Q2-WEEK-1-General-Physics-1 PDF

WEEK 1-4 TOPICS OVERVIEW 2nd QUARTER TOPICS FOR WEEK 1 - ROTATIONAL EQUILIBRIUM AND ROTATIONAL DYNAMICS 1. Moment of inertia 2. Angular position, angular velocity, angular acceleration 3. Torque 4. Torque-angular acceleration relation 5. Static equilibrium TOPICS FOR WEEK 1 - ROTATIONAL EQUILIBR...

WEEK 1-4 TOPICS OVERVIEW 2nd QUARTER TOPICS FOR WEEK 1 - ROTATIONAL EQUILIBRIUM AND ROTATIONAL DYNAMICS 1. Moment of inertia 2. Angular position, angular velocity, angular acceleration 3. Torque 4. Torque-angular acceleration relation 5. Static equilibrium TOPICS FOR WEEK 1 - ROTATIONAL EQUILIBRIUM AND ROTATIONAL DYNAMICS 6. Rotational kinematics 7. Work done by a torque 8. Rotational kinetic energy 9. Angular momentum TOPICS FOR WEEK 2 - GRAVITY 1. Newton’s Law of Universal Gravitation 2. Gravitational field 3. Gravitational potential energy 4. Escape velocity 5. Orbits 6. Kepler’s laws of planetary motion TOPICS FOR WEEK 3 - PERIODIC MOTION 1. Periodic Motion 2. Simple harmonic motion: spring-mass system, simple pendulum, physical pendulum 3. Damped and Driven oscillation 4. Periodic Motion experiment 5. Mechanical waves TOPICS FOR WEEK 4 - MECHANICAL WAVES AND SOUND 1. Sound 2. Wave Intensity 3. Interference and beats 4. Standing waves 5. Doppler effect PRE-TEST TRUE OR FALSE NO. 1 If you want to make an object move, apply force. NO. 2 Torque is the quantity that measures the ability of a force to rotate an object around some axis. NO. 3 Inertia is the measurement of a rotation caused by a force. NO. 4 Fulcrum or pivot point refers to the point of rotation. NO. 5 Positive torque is when turned counterclockwise and negative torque is when turned clockwise. GENERAL PHYSICS 1 ROTATIONAL EQUILIBRIUM AND ROTATIONAL DYNAMICS QUARTER 2, WEEK 1  ROTATIONAL MOTION VS. ROTATIONAL DYNAMICS ROTATIONAL MOTION Refers to the motion of an object as it rotates around a fixed axis. It deals with kinematics of rotation, meaning it focuses on describing how an object moves (without considering the forces causing the motion). ROTATIONAL DYNAMICS Refers to the study of the forces and torques that cause rotational motion. It focuses on explaining why an object rotates the way it does, considering factors like force and inertia. HISTORY OF ROTATIONAL DYNAMICS ROTATIONAL DYNAMICS Rotational dynamics is the branch of classical mechanics that deals with the motion of objects that rotate or spin. CLASSICAL MECHANICS Classical mechanics is a branch of physics that deals with the motion of objects and the forces that act upon them. ANCIENT AND EARLY THEORIES Ancient Greeks (circa 4th century BCE) Early concepts of rotation can be traced back to the Greek philosopher Aristotle, who studied the movement of celestial bodies. Aristotle believed that the heavens were perfect and that celestial bodies moved in perfect circles, which was an early hint at rotational motion. ANCIENT AND EARLY THEORIES Archimedes (circa 287–212 BCE) Known for his work on levers and pulleys, Archimedes contributed indirectly to rotational dynamics. He laid the foundation for the concept of torque, the rotational analog of force, though it wasn't formalized until much later. MEDIEVAL CONTRIBUTIONS Islamic Scholars (9th–12th centuries) Islamic scholars, such as Ibn al-Haytham and Al-Biruni, built upon Greek knowledge and refined the understanding of mechanics. While their focus was primarily on linear motion, they contributed to the eventual study of rotational motion. MEDIEVAL RENAISSANCE AND GALILEO’S INSIGHTS Nicolaus Copernicus (1473–1543) Though not directly focused on rotational dynamics, Copernicus’ heliocentric model of the solar system emphasized the rotation of the Earth, which set the stage for the study of rotating systems. MEDIEVAL RENAISSANCE AND GALILEO’S INSIGHTS Galileo Galilei (1564–1642) Galileo made significant contributions to dynamics, particularly with his studies of pendulums and circular motion. His observations of falling bodies and inclined planes led to early insights about the relationship between angular velocity and linear motion ISAAC NEWTON AND CLASSICAL MECHANICS (17TH CENTURY) Isaac Newton (1642–1727) Newton’s laws of motion, laid out in his 1687 book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), are the cornerstone of rotational dynamics. While Newton’s focus was largely