Chapter 6. Rigid Body Dynamics (geostatics)
Recommended Article : 【Physics】 Physics Table of Contents
3. Non-Uniform Circular Motion
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.
⑶ 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