Physics Notes: Dynamics of Particles and Newton's Laws - PDF

Course Code: PHY 101 Course Title: Inductory mechanics and properties of matter Course Outline: Dynamics of particles, forces and linear motions, conservation of momentum, work, energy, power and efficiency. Elastic and inelastic collision. Rotational Dynamics and gravitation. Motion of rigid bodi...

Course Code: PHY 101 Course Title: Inductory mechanics and properties of matter Course Outline: Dynamics of particles, forces and linear motions, conservation of momentum, work, energy, power and efficiency. Elastic and inelastic collision. Rotational Dynamics and gravitation. Motion of rigid bodies. Fluids at rest and fluid in motion. Definition of Dynamics in Physics Dynamics is the branch of mechanics that deals with the motion of objects and the forces causing that motion. Unlike kinematics, which describes how objects move, dynamics explains why objects move the way they do by analyzing the relationship between forces, masses, and accelerations. In mathematical terms, dynamics is rooted in Newton's Laws of Motion, which form the foundation for understanding the cause-and-effect relationship between force and motion. Importance of Dynamics in Physics Explains Motion: Dynamics helps us understand why objects move or remain stationary. It provides the reasoning behind acceleration, deceleration, and equilibrium in systems. Foundation for Engineering and Technology: Concepts from dynamics are used in designing vehicles, bridges, machinery, and even robots. Engineers use dynamics to predict how forces will affect the motion and stability of structures. Predicts Behavior in Nature: Dynamics allows scientists to predict the movement of celestial bodies, like planets, moons, and asteroids, as well as the flow of air and water in the environment. Real-World Applications: In transportation, dynamics helps design vehicles for safety and efficiency. In sports, it explains how forces and motion determine performance (e.g., why a soccer ball curves). In everyday life, it explains phenomena like why a chair remains stationary unless acted upon by an external force. Lays the Groundwork for Advanced Physics: Dynamics is essential for understanding more complex topics in physics, such as electromagnetism, fluid mechanics, and quantum mechanics. Problem-Solving Skills: By studying dynamics, students develop critical thinking and analytical skills needed to solve practical and theoretical problems. Key Examples of Dynamics in Action The acceleration of a car when the driver steps on the gas pedal. A spacecraft’s trajectory under the influence of gravitational forces. The force required to push a heavy object across a surface. By studying dynamics, we not only gain insights into the physical world but also unlock the tools necessary to innovate, solve practical challenges, and understand the universe better. Newton's Laws of Motion Newton's Laws of Motion are the fundamental principles that describe the relationship between the motion of an object and the forces acting upon it. They form the foundation of classical mechanics. Newton’s First Law of Motion (Law of Inertia) Statement: An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. Key Concept: Inertia: The tendency of an object to resist changes in its state of motion. Examples: A book on a table remains stationary until someone pushes or lifts it (object at rest remains at rest). A passenger in a car feels a jolt forward when the car suddenly stops. The passenger continues moving forward due to inertia. Newton’s Second Law of Motion Statement: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically: 𝐹=𝑚𝑎. Where: 𝐹= net force acting on the object (in Newtons, N) 𝑚= mass of the object (in kilograms, kg) ! 𝑎 = acceleration of the object (in meters per second squared, 𝑚⁄𝑠 ) This equation explains how a force acting on an object causes it to accelerate, and the acceleration depends on the object’s mass. Key Concept: Larger forces produce greater accelerations, but heavier objects (larger mass) accelerate less for the same force. Force: - A push or pull acting on an object. - It can change the motion of an object (speed it up, slow it down, or change its direction). ! - Unit: Newton (𝑁), where 1𝑁 = 1𝑘𝑔 ∙ 𝑚⁄𝑠 Mass: - The amount of matter in an object. - It is a measure of an object’s inertia (resistance to changes in motion). - Larger mass = harder to accelerate for the same force. - Unit: Kilogram (𝑘𝑔). Acceleration: - The rate at which an object’s velocity changes with time. - It is directly proportional to the force applied and inversely proportional to the mass. ! - Unit: 𝑚⁄𝑠 Relationship Between Force, Mass, and