Physics Equations of Motion and Force Diagrams PDF

Equations of Motion Under Constant Acceleration The equations of motion describe the relationship between displacement, velocity, acceleration, and time when an object is moving under constant acceleration. These are derived from Newton’s second law of motion and are fundamental in solving kinematic...

Equations of Motion Under Constant Acceleration The equations of motion describe the relationship between displacement, velocity, acceleration, and time when an object is moving under constant acceleration. These are derived from Newton’s second law of motion and are fundamental in solving kinematic problems in physics. The Three Main Equations First Equation of Motion: 𝑣 = 𝑢 + 𝑎𝑡 Where 𝑣: Final velocity, u: Initial velocity (𝑚/𝑠), 𝑎: Acceleration (𝑚⁄𝑠 ! ), 𝑡: Time (𝑠) Interpretation: Describes how an object’s velocity changes over time under constant acceleration. " Second Equation of Motion: 𝑠 = 𝑢𝑡 + 𝑎𝑡 ! where 𝑠: Displacement (𝑚) ! 𝑢: Initial velocity (𝑚/𝑠), 𝑎: Acceleration (𝑚⁄𝑠 ! ), 𝑡: Time (𝑠) Interpretation: Gives the total displacement of the object after time 𝑡 under constant acceleration. Third Equation of Motion: 𝑣 ! = 𝑢! + 2𝑎𝑠 𝑣: Final velocity (𝑚/𝑠), 𝑢: Initial velocity (𝑚/𝑠), 𝑎: Acceleration (𝑚⁄𝑠 ! ), 𝑠: Displacement (𝑚). Interpretation: Relates the object’s velocities (initial and final), displacement, and acceleration, without involving time. Applications of the Equations First Equation: To calculate the final velocity of an object after a given time of acceleration. Example: A car accelerating from rest to a certain velocity. Second Equation: To determine the total displacement or distance traveled in a given time. Example: Calculating how far a dropped object falls in a certain time under gravity. Third Equation: Useful when time is unknown but the displacement, acceleration, and initial velocity are given. Example: Calculating the speed of a ball just before it hits the ground. Force Diagrams (Free-Body Diagrams) and Vector Addition of Forces 1. Force Diagrams (Free-Body Diagrams): A force diagram (or free-body diagram) is a graphical representation of all the forces acting on a single object. It is an essential tool for solving problems in mechanics. Steps to Draw a Free-Body Diagram: Identify the object: Focus on the object being analyzed. Isolate the object: Imagine it as a point or a block separated from its surroundings. Draw all forces acting on the object: Represent each force as an arrow starting from the object. The direction of the arrow shows the direction of the force, and the length represents its magnitude. Common forces to include: Weight (𝐹# or 𝑚𝑔): Acts downward due to gravity. Normal Force (𝐹$ ): Acts perpendicular to the surface supporting the object. Friction (𝐹% ): Opposes the motion, parallel to the surface. Tension (𝑇): Acts along a rope or cable. Applied Force (𝐹&''()*+ ) : Any external push or pull. Air Resistance (𝐹&), ) : Acts opposite to the direction of motion. Importance of Free-Body Diagrams: They help break down forces into components for analysis. They allow visualization of the net force acting on an object. They simplify complex problems by isolating the object and its forces. Example of a Free-Body Diagram A box on a flat surface: Forces: Weight (𝐹# = 𝑚𝑔, acting downward. Normal force (𝐹$ ), acting upward, equal in magnitude to 𝐹# if no vertical acceleration. Frictional force (𝐹% ), opposing motion (if the box is moving or tends to move). Applied force (𝐹&''()*+ ), acting at an angle or horizontally. A block on an inclined plane: Forces: Weight (𝐹# ), acting vertically downward. Normal force (𝐹$ ), acting perpendicular to the plane. Frictional force (𝐹% ), opposing motion along the plane. Components of weight: 𝐹# 𝑐𝑜𝑠𝜃: Perpendicular to the incline. 𝐹# 𝑠𝑖𝑛𝜃: Parallel to the incline. Vector Addition of Forces In many situations, forces act on an object in different directions. To find the net force or resultant force, we use vector addition. Rules of Vector Addition: For forces in the same direction: Add their magnitudes directly. 𝐹.*/ = 𝐹" + 𝐹! For forces in opposite directions: Subtract the smaller force from the larger force. 𝐹.*/ = 𝐹! − 𝐹" For forces at an angle (non-parallel): Break each force into components (horizontal and vertical). Use trigonometric functions (sin, cos, tan) to resolve forces. Add the components along each axis: 𝐹0 = 𝐹" 𝑐𝑜𝑠𝜃" + 𝐹! 