Particle Dynamics: Newton's Laws, Friction, Circular Motion

House of Physics October 18, 2025
Particle Dynamics: Complete Physics Guide | Newton's Laws, Friction, Circular Motion

Particle Dynamics: Complete Physics Guide

Mastering Newton's Laws, Frictional Forces, Circular Motion, and Non-Inertial Frames
Particle Dynamics Newton's Laws Frictional Forces Circular Motion Centripetal Force Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

Particle dynamics has evolved through centuries of scientific discovery:

  • Galileo Galilei (1564-1642): Pioneered studies of motion and inertia
  • Isaac Newton (1643-1727): Formulated the three laws of motion and universal gravitation
  • Leonhard Euler (1707-1783): Developed mathematical framework for dynamics
  • Joseph-Louis Lagrange (1736-1813): Created Lagrangian mechanics
  • William Rowan Hamilton (1805-1865): Developed Hamiltonian mechanics

These developments established the foundation for classical mechanics and our understanding of particle motion.

Introduction to Particle Dynamics

🔬 What is Particle Dynamics?

Particle dynamics is the branch of classical mechanics that studies the motion of particles under the influence of forces. It builds upon Newton's laws of motion to analyze how objects move and interact with their environment.

In this chapter, we explore applications of Newton's laws, study frictional forces and their consequences, discuss non-constant forces, and show how to solve equations of motion for such forces. We also examine how using non-inertial reference frames produces effects that can be analyzed by introducing inertial forces or pseudo-forces.

📝 Newton's Laws of Motion

The foundation of particle dynamics rests on Newton's three laws:

  1. First Law (Law of Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
  2. Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass: \( \vec{F} = m\vec{a} \).
  3. Third Law: For every action, there is an equal and opposite reaction.

Fundamental Forces in Nature

🔬 Types of Interaction (Forces)

To understand the cause of force, we must have a detailed microscopic understanding of the interaction of objects with their environment. On the most fundamental level, nature appears to operate through a small number of fundamental forces.

Four Fundamental Forces

Gravitational Force

This force originates due to the presence of matter. It's always attractive and acts between all objects with mass.

Electromagnetic Force

This force includes basic electric and magnetic interactions and is responsible for the binding of atoms and the structure of solids.

Weak Nuclear Force

This force causes certain radioactive decay processes and certain reactions among the fundamental particles.

Strong Nuclear Force

This force operates among the fundamental particles (protons and neutrons) and is responsible for binding the nucleus together.

Relative Strengths of Forces

📊 Relative Strengths of Fundamental Forces (for two protons)

Force Type Relative Strength
Strong Force 1
Electromagnetic Force 10-2
Weak Nuclear Force 10-7
Gravitational Force 10-38

💡 Unification of Forces

Physicists have made progress in unifying these fundamental forces:

  • Electroweak Force (1967): Weak and electromagnetic forces are regarded as parts of a single force called electroweak force.
  • Grand Unification Theories: Attempt to combine strong and electroweak forces into a single framework.
  • Theories of Everything: Attempt to unify all four fundamental forces into a single theoretical framework.

Frictional Forces

🔬 What is Friction?

As one object slides against another, a force is produced between the surfaces. This force opposes the motion of the bodies and is called the force of friction.

Frictional forces oppose relative motion. Even when there is no relative motion, frictional forces may exist between surfaces. In an automobile, about 20% of the engine power is used to counteract frictional forces.

💡 Importance of Friction

While friction causes wear and seizing of moving parts (requiring effort to reduce it), it's also essential in daily life:

  • Brings rotating shafts to a halt
  • Enables walking
  • Allows us to hold pencils and write
  • Makes wheeled transportation possible

Force of Static Friction

🔬 Static Friction

The frictional forces acting between surfaces at rest with respect to each other are called forces of static friction.

Consider a block of mass m placed on a horizontal surface. The weight of the block is mg, balanced by the normal force N (reaction of the horizontal surface).

When a force F is applied to the resting block, it's balanced by an equal and opposite force of static friction fs. As F increases, the force of static friction also increases until fs reaches a certain maximum value just before the block begins to slide.

\[ f_s \leq \mu_s N \]
\[ (f_s)_{\text{max}} = \mu_s N \]

Here (fs)max is the maximum value of the force of static friction, just before sliding occurs. μs is called the coefficient of static friction and depends on the nature of the surfaces in contact.

Force of Kinetic Friction

🔬 Kinetic Friction

Once the block starts sliding, the frictional force decreases to a constant value fk, called the force of kinetic friction.

This force opposes the motion and is proportional to the normal force:

\[ f_k = \mu_k N \]

Here μk is the coefficient of kinetic friction, which is generally less than μs.

