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Chapter 6. Rigid Body Dynamics (geostatics)

Recommended Article : 【Physics】 Physics Table of Contents


1. Rotational Motion System

2. Uniform Circular Motion

3. Non-Uniform Circular Motion

4. Tools and Torque



1. Rotational Motion System

Table. 1. Major Physical Quantities of Rotational Motion

⑴ Angular Velocity ω : Rate of change of angular displacement per unit time

① Angular velocity is defined as positive when counterclockwise

② Corresponds to velocity v in translational motion

⑵ Angular Acceleration α : Rate of change of angular velocity per unit time

① Angular acceleration is defined as positive when counterclockwise

② Corresponds to acceleration a in translational motion



2. Uniform Circular Motion

⑴ Centripetal Acceleration and Tangential Acceleration

① Centripetal acceleration: Acceleration towards the center in uniform circular motion

○ (Note) Uniform circular motion involves acceleration because the direction of motion constantly changes

Derivation 1: Derivation using hodograph

Figure. 1. Derivation using hodograph

Derivation 2: Derivation using dimensional analysis

Derivation 3: Derivation using coordinate system analysis

② Tangential acceleration: Acceleration in the direction of the tangent

○ Always 0 in uniform circular motion

○ Represented by α

⑵ Centrifugal Force

① Definition : Centripetal acceleration × Mass of the object

② Acts perpendicular to the object’s motion direction, so it doesn’t increase the object’s energy

③ Centripetal Force

○ Definition : Hypothetical force acting radially outward in an accelerating reference frame rotating with the object

○ From the perspective of an observer rotating at the same speed : The object appears stationary. Equilibrium relationship between centripetal force and centrifugal force



3. Non-Uniform Circular Motion

⑴ Overview

① Definition : Net force acting on an object in the direction not perpendicular to the motion direction in circular motion

② Period, angular velocity, etc., are not observed to be constant

⑵ Theory of Non-Uniform Circular Motion

① Non-uniform circular motion of a single point mass

② Non-uniform circular motion of multiple point masses

○ r vector : Vector from the center of rotation to the point mass

○ Direction of τ : Direction where your thumb points when you curl your fingers from r vector to F vector at a small angle

⑶ Rotational Inertia (Moment of Inertia)

① Definition : Sum of the squares of the individual masses (mi) multiplied by the square of their distances (ri) from the axis of rotation Σmi ri2 (unit : kg·㎡)

② Rotational inertia of multiple objects

Table. 2. Rotational Inertia of Multiple Objects

③ Parallel-Axis Theorem : Formula to calculate rotational inertia when the center of rotation is not at the center of mass

○ ICM : Rotational inertia at the center of mass

○ M : Total mass

○ h : Distance between the axis of rotation and the center of mass

⑷ Angular Acceleration Motion Formula

⑸ Torque

① Definition : Cause of angular acceleration in a rigid body

② Corresponds to force F in translational motion

③ Formulation

⑹ Angular Momentum

① Formulation

② Angular momentum is defined as positive when counterclockwise

③ Corresponds to linear momentum p = mv in translational motion

④ Law of Conservation of Angular Momentum

○ Law of Torques : If there is symmetry, there must be a corresponding conserved quantity

○ The universe has rotational symmetry in space

○ The corresponding conserved quantity is believed to be angular momentum

⑺ Rotational Energy

① Formulation

② Rotational energy corresponds to translational kinetic energy in translational motion

③ When an object has both translational and rotational motion, total kinetic energy is the sum of translational and rotational kinetic energies

⑻ Gyroscopic Effect

① Definition : If an object rotates rapidly, it remains stable

② Principle : Angular momentum

③ Example : Riding a bicycle



4. Tools and Torque

⑴ Lever

① Definition : A tool for gaining mechanical advantage by setting a fulcrum point, an effort point, and a load point on a beam

○ Fulcrum point : The point where the beam rests on the fulcrum base

○ Effort point : The point where the lever applies force to the object

○ Load point : The point where force acts on the lever

② Principle of the lever

③ Types of levers

○ First-class lever : Lever with effort and load points on either side of the fulcrum

○ Examples : Scissors, wrench, balance scale, rope, etc.

○ Second-class lever : Lever with the effort point between the load point and the fulcrum

○ Examples : Bottle opener, wheelbarrow, nail clipper, etc.

○ Third-class lever : Lever with the load point between the effort point and the fulcrum

○ Examples : Tweezers, fishing rod, chopsticks, etc.

⑵ Axle and Wheel

① Definition : A device consisting of two circular wheels of different diameters attached to a common axis, allowing them to rotate together

② Principle of the axle and wheel : Same as the principle of the lever

③ Types of axle and wheel

○ Fixed pulley : Can lift an object with the same force as its weight by pulling at a constant velocity

○ Movable pulley : Similar to a second-class lever when lifting an object at a constant velocity

④ Examples : Car steering wheel, gear shift lever, bicycle gears, etc.

Principle of Work : When using a tool to perform work, even if there is a mechanical advantage, the amount of work remains unchanged due to losses in distance traveled



Input : 2019.03.28 19:55

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