians and has a radius of 0.5 m, the arc length is $s = 0.5 \times 2 = 1$ m.

Translational and Rotational Velocity

Translational velocity describes linear motion, while rotational velocity describes angular motion. The two are related for points on a rotating object.

• Translational Velocity (v): $v = r\omega$

• Translational Acceleration (a_{tan}): $a_{tan} = r\alpha$

• For objects rotating without slipping: $v_{cm} = R\omega$

• Example: A point 0.2 m from the axis rotates at $\omega = 5$ rad/s, so $v = 0.2 \times 5 = 1$ m/s.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and depends on its moment of inertia and angular speed.

• Formula: $K_{rot} = \frac{1}{2} I \omega^2$

• Moment of Inertia (I): Measures how mass is distributed relative to the axis of rotation. Units: kg·m2.

• For a point mass: $I_{point} = mr^2$

• Example: A disk of mass 2 kg and radius 0.5 m has $I = \frac{1}{2}mr^2 = \frac{1}{2} \times 2 \times 0.25 = 0.25$ kg·m2.

Gravitational Potential Energy for Rigid Bodies

Gravitational potential energy is the energy stored due to an object's position in a gravitational field.

• Formula: $U = Mgh_{cm}$

• Where: $M$ is mass, $g$ is gravitational acceleration, $h_{cm}$ is the center of mass height relative to $U_{grav}=0$.

• In practice: $U = Mgh$ is often used for simplicity.

• Example: A 3 kg object at 2 m height: $U = 3 \times 9.8 \times 2 = 58.8$ J.

Total Kinetic Energy for Rigid Bodies

The total kinetic energy of a rigid body includes both translational and rotational components.

• Formula: $K_{tot} = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I \omega^2$

• Where: $v_{cm}$ is the center of mass speed, $I$ is moment of inertia, $\omega$ is angular speed.

• Example: A rolling cylinder with $v_{cm} = 2$ m/s, $I = 0.5$ kg·m2, $\omega = 3$ rad/s.

Torque and Rotational Dynamics

Torque is the rotational analog of force, causing objects to rotate about an axis.

• Torque (\tau): $\tau = Fr\sin\theta$

• Where: $F$ is force, $r$ is distance from axis, $\theta$ is angle between force and lever arm.

• Moment Arm (l): The perpendicular distance from axis to line of action of force.

• Newton's Second Law for Rotation: $\sum \tau = I\alpha$

• Example: A force of 10 N applied 0.3 m from axis at $90^\circ$: $\tau = 10 \times 0.3 \times 1 = 3$ Nm.

Rotational Work and Power

Work and power in rotational motion are analogous to their linear counterparts.

• Rotational Work: $W = \tau \Delta \theta$

• Rotational Power: $P_{avg} = \tau \omega$

• Example: If $\tau = 2$ Nm and $\Delta \theta = 5$ rad, $W = 2 \times 5 = 10$ J.

Angular Momentum

Angular momentum is conserved in the absence of external torques and is a key quantity in rotational dynamics.

• For rigid bodies: $L = I\omega$

• For point masses: $L = mvr\sin\theta$

• Conservation: Angular momentum remains constant if no external torque acts.

• Example: A figure skater pulling in arms increases $\omega$ as $I$ decreases, keeping $L$ constant.

Static Equilibrium

Static equilibrium occurs when the sum of forces and torques on an object is zero.

• Conditions: $\sum F = 0$ and $\sum \tau = 0$

• Choose axis: Select axis of rotation to simplify torque calculations.

• Example: A seesaw balanced when torques from both sides are equal.

Periodic Motion and Simple Harmonic Motion (SHM)

Spring Force and Hooke's Law

Hooke's Law describes the restoring force exerted by a spring when it is displaced from equilibrium.

• Formula: $F_x = -kx$

• Where: $k$ is spring constant, $x$ is displacement from equilibrium.

• Maximum force: $F_{max} = -kA$ (A = amplitude)

• Example: A spring with $k = 100$ N/m stretched 0.05 m: $F_x = -100 \times 0.05 = -5$ N.

Acceleration in SHM

The acceleration of a mass attached to a spring is always directed toward equilibrium and is proportional to displacement.

• Formula: $a_x = -\frac{k}{m}x$

• Maximum acceleration: $a_{max} = -\frac{k}{m}A$

• Acceleration is zero at equilibrium ($x = 0$).

Frequency, Period, and Angular Frequency

Frequency and period describe how often oscillations occur, while angular frequency relates to the rate of oscillation in radians per second.

• Frequency (f): Number of cycles per second. Units: Hz.

• Period (T): Time for one cycle. Units: s.

• Relationship: $f = \frac{1}{T}$

• Angular Frequency (\omega): $\omega = 2\pi f$

• Example: If $T = 0.5$ s, $f = 2$ Hz, $\omega = 4\pi$ rad/s.

Equations of Motion for SHM

Position and velocity as functions of time for an object in simple harmonic motion (SHM).

• Position: $x(t) = A \cos(\omega t)$

• Velocity: $v(t) = -\omega A \sin(\omega t)$

• Example: For $A = 0.1$ m, $\omega = 2$ rad/s, $x(1) = 0.1 \cos(2 \times 1) = 0.1 \cos(2)$

SHM for Mass-Spring and Pendulum Systems

Mass-spring and pendulum systems exhibit SHM, with specific formulas for frequency and period.

• Mass-Spring System:

  • Angular frequency: $\omega = \sqrt{\frac{k}{m}}$

  • Period: $T = 2\pi \sqrt{\frac{m}{k}}$

  • Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

• Pendulum System:

  • Angular frequency: $\omega = \sqrt{\frac{g}{l}}$

  • Period: $T = 2\pi \sqrt{\frac{l}{g}}$

  • Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{g}{l}}$

• Example: A 1 kg mass on a spring with $k = 100$ N/m: $\omega = 10$ rad/s, $T = 0.628$ s.

Energy in SHM

The total mechanical energy in SHM is conserved in the absence of friction, with kinetic and potential energy exchanging as the object oscillates.

• Total Energy: $E = \frac{1}{2} k A^2 = \text{constant}$

• Kinetic energy maximized when $x = 0$ (object at equilibrium).

• Potential energy maximized when $v = 0$ (object at maximum displacement).

• Velocity at position $x$: $v(x) = \pm \omega \sqrt{A^2 - x^2}$

• Example: For $A = 0.1$ m, $k = 100$ N/m, $E = 0.5 \times 100 \times 0.01 = 0.5$ J.

Summary Table: Rotational and SHM Quantities

Quantity	Symbol	Formula	Units
Momentum	p	$p = mv$	kg·m/s
Impulse	J	$J = F_{avg} \Delta t$	N·s
Angular Velocity	\omega	$\omega = \frac{\Delta \theta}{\Delta t}$	rad/s
Moment of Inertia	I	$I = \sum m_i r_i^2$	kg·m2
Rotational Kinetic Energy	$K_{rot}$	$K_{rot} = \frac{1}{2} I \omega^2$	J
Torque	\tau	$\tau = Fr\sin\theta$	Nm
Spring Force	$F_x$	$F_x = -kx$	N
SHM Position	$x(t)$	$x(t) = A \cos(\omega t)$	m
SHM Velocity	$v(t)$	$v(t) = -\omega A \sin(\omega t)$	m/s
SHM Energy	E	$E = \frac{1}{2} k A^2$	J

Additional info: Some formulas and explanations were expanded for clarity and completeness, including standard definitions and example calculations.