𝑐𝑜𝑠𝜃! 𝐹0 = 𝐹" 𝑠𝑖𝑛𝜃" + 𝐹! 𝑠𝑖𝑛𝜃! Find the resultant force: 𝐹.*/ = ;𝐹0! + 𝐹1! 3 Find the direction (𝜃): 𝜃 = 𝑡𝑎𝑛2" ( ! ) 3" Examples of Vector Addition: 1. Forces at Right Angles: A box is pulled by a 10 N force horizontally and a 6 N force vertically. Find the resultant force. 2. Forces at an Angle: Two forces, 50 N at 304 and 40 N at 1204 , act on an object. Find the resultant force. Conservation of Momentum Definition of Momentum 1. Momentum is a fundamental concept in physics that describes the motion of an object in terms of its mass and velocity. It is a vector quantity, meaning it has both magnitude and direction. Mathematical Definition Momentum (𝑝) is defined as the product of an object's mass (𝑚) and its velocity (𝑣): p=mv Where: 𝑝 = momentum (kg·m/s) 𝑚 = mass of the object (kg), 𝑣 = velocity of the object (m/s) Since velocity is a vector, momentum also has direction. The momentum of an object depends on both how fast it is moving and in which direction. 2. Law of Conservation of Momentum The law of conservation of momentum states that: "In a closed and isolated system, the total momentum before an event (such as a collision or explosion) is equal to the total momentum after the event, provided no external forces act on the system." Mathematical Expression If two objects interact in a system, their total momentum before and after the interaction remains constant: 𝑝).)/)&( = 𝑝%).&( For a two-body system: 𝑚" 𝑣") + 𝑚! 𝑣!) = 𝑚" 𝑣"% + 𝑚! 𝑣!% Where: 𝑚" , 𝑚! = masses of objects 𝑣") , 𝑣!) = initial velocities 𝑣"% , 𝑣!% = final velocities This principle applies regardless of whether the interaction is elastic or inelastic. 3. Applications of the Conservation of Momentum (a) Collisions Collisions are common examples where momentum conservation plays a crucial role. Collisions can be elastic or inelastic: 1. Elastic Collisions: o Kinetic energy is conserved. o Example: Collision between gas molecules. 2. Inelastic Collisions: o Kinetic energy is not conserved (some energy is converted into heat, sound, or deformation). o Example: Car crashes, a bullet hitting a block of wood. 3. Perfectly Inelastic Collisions: o The two objects stick together after collision, moving as one mass. o Example: A lump of clay thrown onto a moving cart. In all types of collisions, momentum is always conserved. (b) Explosions In an explosion, a system that was initially at rest breaks apart into different fragments moving in different directions. Since no external forces act on the system, the total momentum before and after the explosion must be equal. Example: A bomb at rest explodes into fragments that move in different directions. A firecracker bursting into multiple pieces. (c) Rocket Propulsion The principle of conservation of momentum explains how rockets move in space: Rockets work by expelling exhaust gases backward at high speed. The momentum of the expelled gases is balanced by the forward momentum of the rocket. This principle follows Newton’s Third Law: For every action, there is an equal and opposite reaction. Example: Spacecraft propulsion in vacuum conditions. 4. Practical Examples of Momentum Conservation A moving train car coupling with another stationary train car. Recoil of a gun when a bullet is fired. A skateboarder pushing against a wall and moving backward. Newton’s cradle (a device with swinging spheres demonstrating momentum transfer). Summary Momentum (𝑝) is the product of mass and velocity: 𝑝 = 𝑚𝑣 The law of conservation of momentum states that total momentum remains constant in a closed system. Momentum conservation applies to collisions, explosions, and rocket propulsion. Understanding conservation of momentum helps explain various real-world phenomena, from car crashes to space travel. Work and Energy 1. Definition of Work Work is a fundamental concept in physics that describes how a force applied to an object causes displacement. Work is only done when there is movement in the direction of the applied force. Mathematical Definition Work (𝑊) is defined as the product of force (𝐹) and displacement (𝑑) in the direction of the force: 𝑊 = 𝐹𝑑cos𝜃 Where: 𝑊 = Work done (Joules, J) 𝐹 = Applied force (Newtons, N) 𝑑 = Displacement of the object (meters, m) 𝜃 = Angle between the force and displacement direction Conditions for Work to Be Done A force must be applied to an object. The object must move in the direction of the applied force. If displacement is zero, no work is done (e.g., holding a heavy object without moving it). Types of Work 1. Positive Work: When force and displacement are in the same direction (e.g., lifting an object). 