Kinematics, Newtonian Mechanics, and Circular Motion: Study Notes

Kinematics: Motion in Two Dimensions

Vectors in Two Dimensions

Understanding vectors is essential for describing motion in two dimensions. Vectors have both magnitude and direction, and can be represented graphically or algebraically.

• Adding and Subtracting Vectors: Use components or the tail-to-head method to combine vectors. Components involve breaking vectors into x and y parts.

• Using Free-Body Diagrams: Draw all forces acting on an object, resolve them into components, and use them to analyze motion.

• Breaking Forces into Components: Forces can be split into perpendicular directions (usually x and y axes) for easier analysis.

Example: To find the resultant of two forces at an angle, resolve each into x and y components, add the components, and use the Pythagorean theorem to find the magnitude.

Kinetic and Static Friction

Friction is a force that opposes motion between surfaces in contact. It can be classified as kinetic (sliding) or static (stationary).

• Definitions: Kinetic friction acts when objects slide; static friction acts when objects are stationary.

• Direction: Friction always acts opposite to the direction of motion or intended motion.

• Equations:

$f_k = ext{coefficient of kinetic friction} imes ext{normal force}$

$f_s ext{ (max)} = ext{coefficient of static friction} imes ext{normal force}$

Projectile Motion

Projectile motion involves objects moving in two dimensions under the influence of gravity.

• Components of Motion: The motion can be separated into horizontal (x) and vertical (y) components.

• Equations:

$x = v_{0x} t$

$y = v_{0y} t - \frac{1}{2} g t^2$

• Instantaneous Velocity: Find by combining x and y velocity components at a given time.

Example: A ball thrown horizontally from a cliff will have constant horizontal velocity and accelerating vertical velocity due to gravity.

Newtonian Mechanics

Forces and Motion on Inclined Planes

Analyzing motion on inclined planes requires resolving forces parallel and perpendicular to the surface.

• Choosing Axes: Axes are often chosen parallel and perpendicular to the incline for easier calculations.

• Equations:

$F_{\text{parallel}} = mg \sin \theta$

$F_{\text{perpendicular}} = mg \cos \theta$

• Friction: Frictional forces act parallel to the surface and oppose motion.

Motion of Connected Objects

Objects connected by pulleys, strings, or masses require analysis of tension and acceleration.

• Free-Body Diagrams: Draw forces acting on each object, including tension, gravity, and friction.

• Matching Acceleration: Connected objects often share the same acceleration.

Circular Motion

Direction of Velocity, Acceleration, and Net Force

Objects moving in a circle experience constant changes in direction, even if speed is constant.

• Velocity: Always tangent to the circle.

• Acceleration: Points toward the center (centripetal acceleration).

• Net Force: Also points toward the center.

$a_c = \frac{v^2}{r}$

$F_c = m \frac{v^2}{r}$

Radial and Tangential Direction

Radial direction points toward the center; tangential direction is perpendicular to the radius.

• Speeding Up/Slowing Down: Tangential acceleration changes speed; radial acceleration changes direction.

Period of Circular Motion

The period is the time taken for one complete revolution.

• Equation:

$T = \frac{2\pi r}{v}$

Newton's Second Law in Circular Motion

Newton's second law applies to circular motion, with the net force providing the required centripetal acceleration.

• Equation:

$F_{\text{net}} = m a_c = m \frac{v^2}{r}$

Apparent Weight in Circular Motion

Apparent weight can differ from true weight due to acceleration in circular motion.

• Example: Riders in a roller coaster feel lighter or heavier depending on the direction of acceleration.

Universal Gravitation and Orbital Motion

Newton's law of universal gravitation explains the force between two masses.

• Equation:

$F_g = G \frac{m_1 m_2}{r^2}$

• Orbital Motion: The gravitational force provides the centripetal force for planetary orbits.

Example: The Moon orbits the Earth due to gravitational attraction.

Relationship Between $g$ and $G$

The acceleration due to gravity at Earth's surface ($g$) is related to the universal gravitational constant ($G$).

• Equation:

$g = G \frac{M_{\text{Earth}}}{R_{\text{Earth}}^2}$

Combining Newton's Law with Circular Motion

Newton's law of universal gravitation can be combined with circular motion equations to analyze planetary orbits.

• Equation: Set gravitational force equal to centripetal force for orbital motion.

$G \frac{M m}{r^2} = m \frac{v^2}{r}$

Example: Calculating the orbital speed of a satellite around Earth.