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Chapter 3. Laws of Motion

Recommended Article : 【Physics】 Physics Index


1. Newton’s Laws of Motion

2. Types of Forces

3. Analysis of Forces

4. Center of Mass


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

Application 1: A stationary object remains at rest

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

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

Electric Force : Force between charged particles

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.

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.

Type 1-3. Weak Nuclear Force : Involved in nuclear reactions.

Type 1-4. Strong Nuclear Force : Binds protons and neutrons.

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.

Example 1. Observer moving upward with an elevator accelerating by a : The observer perceives a downward inertial force of ma.

Example 2. Observer moving downward with an elevator accelerating by a : The observer perceives an upward inertial force of ma.

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

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