📊 Comparison of Static and Kinetic Friction

Property Static Friction Kinetic Friction
Definition Acts when there's no relative motion Acts when there's relative motion
Magnitude 0 ≤ fs ≤ μsN fk = μkN
Coefficient μs (generally larger) μk (generally smaller)
Direction Opposes impending motion Opposes actual motion

Microscopic Basis of Friction

🔬 Why Does Friction Occur?

Even highly polished surfaces are rough on a microscopic scale. When two surfaces are placed in contact, they touch only at a few points. The actual area of contact is much smaller than the apparent area of contact.

At these points of contact, strong adhesive forces develop. When one surface is pulled over the other, these adhesive bonds must be broken, which requires force. This is the origin of friction.

🧪 Experimental Observations

Experiments show that:

  • Friction is independent of the apparent area of contact
  • Friction is proportional to the normal force
  • Static friction is greater than kinetic friction
  • Friction depends on the nature of surfaces in contact
  • Friction is independent of the relative speed between surfaces (approximately)

Sample Problems

Sample Problem 1: Block on Horizontal Surface

A block of mass 10 kg rests on a horizontal surface. The coefficient of static friction is 0.4 and the coefficient of kinetic friction is 0.3. Determine the force of friction acting on the block if a horizontal force of (a) 30 N, (b) 50 N is applied to it. (g = 10 m/s²)

Given:
\[ m = 10 \, \text{kg} \]
\[ \mu_s = 0.4 \]
\[ \mu_k = 0.3 \]
\[ g = 10 \, \text{m/s}^2 \]
Normal force:
\[ N = mg \]
\[ = 10 \times 10 \]
\[ = 100 \, \text{N} \]
Maximum static friction:
\[ (f_s)_{\text{max}} = \mu_s N \]
\[ = 0.4 \times 100 \]
\[ = 40 \, \text{N} \]
Kinetic friction:
\[ f_k = \mu_k N \]
\[ = 0.3 \times 100 \]
\[ = 30 \, \text{N} \]
(a) For F = 30 N:
\[ F < (f_s)_{\text{max}} \]
\[ \text{Block doesn't move} \]
\[ f_s = 30 \, \text{N} \]
(b) For F = 50 N:
\[ F > (f_s)_{\text{max}} \]
\[ \text{Block moves} \]
\[ f_k = 30 \, \text{N} \]
Sample Problem 2: Block on Inclined Plane

A block of mass 2 kg rests on an inclined plane making an angle of 30° with the horizontal. The coefficient of static friction is 0.7. Will the block slide? If not, how much additional force parallel to the plane is needed to (a) just start the block moving up the plane, (b) just prevent it from sliding down? (g = 10 m/s²)

Given:
\[ m = 2 \, \text{kg} \]
\[ \theta = 30^\circ \]
\[ \mu_s = 0.7 \]
\[ g = 10 \, \text{m/s}^2 \]
Components of weight:
\[ mg \sin\theta = 2 \times 10 \times \sin 30^\circ \]
\[ = 20 \times 0.5 \]
\[ = 10 \, \text{N} \]
\[ mg \cos\theta = 2 \times 10 \times \cos 30^\circ \]
\[ = 20 \times 0.866 \]
\[ = 17.32 \, \text{N} \]
Maximum static friction:
\[ (f_s)_{\text{max}} = \mu_s mg \cos\theta \]
\[ = 0.7 \times 17.32 \]
\[ = 12.12 \, \text{N} \]
Will the block slide?
\[ mg \sin\theta = 10 \, \text{N} \]
\[ (f_s)_{\text{max}} = 12.12 \, \text{N} \]
\[ mg \sin\theta < (f_s)_{\text{max}} \]
\[ \text{Block will not slide} \]
(a) Force needed to move up:
\[ F = mg \sin\theta + (f_s)_{\text{max}} \]
\[ = 10 + 12.12 \]
\[ = 22.12 \, \text{N} \]
(b) Force needed to prevent sliding down:
\[ F + (f_s)_{\text{max}} = mg \sin\theta \]
\[ F = mg \sin\theta - (f_s)_{\text{max}} \]
\[ = 10 - 12.12 \]
\[ = -2.12 \, \text{N} \]
\[ \text{No force needed, friction alone prevents sliding} \]

Uniform Circular Motion

🔬 What is Uniform Circular Motion?

When a particle moves in a circle with constant speed, it is said to be in uniform circular motion. Although the speed is constant, the direction of velocity changes continuously, so the particle experiences acceleration.

🧮 Centripetal Acceleration

Step 1: Velocity Vectors

Consider a particle moving in a circle of radius r with constant speed v. At two nearby points, the velocity vectors are perpendicular to the radius vectors.

Step 2: Change in Velocity

The change in velocity Δv is directed toward the center of the circle.

