Chapter 5: Work, Energy, and Power Key Definitions Work: Done when a force causes displacement. Mathematically, work \( W = \mathbf{F} \cdot \mathbf{d} = Fd \cos \theta \), where \( \theta \) is the angle between force (\( \mathbf{F} \)) and displacement (\( \mathbf{d} \)). Energy: The capacity to do work. Types include kinetic (energy […]
Chapter 5 / 5.1 Introduction Work, Energy and Power Read More »
Key Concepts in Mechanics 1. Banked Roads and Circular Motion When a car moves on a banked road, the normal force (N) and friction help provide the centripetal force needed for turning. The maximum speed (v_{\text{max}}) to avoid slipping is: \[ v_{\text{max}} = \sqrt{ Rg \frac{\mu_s + \tan \theta}{1 – \mu_s \tan \theta} } \]
Circular Motion Basics When an object moves in a circle with uniform speed, it experiences centripetal acceleration directed toward the center: \(\boxed{a_c = \frac{v^2}{R}}\) The force causing this acceleration is the centripetal force: \(\boxed{f_c = \frac{mv^2}{R}}\) Examples: Tension in a string (for a rotating stone), gravitational force (for planets), or friction (for cars on roads).
Common Forces in Mechanics Key Types of Forces Gravitational Force: Acts between objects with mass. Always pulls objects toward Earth (e.g., weight = \(mg\)). Contact Forces: Require physical touch. Examples: Normal Reaction (N): Perpendicular to surfaces in contact. Friction (f): Parallel to surfaces. Opposes relative motion. Tension (T): Force in strings/ropes. Assumed constant if massless.
What is Equilibrium of a Particle? A particle is in equilibrium when the net external force acting on it is zero. This means: It is either at rest or moving with constant velocity (no acceleration). Conditions for Equilibrium Two forces: They must be equal in magnitude and opposite in direction. Equation: \( F_1 = -F_2
Conservation of Momentum Key Idea: When no external forces act on a system, the total momentum of the system remains constant. Example: When a bullet is fired from a gun, the bullet and gun have equal and opposite momenta. Total momentum before and after firing is zero. Formula: For two colliding bodies A and B:
Newton’s Third Law of Motion Key Statement: “To every action, there is always an equal and opposite reaction.” Forces always occur in pairs between two interacting bodies. Important Points 💡 Action and reaction forces are equal in magnitude, opposite in direction, and act on different bodies. For example: If body A exerts a force FAB
Newton’s Second Law of Motion Key Idea Force is directly related to how fast an object’s momentum changes. The bigger the force, the faster the change in momentum. Momentum Momentum (\( \mathbf{p} \)) = mass (\( m \)) × velocity (\( \mathbf{v} \)): \( \mathbf{p} = m \mathbf{v} \) Examples: A heavy truck needs more
Newton’s First Law of Motion Key Idea: An object stays at rest or moves at a constant speed in a straight line unless a net external force acts on it. 1. Galileo’s Insight A ball rolling on a horizontal plane stops due to friction. Without friction, it would keep moving forever! Rest and uniform motion
Newton’s First Law of Motion (Law of Inertia) Key Ideas 🔄 Inertia: A body resists changes to its state of rest or motion. 🚫 Aristotle’s Mistake: He thought forces are needed to keep objects moving. Example: A toy car stops when you stop pulling it. 🔧 Galileo’s Insight: Friction is the hidden force that stops
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