2. Negative Work: When force and displacement are in opposite directions (e.g., friction slowing down a moving car). 3. Zero Work: When displacement is zero or force is perpendicular to displacement (e.g., carrying an object horizontally without lifting). 2. Definition of Energy Energy is the ability to do work. An object or system with energy can exert force and cause displacement. Energy is a scalar quantity and is measured in Joules (J). 3. Types of Energy (a) Kinetic Energy (𝐾𝐸) Kinetic energy is the energy of a moving object. The faster an object moves, the " more kinetic energy it has. 𝐾𝐸 = ! 𝑚𝑣 ! Where: 𝐾𝐸 = Kinetic energy (J), 𝑚 = Mass of the object (kg) 𝑣 = Velocity of the object (m/s) Example: A moving car, a rolling ball, or a running person all possess kinetic energy. (b) Potential Energy (𝑃𝐸) Potential energy is stored energy due to an object’s position or configuration. (i) Gravitational Potential Energy (𝑃𝐸# ) Energy stored in an object due to its height above the ground. 𝑃𝐸# = 𝑚𝑔ℎ Where 𝑃𝐸# = Gravitational potential energy (J), 𝑚 = Mass (kg), 𝑔= Acceleration due to gravity (9.81 m/s²), ℎ = Height above the reference point (m) Example: Water in a dam, a book on a shelf, or a stretched bowstring. (ii) Elastic Potential Energy Energy stored in stretched or compressed elastic materials like springs and rubber bands.  1 ! 𝑃𝐸 = 𝑘𝑥 2 Where: 𝑘 = Spring constant (N/m), 𝑥 = Compression or stretch length (m) Example: A stretched bowstring before releasing an arrow. (c) Thermal Energy Thermal energy is the internal energy of a substance due to the motion of its molecules. The higher the temperature, the greater the thermal energy. Example: Boiling water, the heat from the sun, and the warmth from friction. (d) Electrical Energy Energy carried by moving electrons in a conductor. Example: Batteries, lightning, and electric currents in appliances. 4. Law of Conservation of Energy The law of conservation of energy states that: "Energy cannot be created or destroyed; it can only be converted from one form to another." Mathematical Expression 𝐸).)/)&( = 𝐸%).&( Examples of Energy Transformation A falling object: Gravitational potential energy → Kinetic energy. A pendulum: Swings back and forth, converting kinetic energy into potential energy and vice versa. A battery powering a bulb: Chemical energy → Electrical energy → Light and heat energy. Burning fuel in a car engine: Chemical energy → Heat energy → Mechanical energy. 5. Summary Work (𝑊) is the product of force and displacement in the direction of the force. Energy is the ability to do work and exists in different forms like kinetic, potential, thermal, and electrical energy. The law of conservation of energy states that energy is never lost; it only transforms from one form to another. This principle explains many natural and technological processes, from the motion of objects to power generation and everyday activities like walking, driving, and using electrical devices. Power and Efficiency 1. Definition of Power Power is the rate at which work is done or energy is transferred over time. It measures how quickly energy is used or converted from one form to another. Mathematical Definition 5 6 Power (𝑃) is defined as: 𝑃 = / or 𝑃 = / Where: 𝑃 = Power (Watts, W), 𝑊 = Work done (Joules, J) = Energy transferred (Joules, J), 𝑡 = Time taken (seconds, s) SI Unit of Power The SI unit of power is the Watt (W), named after James Watt 789(* 1 𝑊𝑎𝑡𝑡 = 1 :*;8.+ Alternative Expressions of Power If work is done by a force (𝐹) causing displacement (𝑑), we use: 3+ 𝑃= / + Since velocity (𝑣) is displacement per unit time (𝑣 = / ), power can also be expressed as: P = Fv This equation is useful when dealing with moving objects like vehicles or machines. 2. Definition of Efficiency Efficiency is the ratio of useful energy output to total energy input, expressed as a percentage. It measures how effectively a system converts input energy into useful work. Mathematical Definition