Two-Dimensional Motion

Introduction to Two-Dimensional Motion

Motion in a plane, or two-dimensional motion, is a fundamental topic in college physics. It involves analyzing the position, velocity, and acceleration of objects moving in two spatial dimensions, typically represented by the x- and y-axes. This topic is essential for understanding projectile motion, relative velocity, and the application of vector mathematics in physics.

• Position Vector: Describes the location of a particle in the plane relative to the origin.

• Velocity Vector: Represents the rate of change of position with respect to time.

• Acceleration Vector: Represents the rate of change of velocity with respect to time.

Goals for Chapter

• Study and calculate position, velocity, and acceleration vectors in 2D.

• Frame two-dimensional motion as it occurs in the motion of projectiles.

• Use the equations of motion for constant acceleration to solve for unknown quantities for an object moving under constant acceleration in 2D.

• Study the relative velocity of an object for observers in different frames of reference in 2D.

Position, Displacement, and Vectors in 2D

Vectors in two dimensions are used to describe position, displacement, and velocity in the x-y plane.

• Position Vector:

• Magnitude:

• Displacement:

Vector Representation: Cartesian and Polar Coordinates

• Vectors can be expressed in terms of their Cartesian components or as Polar coordinates :

Velocity in a Plane

Average and Instantaneous Velocity

Velocity in two dimensions is a vector quantity describing both the speed and direction of motion.

• Average velocity: The total displacement divided by the total time interval.

• Instantaneous velocity: The velocity at a specific instant, tangent to the path.

Key Point: The instantaneous velocity vector is always tangent to the object's path in the x-y plane.

Example: Motion of a Model Car

• Given: , ,

• Average velocity components:

• Magnitude of average velocity:

Acceleration in a Plane

Average and Instantaneous Acceleration

• Average acceleration:

• Instantaneous acceleration:

Key Point: The acceleration vector always points toward the concave side of the curved path.

Vector Addition and Subtraction

• Addition (Head-to-Tail Method): Place the tail of the second vector at the head of the first; the resultant vector is drawn from the tail of the first to the head of the second.

• Subtraction: Add the opposite of the vector to be subtracted (reverse its direction by 180°).

Projectile Motion

Characteristics of Projectile Motion

Projectile motion is a special case of two-dimensional motion where an object moves under the influence of gravity alone (neglecting air resistance). The path followed is a parabola in the x-y plane.

• The motion can be analyzed by separating it into horizontal (x) and vertical (y) components.

• Horizontal motion: constant velocity (no horizontal acceleration).

• Vertical motion: constant acceleration due to gravity ().

Equations of Motion for Projectiles

Key Quantities in Projectile Motion

• Maximum height: The highest vertical position reached by the projectile.

• Range: The horizontal distance traveled by the projectile.

• Time of flight: The total time the projectile is in the air.

For a projectile launched from ground level ():

• Time of flight:

• Range:

• Maximum height:

Examples and Applications

• Sports: Calculating the trajectory of a baseball, football, or paintball.

• Engineering: Determining the range and height for projectiles in design problems.

Summary Table: Key Equations for Projectile Motion

Additional info: Relative velocity in two dimensions and frames of reference are also important but not detailed in the provided slides. In general, relative velocity is calculated by vector addition or subtraction of velocities as observed from different frames.

Friction

Introduction to Friction

Friction is a fundamental concept in physics that describes the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. It plays a crucial role in everyday phenomena and engineering applications.

• Friction is a force that opposes relative motion between two objects in contact.

• It acts parallel to the contact surface and always in a direction that opposes motion or attempted motion.

• Friction arises due to the microscopic roughness of surfaces and the attractive forces between molecules of the materials in contact.

Types of Friction

• Kinetic Friction: Occurs when two surfaces are in contact and moving relative to each other.

• Static Friction: Acts when surfaces are stationary relative to each other, preventing the onset of motion.

• The magnitudes of static and kinetic friction are generally different for the same pair of surfaces.

Origin and Nature of Friction

• Friction is not only due to surface roughness but also to the attractive forces between the molecules of the two materials in contact. Even highly polished surfaces are not completely free of friction.

• When the normal force (the force pressing the surfaces together) increases, the frictional force also increases.

• At nonzero speeds, kinetic friction is nearly independent of speed for most materials.

Static Friction

• Static friction prevents the initiation of motion between two surfaces. Its magnitude can vary up to a maximum value, depending on the normal force and the nature of the surfaces.

• Static friction adjusts up to its maximum value to prevent motion.

Example: Static Friction Calculation

• A 100 N box rests on a floor with a horizontal force of 30 N applied via a rope. The maximum static friction is 50 N, so the box does not move.