on linear motion, he formulated the basic principles of angular momentum, torque, and rotational inertia. He extended his second law of motion to rotational systems MOMENT OF INERTIA AND TORQUE WHAT IS INERTIA? a. Inertia is a property of matter that describes an object's resistance to any change in its state of motion. b. The greater the mass of an object, the greater its inertia. Example: A heavier object will be harder to move from rest or to stop once it's moving, compared to a lighter object. WHAT MOMENT OF INERTIA? a property that quantifies how difficult it is to change the rotational motion of an object about a specific axis. It is the rotational equivalent of mass for linear motion. WHAT MOMENT OF INERTIA? a. Moment of inertia depends on the distribution of the mass. b. A mass which is at greater distance from the axis of rotation has a greater moment of inertia compared to the same mass which is near the axis of rotation. WHAT MOMENT OF INERTIA? MOMENT OF INERTIA Moment of inertia is a scalar quantity, meaning it has magnitude but no direction. The moment of inertia is highly sensitive to the shape and distribution of mass in an object. Objects with mass concentrated farther from the axis of rotation have larger moments of inertia. The SI Unit for moment of inertia is 𝑘𝑔𝑚2  MOMENT OF INERTIA OF A POINT MASS a.point mass in physics refers to an idealized object that has mass but occupies no space. It is considered to be concentrated at a single point in space. b.The moment of inertia of a point mass is a measure of how resistant the object is to rotational motion about a specific axis. MOMENT OF INERTIA OF A POINT MASS The moment of inertia (I) for a point mass m at a distance r from an axis of rotation is given by the formula: 𝑰=𝒎 𝒓 𝟐 Here's what each variable represents: I is the moment of inertia. m is the mass of the point. r is the perpendicular distance from the axis of rotation to the point mass MOMENT OF INERTIA OF A POINT MASS Example: Suppose you have a point mass of 2𝑘𝑔 located at a distance of 3𝑚 from an axis of rotation. Calculate the moment of inertia for this point mass. MOMENT OF INERTIA OF A POINT MASS For a system made up of several particles, the moment of inertia of the system is the sum of the individual moments of inertia. 𝟐 𝟐 𝟐 𝑰 = 𝒎 𝟏 𝒓 𝟏 + 𝒎𝟐 𝒓 𝟐 + 𝒎𝟑 𝒓 𝟑 + … MOMENT OF INERTIA OF A POINT MASS Example: Calculate the total moment of inertia of a system consisting of four children sitting at different distances from the center of a rotating playground carousel. Child 1: 30kg, 1.5m Child 2: 25kg, 1m Child 3: 20, 2m Child 4: 35kg, 1.8m MOMENT OF INERTIA: RADIUS OF GYRATION a. The radius of gyration gives a measure of how far the mass of a body is distributed from the axis of rotation. b. A larger radius of gyration means that the mass is distributed further away from the axis, resulting in a larger moment of inertia for the same total mass. c. In practical applications, knowing the radius of gyration allows engineers and designers to assess the stability and behavior of structures or mechanical systems under rotational forces. RADIUS OF GYRATION VS. MOMENT OF INERTIA Moment of inertia is the primary quantity describing the distribution of mass. Radius of gyration offers a simplified measure of how far the mass is distributed from the axis. MOMENT OF INERTIA: RADIUS OF GYRATION 𝐼 𝑘= 𝑀 Where: 𝐼 is moment of inertia of the object about the axis 𝑀 is the total mass of the object MOMENT OF INERTIA OF UNIFORM AND REGULAR SHAPED BODIES MOMENT OF INERTIA Example: A uniform circular disk or solid cylinder of radius of 0.5 cm and mass of 2kg is rotating about its central axis. Calculate the moment of inertia (I), assuming the disk has a uniform mass distribution. MOMENT OF INERTIA A thin uniform rod of length of 2m and mass of 4kg is rotating about an axis perpendicular to the rod and passing through one end. Calculate the moment of inertia (I) and the radius of gyration (k) for the rod. MOMENT OF INERTIA (TRY THIS!) A gymnast is performing a routine on a horizontal bar. The bar is a uniform rod of length 3m and mass 5kg. The gymnast is holding the bar at one end and spinning around a vertical axis. Calculate the moment of inertia and the radius of gyration for this system.  