Circular Motion and Newton's Laws: Structured Study Notes

Circular Motion

Definition and Types of Circular Motion

Circular motion refers to the movement of an object along a circular path, characterized by a change in the angular position θ over time. This motion can be classified as either uniform or non-uniform, depending on whether the speed remains constant.

• Uniform Circular Motion: The object moves at a constant speed, with equally spaced positions at equal time intervals along the circular trajectory.

• Non-uniform Circular Motion: The object’s speed changes, resulting in positions that are not equally spaced at equal time intervals. The object may be speeding up or slowing down as it travels the circular path.

Key Measurement Quantities:

• Time (t): Fundamental measurement for motion analysis.

• Angular Position (θ): The angle describing the object's position on the circle, measured in radians. Positive when measured counterclockwise.

Quantities Describing Circular Motion

At any instant, circular motion is described by several key quantities:

• Angular Position (θ): SI unit: radians.

• Angular Displacement (Δθ): Change in angular position.

• Angular Speed (ω): Sometimes called angular velocity. $\omega = \frac{d\theta}{dt}$ (rad/sec).

• Velocity (vector): The direction is always tangent to the circular trajectory at any given time.

Coordinate System for Circular Motion

The preferred coordinate system for analyzing circular motion is the r-t-z axis (radial-tangential-z), which helps separate radial and tangential components.

Velocity in Circular Motion

• For both uniform and non-uniform circular motion, the velocity vector is tangent to the path and changes over time.

• If the velocity changes only in direction (not magnitude), the motion is uniform circular motion.

• If the velocity changes in both direction and magnitude, the motion is non-uniform circular motion.

Time Period and Linear Speed

• Time Period (T): The time for one complete revolution in uniform circular motion. $T = \frac{2\pi}{\omega}$

• Linear Speed (v): Depends on the distance (r) from the axis of rotation. $v = r\omega$

Acceleration in Circular Motion

• Centripetal (Radial) Acceleration: Associated with the change in direction of velocity. Always directed toward the center of the circle. $a_c = \frac{v^2}{r}$

• Tangential Acceleration: Associated with the change in the magnitude of velocity (speed). $a_t = \alpha r$ (where $\alpha$ is angular acceleration)

Angular Acceleration

• Angular Acceleration (α): Rate of change of angular speed. $\alpha = \frac{d\omega}{dt}$ (SI units: rad/s2)

Sign Conventions for ω and α

• Clockwise and speeding up: ω negative, α negative

• Clockwise and slowing down: ω negative, α positive

• Counterclockwise and speeding up: ω positive, α positive

• Counterclockwise and slowing down: ω positive, α negative

Graphical Analysis

• From ω vs. t graph: Slope gives α.

• From θ vs. t graph: Slope gives ω.

• From α vs. t graph: Area under curve gives change in ω.

• From ω vs. t graph: Area under curve gives change in θ.

Kinematic Equations

• Uniform Circular Motion:

  • $\theta = \theta_0 + \omega t$

  • $\omega = \text{constant}$

• Non-uniform Circular Motion:

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

  • $\omega = \omega_0 + \alpha t$

  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Conversions and Units

• 1 revolution = $2\pi$ radians

• To convert rpm to rad/s: $1\ \text{rpm} = \frac{2\pi}{60}\ \text{rad/s}$

Newton's Laws of Motion

Newton's Second Law

Newton's second law relates the net force acting on an object to its mass and acceleration.

• Vector form: $\vec{F}_{\text{net}} = m\vec{a}$

• Component form:

  • $F_{x,\text{net}} = ma_x$

  • $F_{y,\text{net}} = ma_y$

  • $F_{z,\text{net}} = ma_z$

Problem-Solving Steps Using Newton's Laws

• Identify all forces acting on the object.

• Draw a free-body diagram.

• Choose a coordinate system and resolve forces into components.

• Apply Newton's second law to each direction.

• Solve for the unknowns.

Free-Body Diagrams

A free-body diagram is a sketch showing all the forces acting on an object. It is essential for analyzing forces in word problems.

• Practice drawing free-body diagrams for various scenarios, such as objects on inclined planes, pulleys, and ropes.

Vector Components

• If a vector is not aligned with a coordinate axis, its component along an axis is found using the angle θ:

• Component along axis = (magnitude of vector) $\cos\theta$

Types of Equilibrium

• Static Equilibrium: Object at rest; net force and net torque are zero.

• Dynamic Equilibrium: Object moves with constant velocity; net force is zero.

Newton's Third Law

• States that for every action, there is an equal and opposite reaction.

• Applies when two objects interact; the forces are equal in magnitude and opposite in direction.

Weightlessness

• Occurs when the normal force acting on an object is zero, such as in free fall or orbit.

Massless Ropes and Frictionless Pulleys

• In idealized problems, ropes are considered massless and pulleys frictionless to simplify force analysis.

Friction

• Static Friction: The force that prevents relative motion between surfaces at rest.

• Kinetic Friction: The force that opposes motion once surfaces are sliding.

• Maximum static friction is greater than kinetic friction.

Centripetal Force

Definition and Properties

Centripetal force is the net force required to keep an object moving in a circular path, directed toward the center of the circle.

• Magnitude: $F_c = \frac{mv^2}{r}$

• Direction: Always points toward the center of the circular path.

Applications

• Examples include tension in a string for a swinging ball, gravitational force for planetary orbits, and friction for a car turning on a road.

Type of Friction	Definition	Relative Magnitude
Static Friction	Prevents motion between surfaces at rest	Maximum value is greater than kinetic friction
Kinetic Friction	Opposes motion between sliding surfaces	Lower than maximum static friction