Acceleration Force is proportional to acceleration: If the same mass is involved, increasing the force increases acceleration. Example: Pushing a toy car gently causes slow acceleration, but pushing it harder makes it accelerate faster. Force is proportional to mass: For the same acceleration, a heavier object requires a larger force. Example: Lifting a small box is easier than lifting a heavy box. Acceleration is inversely proportional to mass: If the force remains constant, an object with greater mass will accelerate less. Example: A small bicycle accelerates faster than a heavy motorcycle when the same force is applied. Practical Examples Driving a car: - Pressing the gas pedal increases the force (engine power), which accelerates the car. - A heavier car (greater mass) requires more force to achieve the same acceleration as a lighter car. Pushing a grocery cart: - An empty cart (less mass) accelerates quickly with a small push. - A full cart (more mass) requires greater force to move at the same speed. - Pushing a shopping cart: A cart with groceries (greater mass) requires more force to accelerate than an empty cart. Throwing a ball: - A tennis ball (less mass) accelerates more when thrown with the same force compared to a basketball (greater mass). - Kicking a football: The harder you kick, the greater the acceleration of the ball. Applications of 𝐹=𝑚𝑎 Rocket propulsion: A rocket’s engine produces a large force to accelerate its massive body upward. Sports: In soccer, a harder kick (more force) causes the ball to accelerate faster. Lifting objects: Machines like cranes apply significant force to lift heavy loads. Questions 1. What happens to acceleration if you double the mass but keep the force constant? 2. If you increase the force applied to an object, how does its motion change? Newton’s Third Law of Motion Statement: For every action, there is an equal and opposite reaction. Key Concept: Forces always come in pairs. If object A exerts a force on object B, object B exerts an equal force in the opposite direction on object A. Examples: A swimmer pushes water backward, and the water pushes the swimmer forward with an equal force. A rocket launch: The exhaust gases are expelled downward with force, and the rocket is propelled upward with an equal and opposite force. Jumping off a boat: When you jump forward, the boat moves backward due to the opposite force. Types of Forces A force is a push or pull acting on an object, and it can be classified based on its origin and effect. Below are the major types of forces: 1. Gravitational Force Definition: The attractive force exerted by any object with mass on another object with mass. This is the force that keeps objects grounded on Earth and governs planetary motion. #$! $" Formula: 𝐹" = %" Where: 𝐹" = gravitational force, G = gravitational constant (6.674 × 10&'' 𝑁𝑚! ⁄𝑘𝑔! , 𝑚' 𝑚! = masses of two objects, 𝑟= distance between the centers of the two objects Examples: An apple falling from a tree, The Moon orbiting the Earth. 2. Frictional Force Definition: A force that opposes the relative motion or tendency of motion between two surfaces in contact. It acts parallel to the surface of contact. Types of Friction: Static friction: Prevents motion when the object is stationary. Kinetic friction: Acts when the object is moving. Formula: 𝐹( = 𝜇𝐹) Where: 𝐹( = frictional force, 𝜇 = coefficient of friction (depends on the surfaces in contact), 𝐹) = normal force Examples: A car slowing down due to brake friction. Sliding a book across a table. 3. Tension Force Definition: A pulling force exerted by a string, rope, or cable when it is stretched by forces acting at either end. Characteristics: Always directed along the string or rope. Acts away from the object attached to the string. Examples: A bucket hanging from a rope. A person pulling an object using a rope. 4. Normal Force Definition: A support force exerted by a surface perpendicular to the object resting on it. It counteracts the gravitational force acting on the object. Formula: 𝐹) = 𝐹" = 𝑚𝑔. (if no other forces are acting vertically) Examples: A book resting on a table experiences a normal force from the table. A person standing on the ground.  5. Applied Force Definition: A force that is applied to an object by a person or another object. Examples: Pushing a door open, Pulling a suitcase. 6. Air Resistance (Drag Force) Definition: A type of frictional force that opposes the motion of an object as it moves through a fluid (air or liquid). Characteristics: - Depends on the speed, shape, and surface area of the object. - Increases with speed. Examples: A skydiver falling through the atmosphere. A car moving at high speed experiences air resistance. 