RECALL The moment of inertia also known as rotational inertia is the measure of the resistance of a body to a change in its rotational motion. The larger the moment of inertia of a body, the more difficult it is to change its rotational motion. Radius of gyration (k) is the distance from an axis of rotation where the mass of a body may be assumed to be concentrated without altering the moment of inertia of the body about that axis. Before solving for the moment of inertia, consider first the shape of the object and its specified axis of rotation. TORQUE The effectiveness of a force in rotating a body on which it acts is called torque or moment of force. The Greek letter tau τ is usually used to represent torque. TORQUE τ=(F)(r)(sin(θ)) where: τ is the torque (Nm) F is the force applied (N) r is the distance from the axis of rotation to the point where the force is applied (m) 𝜃 is the angle between the force vector and the lever arm. TORQUE Torque may also be positive or negative depending on the sense of rotation. By convention, torque is positive if it tends to produce counterclockwise rotation. It is negative if it tends to produce clockwise rotation. 𝜃 ≤90∘: Positive torque (counterclockwise rotation). 𝜃>90∘: Negative torque (clockwise rotation) TORQUE You are tightening a nut with a wrench. If you apply a force of 15 Newtons at the end of a wrench that is 0.2 meters long, and the angle between the force and the wrench is 30 degrees, what is the torque applied to the nut? TORQUE You are trying to close a door with a hinge on the left by pulling it with a force of 30 Newtons at the edge of the door, using a lever that is 0.5 meters long. If the angle between the force and the lever is 150 degrees, what is the torque applied to the door?  ACTIVITY – IDENTIFICATION (1/2 SHEET OF PAPER) 1. What do you call the motion of an object as it rotates around a fixed axis? 2. It focuses on explaining why an object rotates the way it does, considering factors like force and inertia. 3. Who believed that celestial bodies moved in perfect circles? 4. Who laid the foundation for the concept of torque? 5. Who believed that the Sun is at the center of the solar system, with the Earth rotating around it?  ACTIVITY – IDENTIFICATION (1/2 SHEET OF PAPER) 6. Who wrote the book Mathematical Principles of Natural Philosophy, which is considered the cornerstone of rotational dynamics? 7. What do you call the property of matter that describes an object's resistance to any change in its state of motion? 8. It is the rotational equivalent of mass for linear motion. 9. What is the SI unit of moment of inertia? 10. It is considered to be concentrated at a single point in space. 11. It gives a measure of how far the mass of a body is distributed from the axis of rotation. 12. What Greek letter is usually used to represent torque? ROTATIONAL QUANTITIES AND STATIC EQUILIBRIUM WHAT IS ROTATION? Rotation refers to the motion of a body turning about an axis, where each particle of the body moves along a circular path. TRANSLATIONAL MOTION Translational motion refers to the movement of an object in which every point of the object moves by the same distance in a given direction. Example: distance, velocity, and acceleration in kinematics (linear quantities) CONNECTION OF ANGULAR QUANTITIES TO ROTATION Angular quantities are a way to describe rotational motion. They are analogous to the linear quantities used to describe translational motion. ROTATIONAL KINEMATICS The kinematics of rotation deals with the motion of objects that rotate about an axis. It involves the description of rotational displacement, velocity, and acceleration, similar to how kinematics in translational motion describes these quantities for objects moving in a straight line.  Kinematics ↓ Translational Motion ↓ Linear Quantities Rotational Kinematics ↓ Rotational Motion ↓ Angular Quantities KINEMATICS ROTATION Angular displacement refers to the change in the angular position of a rotating object. It measures how much an object has rotated about a fixed axis, considering both the magnitude and direction of the rotation.  