Physics: Newton's Laws of Motion

• Physics covers Laws of Motion
• Aristotle's law states that external forces are needed to maintain the motion of a body.
• External forces are necessary to overcome frictional forces in solids and viscous forces in fluids.
• Newton created three laws of motion based on Galileo's ideas.

Newton's First Law of Motion

• An object remains at rest or in uniform motion unless compelled by an external force.
• It is also known as the Law of Inertia.
• To find the cause of motion for a body initially at rest:
  • First, consider the first law of motion.
  • Check the options by applying the law.
  • If only one option matches, that option is correct.
• Passengers in a starting bus fall backward because their body opposes the bus's motion due to inertia.
• Dust particles separate from a carpet when beaten with a stick due to inertia.

Newton's Second Law of Motion

• The rate of change of momentum is proportional to the applied external force.
• The change occurs in the direction of the external force.
• Force (F) is proportional to the rate of change of momentum divided by time.
• Acceleration is defined as the change in velocity with respect to time: a = (v-u)/t
• Momentum is defined as the product of mass and velocity: p = m × v
• F = K * (p₂ - p₁) / t = K * (mv - mu) / t = K * m(v - u) / t = Kma, where K is a constant of proportionality.
• A cricket player moves hands backward to catch a fast ball, following Newton's second law.
• High jumping athletes land on cushions or sand, as another application of Newton's Second Law.

Law of Conservation of Linear Momentum

• Conservation occurs when no external force acts on a system.
• The total linear momentum remains constant.
• Total linear momentum is the vector sum of the linear momentum of all particles.
• Consider an isolated system (no external force) with m particles of masses m₁, m₂, m₃...mₙ and velocities v₁, v₂, v₃...vₙ.
• The total linear momentum of the system is p = m₁v₁ + m₂v₂ + m₃v₃ + ... + mₙvₙ.
• p = p₁ + p₂ + p₃ + ... + pₙ
• F = dp/dt according to Newton's second law, where F is an external force.
• For an isolated system, F = 0, so dp/dt = 0.
• Therefore p₁ + p₂ + p₃ + ... + pₙ = constant
• The total linear momentum of the system is constant.

Newton's Third Law of Motion

• When two bodies interact, they exert equal and opposite forces on each other.
• It is also known as the law of action and reaction.
• During walking, a person pushes the ground backward, and the ground pushes the person forward with equal force, per the third law.
• When a gun fires a bullet, the bullet pushes the gun backward with an equal force, causing recoil.

Equilibrium of a Particle

• A particle is in equilibrium when it does not move under the action of two or more forces.
• The resultant force is zero, and the particle remains stationary.
• ΣF = 0, implying ΣFx = 0, ΣFy = 0, and ΣFz = 0

Stable Equilibrium

• A body tends to regain its original position after slight displacement.
• Potential energy (P.E.) is minimum, d²u/dr² = +ve
• The center of gravity is at its lowest point.

Unstable Equilibrium

• A body moves in the direction of displacement after being displaced.
• Potential energy is maximum, d²u/dr² = -ve
• The center of gravity is at its highest point.

Impulse

• Impulse is defined as a force that acts for a very short time and produces a large change in momentum.
• Impulse (I) = Force × Time
• Its unit is Newton second.

Friction

• Friction opposes relative motion between two objects in contact.

Types of Friction

• There are three types
• Static Friction
  • It is an opposing force that comes into play when an object tends to move but does not.
  • For example, a block resting on a table.
• Limiting Friction
  • Maximum static friction when an object is about to slide.
• Kinetic Friction
  • Opposing force when an object is moving over a surface
• Also called dynamic friction
• Kinetic friction is of two types: Sliding and Rolling.

Sliding Friction

• Occurs when one object slides over another.

Rolling Friction

• Occurs when one object rolls over another.
• Force is an interaction that alters a body's state of rest or motion.

Types of Forces

• Conservative Force
  • Actions are independent of the path taken.
  • Work depends only on starting and ending points.
  • A conservative force in a closed loop does no work.
  • Examples: gravity, static electricity, elastic force.
• Non-Conservative Force
  • Work depends on the path taken.
  • Examples: friction, air drag.
• Constant Force
  • Remains constant over time and space.
• Variable Force
  • Varies in time and distance, depends on factors like time, space, velocity, etc.
• Contact Force
  • Requires physical contact.
• Non-Contact Force
  • Does not require physical contact.
  • Gravity is the primary non-contact force in mechanics.

Types of Contact Forces

• Tension Force
  • The mass connected to a string is balanced by the tension in the string.
• Friction Force
  • Arises when two surfaces are in contact and one moves or tends to move.
  • Prevents relative motion of solid surfaces.
  • Example: friction occurs between a car's tires and the road.
• Spring Force
• Drag Force
• Pseudo Force

Friction Denotation

• Friction is denoted by "f".
• Friction force (f) = μN, where N is the normal force and μ is the coefficient of friction.