7. Spring Force Definition: A restoring force exerted by a compressed or stretched spring, attempting to bring it back to its equilibrium position. Hooke’s Law: 𝐹* = −𝑘𝑥 Where: 𝐹* = spring force, 𝑘= spring constant (stiffness of the spring), 𝑥 = displacement from the equilibrium position Examples: A stretched rubber band. A compressed spring in a mechanical toy. 8. Electromagnetic Force Definition:A force caused by electric charges, both at rest and in motion. This includes electric and magnetic forces. Examples: A charged balloon attracting small pieces of paper (electric force). A magnet attracting iron nails (magnetic force).  9. Buoyant Force Definition: The upward force exerted by a fluid on an object immersed in it, countering the object’s weight. Archimedes’ Principle: The buoyant force is equal to the weight of the fluid displaced by the object. Examples: A boat floating on water. A helium balloon rising in the air. 10. Centripetal Force Definition: A force that keeps an object moving in a circular path, directed toward the center of the circle. $, " Formula: 𝐹+ = % Where: 𝐹+ = centripetal force, 𝑚 = mass of the object, 𝑣 = velocity, 𝑟 = radius of the circular path Examples: A car turning around a curve. A satellite orbiting Earth. Free-Body Diagrams (FBDs) Definition: A Free-Body Diagram (FBD) is a simplified graphical representation used to visualize all the forces acting on a single object or body. It isolates the object from its surroundings and shows the direction and magnitude of forces acting upon it. Importance of Free-Body Diagrams - Simplifies Complex Problems: Breaking down forces acting on an object helps in solving problems involving dynamics, equilibrium, and motion. - Visual Representation: FBDs give a clear picture of all the forces acting on the object, making it easier to analyze and understand the situation.  - Critical in Problem Solving: FBDs are essential for applying Newton’s laws of motion and solving for unknowns like acceleration, tension, friction, or net force. - Identifies Force Directions: FBDs explicitly show the direction of forces, which is crucial for determining net force and resulting motion. - Foundation for Physics Applications: They are widely used in mechanics, engineering, and real-world applications like designing structures or understanding vehicle dynamics. Steps to Draw a Free-Body Diagram - Identify the Object: Choose the object for which you need to analyze the forces. - Isolate the Object: Imagine the object separated from its environment. Ignore unnecessary details like surroundings. - Draw the Object as a Point or Simplified Shape: Represent the object as a box or a dot at the center. - Show All Forces Acting on the Object: Use arrows to represent forces. - Each arrow starts from the object and points in the direction of the force. - The length of the arrow represents the magnitude of the force (optional but helpful). - Label Each Force: Clearly label forces like 𝐹" (gravitational force), 𝐹( (friction), 𝐹- (tension), etc. - Consider Directions and Angles: Use axes (e.g., 𝑥-axis and 𝑦-axis) to indicate directions. Break forces into components if they act at an angle. - Double-Check: Ensure all forces acting on the object (contact and non- contact) are included. Common Forces to Include in an FBD Gravitational Force (𝐹" ): Acts downward, towards the center of the Earth. Equal to 𝐹" = 𝑚𝑔 where m = mass, 𝑔= 9.8𝑚⁄𝑠 ! Normal Force (𝐹) ): Perpendicular to the surface the object is resting on. Frictional Force (𝐹( ): Opposes the motion or potential motion between surfaces. Applied Force (𝐹. ): Any external force acting on the object. Tension Force (𝐹- ): Pulling force through a rope, string, or cable. Air Resistance (𝐹./% ): Opposes motion when moving through air. Examples of Free-Body Diagrams Example 1: A Book Resting on a Table Forces Acting: 𝐹" : Weight of the book (downward). 𝐹) : Normal force exerted by the table (upward). Diagram: A box with an upward arrow (𝐹) ) and a downward arrow (𝐹" ) of equal lengths. Example 2: A Box Sliding Down an Inclined Plane Forces Acting: 𝐹" : Weight of the box (acts vertically downward). 𝐹) : Normal force (perpendicular to the incline). 𝐹( : Frictional force (opposes motion, parallel to the incline). Breakdown: The weight (𝐹" ) is split into two components: 𝐹" , ∥ =mgsinθ (parallel to incline). 𝐹" , ⊥ =mgcosθ (perpendicular to incline). Diagram: A box on an incline with arrows showing 𝐹) , 𝐹( , and the components of 𝐹". Example 3: An Object Hanging by a Rope Forces Acting: 𝐹" : Weight of the object (downward). 𝐹- : Tension in the rope (upward). Diagram: A dot with an upward arrow (𝐹- ) and a downward arrow (𝐹" ).