ANGULAR DISPLACEMENT Basic Formula for Angular Displacement: 𝑠 𝜃= 𝑟 𝜃 is the angular displacement in radians or degree 𝑠 is the arc length (distance traveled along the circular path) 𝑟 radius of the circular path Take note: It is measured in radians or degrees. ANGULAR DISPLACEMENT A bicycle wheel has a radius of 0.5 meters. If a point on the tire travels 2 meters along the circumference of the wheel, what is the angular displacement of the wheel? ANGULAR DISPLACEMENT If the motion is in the context of circular motion where the object is moving with a period of revolution 𝐶 2𝜋𝑟 𝑠= or 𝑠 = 𝑡 𝑡 𝐶 is circumference of the circle (𝑚) 𝑡 is time in seconds ANGULAR DISPLACEMENT Sara is playing on a merry-go-round with a radius of 2 meters. She starts at the topmost position and spins counterclockwise for 45 seconds. Calculate the angular displacement of Sara during this time. KINEMATICS ROTATION Angular distance refers to the total rotation an object undergoes around a fixed axis and is typically measured in radians or degrees. It represents the cumulative amount of "path" traveled along the circumference of a circle, regardless of the direction of rotation. It's essentially the absolute value of angular displacement if the motion is straightforward (one rotation). ANGULAR DISPLACEMENT VS. DISTANCE Angular displacement can be positive or negative depending on the direction of rotation (clockwise or counterclockwise). Angular distance is always a positive value, as it measures the total amount of rotation without regard to direction, much like linear distance measures total travel regardless of forward or backward motion.  KINEMATICS ROTATION Angular velocity is a measure of how fast an object rotates or spins around a central point or axis. It describes the rate of change of angular displacement over time, meaning how quickly the angle (in radians or degrees) changes as the object rotates. ω - lowercase Greek letter omega ANGULAR VELOCITY The formula for angular velocity is: Δ𝜃 2𝜋(𝑟𝑒𝑣) ω= 𝑜𝑟 Δ𝑡 Δ𝑡 where: ω is the angular velocity (𝑟𝑎𝑑/𝑠) Δ𝜃 is the change in angular displacement (𝑟𝑎𝑑) Δt is the change in time 𝑟𝑒𝑣 is the number of revolution. ANGULAR VELOCITY Amy is twirling a baton, making continuous circular motions. In a 40- second performance, the baton completes 15 full revolutions. Calculate the angular velocity of the baton. ANGULAR VELOCITY John is spinning a wheel on a toy car. The wheel has a radius of 0.2 meters, and it completes 3 full rotations in 6 seconds. Calculate the angular velocity of the wheel. KINEMATICS ROTATION Angular acceleration is the rate at which an object's angular velocity changes over time. It describes how quickly an object is speeding up or slowing down its rotation. Like linear acceleration measures changes in linear velocity, angular acceleration measures changes in angular velocity. α- It is a lowercase Greek letter alpha ANGULAR ACCELERATION The formula for Angular acceleration (α): Δω α= Δt where: α is the angular acceleration (𝑟𝑎𝑑/𝑠 2 ) 𝑟𝑎𝑑 Δω is the change in angular velocity ( ) 𝑠 Δt is the change in time. ANGULAR ACCELERATION Tom is riding a Ferris wheel, and the wheel starts from rest. Over the first 20 seconds, the Ferris wheel completes 5 full revolutions, reaching an angular velocity of 2π radians per second. Calculate the angular acceleration of the Ferris wheel during this time. ANGULAR ACCELERATION Dave is rotating a wheel. At time t=0, the wheel 𝜋 starts with an initial angular velocity of radians 2 per second. Over the next 6 seconds, the wheel undergoes uniform angular acceleration and reaches a final angular velocity of 2π radians per second. Calculate the angular acceleration of the wheel. STATIC EQUILIBRIUM It occurs when an object remains completely at rest, with no movement or rotation, because all the forces and torques acting on it are perfectly balanced. STATIC EQUILIBRIUM CHARACTERISTICS Net force is zero This means the total forces in all directions (horizontal and vertical) cancel each other out, so the object doesn't move. STATIC EQUILIBRIUM CHARACTERISTICS Net Torque is zero This means there is no rotational motion because all the torques (forces causing rotation) cancel each other out. STATIC EQUILIBRIUM CHARACTERISTICS No Linear nor Angular Acceleration The object remains stationary because the forces acting on it are balanced.  STATIC EQUILIBRIUM EXAMPLE A ladder leaning against a wall The ladder stays at rest if the following conditions are met: 1. The force of gravity pulls the ladder downwards. 2. The wall provides a horizontal force to prevent the ladder from slipping. 