뒤로Fundamental Concepts and Problems in Classical Mechanics
스터디 가이드 - 스마트 노트
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Motion in One Dimension
Kinematic Equations
Motion in one dimension describes the movement of objects along a straight line, characterized by position, velocity, and acceleration.
Position as a function of time:
Velocity as a function of time:
Velocity as a function of position:
Constant acceleration: Acceleration remains unchanged over time.
Example: A car accelerates from rest at for $5v = 0 + 2 \times 5 = 10\,\text{m/s}$.
Circular Motion and Centripetal Acceleration
Centripetal Acceleration
Objects moving in a circle experience an acceleration directed toward the center of the circle, called centripetal acceleration.
Formula:
Where: is the speed of the object, is the radius of the circle.
Example: A ball moving at in a circle of radius has .
Quadratic Equation
The quadratic equation is used to solve for variables in kinematic equations and other physics problems.
General form:
Solution:
Spring Force
Hooke's Law
The force exerted by a spring is proportional to its displacement from equilibrium.
Formula:
Where: is the spring constant, is the displacement.
Gravitational Force
Newton's Law of Universal Gravitation
Describes the attractive force between two masses.
Formula:
Where: is the gravitational constant, and are masses, is the distance between centers.
Center of Mass
The center of mass of a system is the weighted average of the positions of all the masses.
Formula:
Rotational Dynamics
Rotational Form of Newton's Second Law
Relates torque to angular acceleration.
Formula:
Where: is torque, is moment of inertia, is angular acceleration.
Rolling Without Slipping
Describes the motion of objects that roll without sliding.
Relationship:
Where: is linear speed, is radius, is angular speed.
Tensile/Compressive Stress
Stress is the force per unit area applied to a material.
Formula:
Where: is force, is cross-sectional area.
Selected Physics Problems and Solutions
Orbital Mechanics and Gravitational Force
Problems involving gravitational force, orbital radius, and mass calculations for celestial bodies.
Body | Mass | Radius | Orbital radius | Orbital period |
|---|---|---|---|---|
Moon A | 4.0 × 1020 kg | unknown | 2.0 × 108 m | 4.0 × 105 s |
Moon B | 1.5 × 1020 kg | 2.0 × 105 m | 3.0 × 108 m | unknown |
Mithra | unknown | unknown | 3.0 × 108 m | unknown |
Application: Use and orbital period formulas to solve for unknowns.
Rotational Motion and Torque
Problems involving angular acceleration, torque, and moment of inertia.
Torque:
Angular acceleration:
Spring and Elasticity Problems
Problems involving spring stretching and Young's modulus.
Spring stretch:
Young's modulus:
Statics and Equilibrium
Problems involving forces, tension, and equilibrium of rigid bodies.
Equilibrium condition: ,
Application: Used to solve for unknown forces and tensions in signs, ladders, and beams.
Sample Problem: Ladder Against Wall
A ladder of length and mass is supported horizontally. Find the tension in the supporting wire.
Solution: Use torque equilibrium about the wall attachment point.
Sample Problem: Teeter-Totter
A child sits on a teeter-totter. Find the minimum distance for equilibrium given a force applied on the opposite side.
Solution: Set torques about the pivot equal for equilibrium.
Additional info:
Some problems involve conversion between angular and linear quantities: .
Stress and strain concepts are applied in elasticity problems.
All equations are standard for introductory college physics.