Kinetic Friction

• Kinetic friction acts when two surfaces are sliding past each other. Its magnitude is generally less than the maximum static friction for the same surfaces.

• The force of kinetic friction is given by: Where is the coefficient of kinetic friction and is the normal force.

• Kinetic friction remains approximately constant for a wide range of speeds.

Example: Sliding with Kinetic Friction

• A penguin slides down an incline of 10.0° with a mass of 32.0 kg. The kinetic friction force is .

• To find the acceleration, use Newton's second law, accounting for gravity, normal force, and friction.

Direction of Frictional Forces

• The direction of the frictional force is always opposite to the direction of actual or attempted motion.

• For an object at rest, static friction acts to oppose any applied force attempting to move it.

• For a moving object, kinetic friction acts opposite to the direction of motion.

Factors Affecting Friction

• The normal force between the surfaces: Increasing the normal force increases friction.

• The nature of the surfaces: Different materials have different coefficients of friction.

• Surface area does not significantly affect friction for most solid objects.

Coefficients of Friction

The coefficients of friction are dimensionless numbers that characterize the interaction between two surfaces.

Additional info: Typical values for μ range from 0.1 (ice) to 1.0 (rubber or dry concrete).

Sample Problems and Applications

• Pinned Against a Wall: A block is held against a vertical wall by a horizontal force. The frictional force opposes gravity, and the maximum static friction determines whether the block will slide.

• Trapped Hiker: A hiker uses a rope and a rock to prevent sliding. The maximum static friction between the rock and the ground determines the safety of the setup.

• Toy Chest: A child pulls a toy chest with a rope. The force required to start moving the chest is determined by the maximum static friction.

• Pushing a Sled: The maximum force that can be applied to a sled before a person sitting on it starts to slide depends on the static friction between the person and the sled.

Quick Conceptual Questions

• Does friction cause an object to speed up? Friction always acts to oppose relative motion; it cannot cause an object to speed up on its own.

• Effect of Normal Force: Increasing the normal force increases the frictional force, and vice versa.

• Direction of Friction: The direction of the frictional force is always opposite to the direction of motion or attempted motion.

Force and Motion: Newton's Laws

Introduction to Dynamics

Dynamics is the study of the forces that cause objects to move or change their motion. The laws governing dynamics are universal and apply to all physical systems, whether on Earth or in space.

• Definition: Dynamics is the study of the forces that cause objects to move or change their motion.

• Universal Laws: Newton's laws of motion are foundational to dynamics and are considered universal, meaning they apply to all objects and situations.

• Application: These laws are used to analyze and predict the motion of objects under various force conditions.

Isaac Newton's Monumental Work

• Sir Isaac Newton's work, Philosophiae Naturalis Principia Mathematica, published in 1687, laid the foundation for classical mechanics. Newton proposed scientific laws that remain central to our understanding of motion.

• Historical Context: Newton's Principia introduced the three laws of motion and the law of universal gravitation.

• Impact: These laws are still used today to describe and predict the motion of objects.

Development of the Force Concept

• Definition: A force is any interaction that, when unopposed, will change the motion of an object.

• Vector Nature: Forces are vectors and must be added using vector addition rules.

• Example: In the overhead view of skaters pushing each other, the total force is the vector sum of individual forces.

Free-Body Diagrams

• A free-body diagram is a schematic representation used to visualize the forces acting on a single object or system. It helps in analyzing the net force and predicting the resulting motion.

• Forces are represented as arrows.

• External Forces: Only forces from outside the system are considered.

• Net External Force: The sum of all external forces determines the acceleration of the object.

Equation for Net External Force

Types of Forces

• Weight: The gravitational force exerted by Earth on an object.

• Normal Force: The support force exerted perpendicular to the surface of contact.

• Tension: The force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends.

System of Interest

• When analyzing forces, it is important to clearly define the system of interest. This allows for accurate identification of external forces and proper construction of free-body diagrams.

• System: The object or group of objects being analyzed.

• External Forces: Only forces acting from outside the system are included in the analysis.

Example: Vector Addition of Forces

• Consider three skaters pushing each other. The total force on one skater is the vector sum of the individual forces applied by the other two skaters. The direction and magnitude of the resulting force determine the skater's acceleration.

• Application: Use vector addition to find the net force and predict the motion.

Summary Table: Types of Forces

Additional info: The notes cover introductory concepts in two-dimensional motion, friction, and Newton's laws, suitable for a college-level physics course. For deeper study, students should review vector analysis, free-body diagrams, and the mathematical derivation of Newton's laws.