3. The ground provides an upward normal force and a horizontal frictional force to prevent sliding. STATIC EQUILIBRIUM EXAMPLE INVOLVING TORQUE Two people on a seesaw To achieve static equilibrium, their torques must balance. If one person is heavier, they need to sit closer to the fulcrum, while the lighter person sits farther away. The product of force and distance must be the same on both sides for equilibrium. 𝒎𝟏 × 𝒅𝟏 = 𝒎𝟐 × 𝒅𝟐  STATIC EQUILIBRIUM EXAMPLE INVOLVING TORQUE Imagine a seesaw in a playground. On one end, there's a 60 kg person, and on the other end, there are two children. One child weighs 25 kg and is sitting 2 meters away from the fulcrum, and the other child weighs 30 kg and is sitting 1.5 meters away from the fulcrum. The seesaw itself has a mass of 10 kg. At what distance from the fulcrum should the 60 kg person sit to maintain static equilibrium? ROTATIONAL KINEMATIC RELATIONS FOR SYSTEMS WITH CONSTANT ANGULAR ACCELERATIONS ROTATIONAL KINEMATIC RELATIONS FOR SYSTEMS WITH CONSTANT ANGULAR ACCELERATIONS Rotational kinematic relations describe the motion of rotating objects under constant angular acceleration, similar to linear kinematics. These equations can be used to determine quantities like angular displacement, angular velocity, and time.  ROTATIONAL KINEMATIC RELATIONS FOR SYSTEMS WITH CONSTANT ANGULAR ACCELERATIONS Angular Displacement Final angular velocity 1 2 ω𝑓 =ω𝑖 +​ αt 𝜃𝑓 = 𝜃𝑖 + 𝜔𝑖 𝑡 + (𝛼)(𝑡 ) 2 Angular displacement in terms of Final angular velocity squared angular velocity ω𝑓2 = ω2𝑖 + 2α(𝜃𝑓 − 𝜃𝑖 ) ω𝑖 + ω𝑓 𝜃𝑓 = 𝜃𝑖 + 𝑡 2 Angular displacement in terms of angular velocity and acceleration 1 2 𝜃 = ω𝑖 𝑡 + α𝑡 2 ROTATIONAL KINEMATIC RELATIONS FOR SYSTEMS WITH CONSTANT ANGULAR ACCELERATIONS A ceiling fan starts from rest and accelerates with a 𝑟𝑎𝑑 constant angular acceleration of 0.5 2. 𝑠 a. How long will it take for the fan to reach an 𝑟𝑎𝑑 angular velocity of 5 ? 𝑠 b. How many revolutions will the fan complete during this time? ROTATIONAL KINETIC ENERGY RECALL: KINETIC ENERGY Kinetic energy is the energy that an object possesses due to its motion. 1 2 𝐾𝐸 = 𝑚𝑣 2 ROTATIONAL KINETIC ENERGY Rotational kinetic energy is the kinetic energy associated with the rotation of an object around an axis. ROTATIONAL KINETIC ENERGY The formula for rotational kinetic energy depends on the moment of inertia (I) and the angular velocity (ω) of the rotating object. 𝟏 𝟐 𝑲𝒓𝒐𝒕 = 𝑰ω 𝟐 Where: 𝐾𝑟𝑜𝑡 : Rotational kinetic energy. (J) I: Moment of inertia of the rotating object. ω: Angular velocity of the rotating object. ROTATIONAL KINETIC ENERGY Consider a flywheel, a rotating mechanical device, with a moment of inertia (I) of 5 kg·m². The flywheel is initially at rest but is accelerated to an angular velocity (ω) of 10 radians per second in 4 seconds. Calculate the rotational kinetic energy gained by the flywheel during this acceleration. ROTATIONAL KINETIC ENERGY merry-go-round at a playground has a moment of inertia (I) of 200 kg·m². It starts from rest and accelerates uniformly to an angular velocity (ω) of 4 radians per second in a time (t) of 10 seconds. Calculate the rotational kinetic energy acquired by the merry- go-round during this acceleration. ANGULAR MOMENTUM LET’S RECALL: MOMENTUM Momentum is a measure of the motion of an object and is defined as the product of its mass and velocity. It represents how difficult it is to stop or change the direction of a moving object. 𝒑 = 𝒎𝒗 ANGULAR MOMENTUM Angular momentum is the rotational equivalent of linear momentum and represents the quantity of rotational motion an object has around an axis. ANGULAR MOMENTUM The formula for angular momentum (L) is given by: 𝐿 = 𝐼𝜔 Where: 𝑘𝑔𝑚2 L is the angular momentum ( ) 𝑠 I is the moment of inertia (a measure of an object's resistance to changes in its rotation) ω is the angular velocity (the rate at which an object rotates) ANGULAR MOMENTUM EXAMPLE A cylinder with a mass of 5 kg and a radius of 10 cm is rotating about its central axis with an angular velocity of 12 rad/s. Calculate the angular momentum of the cylinder. MOMENT OF INERTIA OF UNIFORM AND REGULAR SHAPED BODIES ANGULAR MOMENTUM EXAMPLE Imagine a figure skater spinning on ice. Initially, her moment of inertia (I) with her arms extended is 3 kg·m², and her angular velocity (ω) is 4 rad/s. She then pulls her arms in, reducing her moment of inertia to 1 kg·m². What is her final angular velocity after pulling her arms in?