Units, Physical Quantities, and Vectors

Units and Significant Figures

Physical quantities are measured using standardized units, and calculations must maintain unit consistency and proper precision.

• SI Units: Length (meter, m), Time (second, s), Mass (kilogram, kg).

• Prefixes: Used to scale units by powers of ten. See table below.

• Unit Consistency: Only quantities with the same units can be added, subtracted, or equated.

• Unit Conversion: Use conversion factors (e.g., ) and raise the entire factor to a power when converting squared or cubed units.

• Significant Figures: For multiplication/division, keep as many significant figures as the least precise factor; for addition/subtraction, round to the least precise decimal place. Round only the final result.

Scalars and Vectors

Scalars are single numbers with units; vectors have both magnitude and direction.

• Scalars: Examples include mass, time, distance, speed.

• Vectors: Examples include displacement, velocity, acceleration, force.

• Vector Operations: Vector ± vector and scalar × vector yield vectors; dot and cross products combine two vectors.

• Dot Product: yields a scalar.

• Cross Product: yields a vector.

Components, Direction Angles, and Unit Vectors

Vectors are described by their components along chosen axes and unit vectors.

• Component Form:

• Unit Vectors: , , point along +x, +y, +z axes.

• Magnitude:

• Direction Angles: , (for xy-plane).

• Compass Directions: "30° east of north" means start from north and rotate 30° toward east.

Adding and Subtracting Vectors

Vectors add graphically head-to-tail and algebraically by components.

• Component Addition:

• Magnitude Bounds:

• Magnitude of Sum:

The Dot (Scalar) Product

The dot product measures how much two vectors point in the same direction.

• Formula:

• Applications: Angle between vectors, test for perpendicularity (), work ().

The Cross (Vector) Product

The cross product yields a vector perpendicular to both original vectors.

• Magnitude:

• Direction: Right-hand rule.

• Component Formula:

• Properties: Reversing order reverses sign; vanishes for parallel vectors.

• Scalar Triple Product: gives volume of parallelepiped.

Motion Along a Straight Line

Position, Velocity, and Acceleration

Motion along a line is described by position, velocity, and acceleration as functions of time.

• Position:

• Displacement:

• Average Velocity:

• Average Speed:

• Instantaneous Velocity:

• Instantaneous Acceleration:

Calculus and Initial Conditions

Integration relates acceleration, velocity, and position, with constants determined by initial conditions.

• Velocity from Acceleration:

• Position from Velocity:

• Average Velocity:

Constant Acceleration and Free Fall

For constant acceleration, motion equations simplify and describe free fall near Earth's surface.

• Equations:

• Free Fall: (with )

• Time to Top:

• Maximum Height:

Motion in Stages and Two Objects

When acceleration changes, apply constant-acceleration equations to each stage. For two objects, use one axis and clock.

• Relative Velocity:

• Relative Acceleration:

Graphs of Motion and Motion Diagrams

Graphical representations help visualize motion.

• x–t Graph: Height = position; slope = velocity.

• v_x–t Graph: Slope = acceleration; area under curve = displacement.

• a_x–t Graph: Area = change in velocity.

• Motion Diagrams: Spacing between positions indicates speeding up or slowing down.

Motion in Two or Three Dimensions

Position, Velocity, and Acceleration Vectors

Motion in space is described by vector quantities.

• Position:

• Velocity:

• Acceleration:

• Speed Change: Parallel component of changes speed; perpendicular component changes direction.

Projectile Motion

Projectile motion combines independent horizontal and vertical motions under gravity.

• Horizontal Acceleration:

• Vertical Acceleration:

• Equations:

• Time of Flight (equal heights):

• Range:

• Path: Parabolic.

Relative Velocity

Relative velocity describes how motion appears from different reference frames.

• General Formula:

• Applications: Boat in current, airplane in wind, launch from moving vehicle.

Circular Motion

Circular motion involves constant turning of velocity, requiring inward acceleration.

• Uniform Circular Motion:

  • Radial acceleration:

  • Speed:

• Nonuniform Circular Motion:

  • Tangential acceleration:

  • Total acceleration:

  • Angle:

• Released Object: Leaves along tangent with instantaneous velocity.

Newton’s Laws of Motion

Force and Newton’s First Law

Force is a vector quantity; Newton's first law defines inertial frames.

• Force: Push or pull; measured in newtons ().

• Net Force:

• First Law:

• Inertial Frame: Frame where Newton's laws hold.

Newton’s Second Law

Acceleration is proportional to net force and inversely proportional to mass.

• Formula:

• Component Form: , ,

• Mass: Positive scalar measuring inertia.

Newton’s Third Law

For every action, there is an equal and opposite reaction.

• Law:

• Third-Law Pair: Forces act on different objects, are equal in magnitude and opposite in direction.

• Examples: Rocket and exhaust, walker and ground.

Weight, Normal Force, and Apparent Weight

Weight is the gravitational force; normal force is the perpendicular push of a surface.

• Weight:

• Normal Force: ; determined by Newton's second law perpendicular to surface.

• Apparent Weight: What a scale reads; in an elevator.

Free-Body Diagrams

Free-body diagrams show all forces acting on an object.

• Draw each force, label by type and agent.

• Choose axes, resolve forces into components.

• Apply Newton's second law along each axis.

• Add constraints and friction relations.

Tension and Connected Objects

Tension is the pull along a rope; ideal ropes and pulleys simplify analysis.

• Tension: ; same throughout ideal rope.

• Connected Objects: Share acceleration; write Newton's second law for each.

• Examples: Atwood machine, movable pulley, blocks in contact.

Inclines and Equilibrium

On an incline, resolve forces along and perpendicular to the slope.

• Gravity Components: down slope, into slope.

• Normal Force:

• Acceleration: (frictionless)

• Equilibrium:

Applying Newton’s Laws

Static and Kinetic Friction

Friction opposes sliding or the tendency to slide; depends on normal force.

• Kinetic Friction:

• Static Friction:

• Direction: Opposes motion or tendency to move.

• Transition: Static friction grows with applied force up to maximum; then kinetic friction applies.

Friction on an Incline

Friction acts along the slope, opposing sliding or tendency to slide.

• Normal Force:

• At Rest: (must be )

• Sliding: Kinetic friction balances downhill component at constant velocity.

Connected Objects with Friction

Friction is considered separately for each object; analyze using Newton's second law.

• Block on Table and Hanging Block: Movement starts if hanging weight exceeds maximum static friction.

• Stacked Blocks: Each block's free-body diagram includes all surfaces it touches; friction limits acceleration.

Dynamics of Circular Motion

Circular motion requires an inward (radial) net force; tangential force changes speed.

• Radial Force:

• Tangential Force:

• Flat Curve: Static friction supplies inward force; maximum static friction must be at least .

• Banked Curve: Normal force tilts toward center; at design speed, horizontal component supplies inward force.

• Conical Pendulum: Vertical component of tension balances weight; horizontal component supplies inward force.

• Vertical Circle: At top, gravity and tension point toward center; minimum speed required for tension to be nonzero.

• Apparent Weight: Over a hill, normal force is less than ; at high speed, object may lose contact.

Appendix: Optional Extensions

Volume from a Scalar Triple Product

The volume of a parallelepiped spanned by three vectors is given by the absolute value of the scalar triple product.

• Formula:

Radius of Curvature of a General Path

Any curved path can be locally approximated by a circle; the radius of curvature relates to perpendicular acceleration.

• Formula: ,

• Radius:

Drag and Terminal Speed

Drag opposes motion through a fluid; terminal speed is reached when drag balances weight.

• Linear Drag:

• Quadratic Drag: