Chapter 3. Laws of Motion
Recommended Article : 【Physics】 Physics Index
a. A person pushing a trolley gains a mechanical advantage of 2.
1. Newton’s Laws of Motion
⑴ Law 1 : Law of Inertia
① Inertia : The property of an object to maintain its state of motion
② Application 1: A stationary object remains at rest
③ Application 2: A moving object tends to move with constant velocity
④ Applies when no factors are influencing the object
⑵ Law 2 : Law of Acceleration
① Definition of force : The cause that changes the shape or state of motion of an object
② Formulation : Sometimes mass is not constant, so it’s best defined in terms of the rate of change of momentum
③ Momentum p = m × v
④ The m in F = ma is called the inertial mass : Unlike gravitational mass, it’s the source of inertia
⑶ Law 3 : Action-Reaction Law
① Every force has a pair
② Equal in magnitude and opposite in direction
2. Types of Forces
⑴ Type 1 : External Force
Force applied from outside. There are exactly four types of forces in physics
⑵ Type 1-1 : Universal Gravitation
① Universal Gravitation : Force between two objects with mass
Note: r : distance between objects, G : universal gravitational constant, 6.67 × 10-11 m3/s2·kg
○ Universal gravitation is interactive : If force is proportional to mass, it’s proportional to the product of masses
○ Universal gravitation inversely proportional to distance squared : Due to flux of gravitational field, which follows the inverse square law of distance
○ Distance dependence of universal gravitation assessed through analysis of celestial motion
○ Mass in universal gravitation referred to as gravitational mass : Unlike inertial mass, source of force
② Gravity
○ Gravity : The universal gravitation on Earth’s surface (r = 6370 km, M = 5.98 × 1024 kg)
○ Mass and weight
○ Mass : Inherent quantity of an object, constant regardless of location
○ Weight : Force due to mass, varies with location
○ Weightless state : State where net force is 0 (e.g., elevator in free fall)
○ Small-scale Earth movement has relatively constant gravity due to nearly constant object-Earth distance : g = 9.81 m/s2
③ Buoyancy : Force resulting from combination of gravity and Pascal’s principle
○ Buoyancy : Force on an object submerged in a fluid, tends to make it rise
○ Magnitude of buoyancy : Equal to the weight of fluid displaced by the object
⑶ Type 1-2 : Electromagnetic Force
① Electric Force : Force between charged particles
② Magnetic Force : Force on a magnetic object
③ Elastic Force : Force due to interatomic repulsion, making object return to original state
○ Elasticity : Property of returning to initial state after deformation
○ Elastic body : An object with elasticity (e.g., rubber used in springs and levers)
○ Elastic Force : Measure of force due to elasticity
○ Elastic limit : Point where elastic body can’t return to initial state
○ Hooke’s Law : Law stating force is proportional to displacement when stretching a spring
○ Formula : F ∝ x (displacement) → F = -kx
○ Negative sign signifies opposite direction of elongation
○ Spring constant (elastic modulus) : Force required to extend spring by 1 m
○ Compliance : Reciprocal of spring constant
○ Changes in spring constant in series connection : Easier elongation, smaller constant
○ Changes in spring constant in parallel connection : Harder elongation, larger constant
○ Simple Harmonic Motion due to elastic force
④ Vertical Force (Normal Force) : Essentially elastic force due to compression
○ All objects can be compressed like a spring. Hook’s law holds for all objects like for a spring
○ Vertical force only acts along the length of a bar
⑤ Tension
○ Definition : Force pulling adjacent parts of a string towards each other
○ Characteristic 1: External force acts in the direction of pulling the ends of the string : Tension only pulls the ends of the object
○ Characteristic 2: Tension on objects at both ends is the same
○ Tension participates in creating conditions for maximum and minimum in Mechanical Equilibrium
⑥ Friction Force : Force arising from vertical force that hinders object’s motion
f = μN
○ Friction proportional to vertical force
Figure 1. Reason why friction is proportional to vertical force
○ Case 1: Object’s shape remains constant
○ If vertical force doubles, object compresses twice as much
○ Deeper compression increases the contact area with the concave floor surface
○ In stationary object, friction proportional to the contact area hinders motion
○ In moving object, object has to push against surface in proportion to the contact area
○ Case 2: Object’s width doubles
○ If vertical force remains constant, compressed height is halved
○ Reason : Surface area doubled, stress halves
○ Surface area doubled, compressed height halved, so contact area with floor remains unchanged
○ Conclusion : Object’s shape doesn’t affect vertical force
○ Limits of this thought experiment
○ Limit 1: Neglects consideration of the object’s contact with the floor : Considers only the object’s side
○ Limit 2: Applies Hooke’s law as an approximation for the floor : Doesn’t consider floor as a separate object
○ The proportionality of friction to vertical force is an approximate conclusion.
○ Type 1. Static Friction : The friction when an object is at rest.
○ The net force on a stationary object is 0, so it is in equilibrium with external forces.
○ Type 1-1. Maximum Static Friction : The limit of static friction of an object, which is the friction just before the object starts to move.
○ Maximum static friction is the largest among friction forces.
○ Vertical force on the incline forming an angle θ with the ground.
○ Horizontal force on the incline forming an angle θ with the ground.
○ Maximum static friction on the incline forming an angle θ with the ground.
○ Type 2. Kinetic Friction : The friction when an object is in motion.
○ The coefficient of kinetic friction is smaller than the coefficient of static friction.
○ Type 3. Rolling Friction
○ Definition : Friction that occurs while rolling on a deformable horizontal surface.
○ Derivation of rolling friction magnitude.
Figure 2. Rolling friction of a rigid sphere on a deformable surface.
○ W : Weight of the object.
○ r : Radius of the object.
○ F : Minimum external force required for the object to roll. Also known as rolling friction or rolling resistance.
○ c : Coefficient of rolling resistance.
○ The minimum conditions for the object to roll and zero net torque are intuitively equivalent. ○ Generally, if the surface is rigid (not deformable), c = 0 and rolling friction does not occur.
○ Application 1. Slip ratio : 1 - (Actual distance moved by the center of mass in one rotation of the sphere) / (Distance moved by the center of mass in one rotation of the sphere when no slipping occurs)
○ Slip ratio is important in discussions of car wheels.
○ Relationship between friction and slip ratio.
Figure 3. Relationship between friction and slip ratio.
○ Friction is strongest when the slip ratio is between 10-15%.
○ When the slip ratio is 0, only static friction is present, similar to rotational motion of a rigid body.
○ From slip ratio 0 to maximum friction, static friction is dominant. Beyond that, kinetic friction becomes dominant.
○ Type 4. Internal Friction
○ Measurement of Friction Coefficients
○ Inclined Plane Method : Measures the coefficient of static friction using the moment when an object starts moving on an inclined plane.
○ Moving Surface Method : Measures both static and kinetic friction coefficients simultaneously.
⑦ Air Resistance : Friction due to air.
○ Terminal Velocity : The speed at which a freely falling object stops accelerating due to air resistance.
○ Reynolds Number(Re) < 1, indicating small or slow-moving objects : Resistance is primarily viscous resistance (∝ v).
○ Reynolds Number(Re) > 1,000, indicating large or fast-moving objects : Resistance is primarily pressure resistance (∝ v^2).
○ Air resistance generates both thrust and lift.
○ Thrust : Force that propels an object forward, e.g., by rotating a propeller or ejecting matter.
○ Lift : Force experienced vertically by an object immersed in a fluid.
⑷ Type 1-3. Weak Nuclear Force : Involved in nuclear reactions.
⑸ Type 1-4. Strong Nuclear Force : Binds protons and neutrons.
⑹ Type 2. Inertial Force
① Definition : The relative effect observed when an inertial frame of reference is accelerated; a virtual force.
② Magnitude : Mass of an object × acceleration of the inertial frame.
③ Direction : Opposite to the direction of acceleration of the inertial frame.
④ Example 1. Observer moving upward with an elevator accelerating by a : The observer perceives a downward inertial force of ma.
⑤ Example 2. Observer moving downward with an elevator accelerating by a : The observer perceives an upward inertial force of ma.
⑥ Example 3. Observer moving with a circular motion : The observer perceives a centripetal force of ma.
3. Analysis of Forces
⑴ Free Body Diagram
⑵ Composition of Forces
① Use the parallelogram law for combining two forces.
② Resultant Force : The sum of all forces acting on an object.
⑵ Resolution of Forces
① Consider any single force as the combination of two different forces.
② Resolution of forces on an incline
○ For an incline making an angle θ with the ground,
○ Force acting along the incline direction on the object : Assuming the object’s mass is m
○ Force acting perpendicular to the incline direction on the object : Assuming the object’s mass is m
⑷ Equilibrium of Forces
① Definition : When two forces of equal magnitude but opposite direction act on an object.
② Phenomenon : Net force is 0, leading to equilibrium.
③ (Distinct Concept) Action-Reaction Law
○ Equilibrium of forces applies to one object.
○ Action-reaction applies to two objects.
4. Center of Mass (Center of Gravity)
⑴ Overview
① Definition : The point where the total gravity of an object can be considered to act.
② Formulation
③ Example
Figure 4. Example of the center of mass.
○ Situation : A quarter of the area of a steel plate, labeled A, is cut out.
○ Coordinates of the center of mass
⑵ Position of the Center of Mass and Stability
① Stability
○ Definition : The property of an object to return to its original equilibrium state when displaced from equilibrium.
○ Stable equilibrium
○ Definition : When the potential energy forms a concave downward function.
○ The object tends to move to where the potential energy is lowest → Stability.
○ Unstable equilibrium
○ Definition : When the potential energy forms a convex upward function.
○ The object tends to move to where the potential energy is highest → No stability.
② Changing the Stability of an Object
Figure 5. Changing the position of the center of mass and the stability of an object.
(가) Maintains equilibrium; (나) Does not maintain equilibrium.
○ Maintaining equilibrium : When the vertical line from the center of mass of the object intersects the ground, a restoring torque acts to return to the original position.
○ Losing equilibrium : When the vertical line from the center of mass of the object does not intersect the ground, a turning force away from the original position occurs.
③ Methods to Increase Object Stability
○ Lowering the center of mass improves stability.
○ Increasing the base area improves stability.
Input: 2016.06.26 21:05