Step 3: Centripetal Acceleration

\[ a_c = \frac{v^2}{r} \]

This acceleration is always directed toward the center of the circle and is called centripetal acceleration.

Step 4: Centripetal Force

By Newton's second law, there must be a net force toward the center:

\[ F_c = m a_c = \frac{m v^2}{r} \]

This is called the centripetal force.

Centripetal Force Examples

Conical Pendulum

A small bob suspended from a fixed point by a string moves in a horizontal circle. The centripetal force is provided by the horizontal component of tension.

The Rotor

A person stands against the wall of a rotating cylindrical chamber. The centripetal force is provided by the normal reaction of the wall.

Banked Curves

Roads are banked at curves so that the horizontal component of the normal force provides the centripetal force required for circular motion.

Conical Pendulum

🧮 Analysis of Conical Pendulum

Step 1: Forces Acting

For a bob of mass m moving in a horizontal circle of radius r with speed v, suspended by a string of length L making angle θ with vertical:

  • Tension T in the string
  • Weight mg downward

Step 2: Vertical Equilibrium

\[ T \cos\theta = mg \]

Step 3: Horizontal Force (Centripetal)

\[ T \sin\theta = \frac{m v^2}{r} \]

Step 4: Relationship Between Variables

\[ \tan\theta = \frac{v^2}{r g} \]
\[ r = L \sin\theta \]
\[ v = \sqrt{r g \tan\theta} \]

Step 5: Time Period

\[ T = \frac{2\pi r}{v} \]
\[ = 2\pi \sqrt{\frac{L \cos\theta}{g}} \]

The Rotor

🧮 Analysis of The Rotor

Step 1: Forces Acting

For a person of mass m pressed against the wall of a rotor of radius R rotating with angular velocity ω:

  • Normal reaction N from the wall (provides centripetal force)
  • Weight mg downward
  • Friction f upward (prevents sliding down)

Step 2: Horizontal Force (Centripetal)

\[ N = \frac{m v^2}{R} = m \omega^2 R \]

Step 3: Vertical Equilibrium

\[ f = mg \]

Step 4: Condition for No Sliding

\[ f \leq \mu_s N \]
\[ mg \leq \mu_s m \omega^2 R \]
\[ \omega \geq \sqrt{\frac{g}{\mu_s R}} \]

Banked Curves

🧮 Analysis of Banked Curves

Step 1: Forces Acting

For a vehicle of mass m moving on a curved road banked at angle θ with the horizontal, with no friction:

  • Normal reaction N perpendicular to the road surface
  • Weight mg downward

Step 2: Vertical Equilibrium

\[ N \cos\theta = mg \]

Step 3: Horizontal Force (Centripetal)

\[ N \sin\theta = \frac{m v^2}{r} \]

Step 4: Banking Angle

\[ \tan\theta = \frac{v^2}{r g} \]

This gives the optimum banking angle for a given speed and radius.

Equations of Motion

🔬 Solving Equations of Motion

The equation of motion for a particle is given by Newton's second law: \( \vec{F} = m\vec{a} \). In component form:

\[ F_x = m \frac{d^2 x}{dt^2} \]
\[ F_y = m \frac{d^2 y}{dt^2} \]
\[ F_z = m \frac{d^2 z}{dt^2} \]

These are second-order differential equations that can be solved given initial conditions.

Motion Under Constant Force

🧮 Constant Force Equations

Step 1: Acceleration

\[ \vec{a} = \frac{\vec{F}}{m} = \text{constant} \]

Step 2: Velocity

\[ \vec{v} = \vec{v}_0 + \vec{a} t \]

Step 3: Position

\[ \vec{r} = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a} t^2 \]

Motion Under Non-Constant Forces

🔬 Variable Forces

When forces depend on position, velocity, or time, the equations of motion become more complex and require integration techniques.

Examples include:

  • Spring force: F = -kx (depends on position)
  • Drag force: F = -bv (depends on velocity)
  • Time-dependent forces: F = F(t)

Drag Forces

🧮 Motion with Drag Force

Step 1: Drag Force Model

For small objects at low speeds, drag force is proportional to velocity:

\[ \vec{F}_d = -b \vec{v} \]

where b is the drag coefficient.

Step 2: Equation of Motion

\[ m \frac{dv}{dt} = -b v \]

Step 3: Solution

\[ \frac{dv}{v} = -\frac{b}{m} dt \]
\[ \int_{v_0}^v \frac{dv}{v} = -\frac{b}{m} \int_0^t dt \]
\[ \ln\left(\frac{v}{v_0}\right) = -\frac{b}{m} t \]
\[ v = v_0 e^{-(b/m) t} \]

Projectile Motion

🔬 What is Projectile Motion?

Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. The path followed by a projectile is called its trajectory.