Types of Friction Category Breakdown

• Friction is divided by state of motion and state of rest.
• Static Friction
  • Exists between surfaces of two bodies at rest but with the tendency to move.
  • For example, trying to move a resting block on a table.
• Friction is adjusted this way.
• Static friction increases with applied force up to a maximum value where the value is known as the limiting value.
• Static friction becomes a factor when an external force is applied.
• It does not exist if the body is at rest and there is no attempt to move it.
• Static friction has a limit value independent of contact area and dependent on the composition of surface.
• (fs)max = μsN, where μs is the coefficient of static friction.
• The force of static friction is expressed by  f s max   s N.
• If the external force (F) exceeds the limiting friction (fₛ)max, then the block starts moving.
• Rolling friction occurs between a body's surface and a surface when rolling.
• With it, the point of contact is not moving in relation to the surface on which the body is rolling.
• Rolling friction is denoted by  f R .
• Example is the friction between a road and a vehicle's wheel.

Points to Remember About Friction

• Static and kinetic coefficients of friction are different.
• Surface type and temperature affect friction coefficient.
• Rolling friction is less than static friction.
• Rolling friction produces no effort, because of no relative motion at the point of contact.

Pseudo Forces

• Force exerted on an object in relation to an accelerating body.
• Also known as Imaginary Force
• Pseudo force  FP   ma, where a is object's acceleration and m is object's mass.

Types of Non-Contact Forces

• Gravitational Force
  • Exerted at a distance
  • The gravitational force is the pull of gravity that one object experiences due to their respective masses.
  • Gravitational force: F ∝ (m₁m₂)/r², where F is gravitational force.
• Electrical Force
  • Interaction between charged bodies (attractive or repulsive).
  • Magnitude is inversely proportional to the distance squared, and directly proportional to charges.
  • Force is F= K*(q₀q₁/r²)*r^, with K proportionality constant
• Magnetic Force
  • The attraction or repulsion between electrically charged particles due to their motion
  • F = qvBsinθ

The Expression for Velocity of Body Moving in a Vertical Circle

• Characterized by a small object with mass m.
• The object is being strung up on one end and rotated in a vertical circle with radius r.
• In this, the body's acceleration rises as it moves up the vertical circle, and falls as it moves down.
• This means the body's speed is constantly changing.
• "mg" works vertically downward
• Energy at point P = Energy at point L
• ½mv² + mgh = ½mu²
• Thus v² + 2gh = u² results in:
• v² = u² - 2gh results in:
• v = √(u² - 2gh) which is expression for velocity of a particle at any point performing a circular motion in a vertical circle.

Tension in the String Formula

• Considers the centripetal force at point P and results in:
• T - mgcosθ = mv²/r
• The final formula can be derived from this to:
• T= m/r * (u² - 3gh + gr)
• Tension can change, depending on the position of the body
• Case 1: When the Body is at the Lowermost Position (h = 0, at point L)
  • T= m/r(u² -3gh + gr)
  • Results in a formula of: T = m/r * (u² + gr)
• Case 2: When the Body is at Uppermost Position (h = -2r, at point H)
  • TH = m/r * (u² - 5gr)
• Case 3: When the Body is at Horizontal Position (h = -r , at point M)
  • T= m/r * (u² - 2gr)

Relation B/w Tension at Highest and Lowest Points

• T = 6mg
• Lowest Point, where h = 0:
  • This is the least speed at which the body needs to move in order to full loop once
  • Requires that the object's highest point has to be greater or equal to zero: T ≥0
  • TH ≥ 0 at its highest point - Requires m
    • m/r * (u² - 5gr) ≥ 0, thus:
      • u ≥ √5gr
• Tension in String and Acceleration of Particle
  • The body has a force to be able to maintain motion with tension at the highest point as such: - TH = mv²/r + mg
• With an acceleration of:
  • a = v²/r
• Banking of Roads
  • Raises the outer borders of roads so that cars can maintain turning safely with required centripetal force.

Formula for Vehicles

• Ncosθ = mg +ƒsin(θ)
• mg = Ncosθ - ƒsin(θ)
• Where F is friction. Subbing for mg yields relation: - mv²/r = Nsinθ + ƒcosθ - then becomes: -v²/rg = (Nsinθ + µNcosθ)/(Ncosθ + µNsinθ) - results in: -v² = rg(tanθ + µ)/(1 - µrtanθ)
• Important Formulas
  • Static Friction = µmg
  • Kinetic Friction = µN
  • Rolling Friction = µrN
• Minimum Force
• At and angle