Basic Equations of Projectile Motion

🧮 Projectile Motion Analysis

Step 1: Initial Conditions

A projectile is launched with initial speed v0 at angle θ with the horizontal.

\[ v_{0x} = v_0 \cos\theta \]
\[ v_{0y} = v_0 \sin\theta \]

Step 2: Horizontal Motion (Constant Velocity)

\[ a_x = 0 \]
\[ v_x = v_0 \cos\theta \]
\[ x = (v_0 \cos\theta) t \]

Step 3: Vertical Motion (Constant Acceleration)

\[ a_y = -g \]
\[ v_y = v_0 \sin\theta - g t \]
\[ y = (v_0 \sin\theta) t - \frac{1}{2} g t^2 \]

Trajectory and Range

🧮 Projectile Trajectory and Range

Step 1: Equation of Trajectory

Eliminate time from the equations:

\[ t = \frac{x}{v_0 \cos\theta} \]
\[ y = (v_0 \sin\theta) \left(\frac{x}{v_0 \cos\theta}\right) - \frac{1}{2} g \left(\frac{x}{v_0 \cos\theta}\right)^2 \]
\[ y = x \tan\theta - \frac{g x^2}{2 v_0^2 \cos^2\theta} \]

Step 2: Time of Flight

When the projectile returns to ground level (y=0):

\[ 0 = (v_0 \sin\theta) T - \frac{1}{2} g T^2 \]
\[ T = \frac{2 v_0 \sin\theta}{g} \]

Step 3: Maximum Height

At maximum height, vy = 0:

\[ 0 = v_0 \sin\theta - g t_h \]
\[ t_h = \frac{v_0 \sin\theta}{g} \]
\[ H = (v_0 \sin\theta) t_h - \frac{1}{2} g t_h^2 \]
\[ H = \frac{v_0^2 \sin^2\theta}{2g} \]

Step 4: Horizontal Range

\[ R = (v_0 \cos\theta) T \]
\[ = (v_0 \cos\theta) \left(\frac{2 v_0 \sin\theta}{g}\right) \]
\[ R = \frac{v_0^2 \sin 2\theta}{g} \]

Non-Inertial Frames and Pseudo Forces

🔬 Inertial vs Non-Inertial Frames

An inertial frame of reference is one in which Newton's first law holds. A frame that is accelerating with respect to an inertial frame is called a non-inertial frame.

In non-inertial frames, Newton's laws don't hold unless we introduce additional forces called pseudo forces or inertial forces.

🧮 Pseudo Forces

Step 1: Accelerating Frame

Consider a frame S' accelerating with acceleration \(\vec{A}\) relative to an inertial frame S.

Step 2: Position Relation

\[ \vec{r} = \vec{r}' + \vec{R} \]

where \(\vec{r}\) is position in S, \(\vec{r}'\) is position in S', and \(\vec{R}\) is position of origin of S' relative to S.

Step 3: Acceleration Relation

\[ \frac{d^2 \vec{r}}{dt^2} = \frac{d^2 \vec{r}'}{dt^2} + \frac{d^2 \vec{R}}{dt^2} \]
\[ \vec{a} = \vec{a}' + \vec{A} \]

Step 4: Modified Newton's Law

\[ \vec{F} = m \vec{a} = m (\vec{a}' + \vec{A}) \]
\[ \vec{F} - m \vec{A} = m \vec{a}' \]

The term \(-m \vec{A}\) is the pseudo force.

💡 Examples of Pseudo Forces

  • Centrifugal Force: Appears in rotating frames, directed away from the center of rotation
  • Coriolis Force: Appears when objects move in rotating frames, perpendicular to velocity
  • Elevator Forces: Additional forces felt when elevators accelerate up or down

Frequently Asked Questions

❓ Why is static friction greater than kinetic friction?

Static friction is generally greater than kinetic friction due to microscopic interactions between surfaces:

  • Surface Adhesion: When surfaces are at rest, microscopic irregularities interlock more strongly, creating more adhesive bonds that need to be broken to initiate motion.
  • Molecular Bonding: At the atomic level, atoms form temporary bonds when in close contact. These bonds take time to form, so stationary surfaces have more established bonds.
  • Plowing Effect: Once motion begins, the surfaces slide over each other rather than having to overcome the initial interlocking.

This difference explains why it's harder to start moving an object than to keep it moving.

❓ What is the difference between centripetal and centrifugal force?

Centripetal and centrifugal forces are often confused but have fundamentally different natures:

  • Centripetal Force: A real force that acts toward the center of circular motion. It's required to keep an object moving in a circle. Examples include tension in a string, gravitational force, or friction.
  • Centrifugal Force: A pseudo force that appears in rotating reference frames. It acts outward from the center of rotation and is not a real force but rather an effect of inertia.

In an inertial frame, only centripetal force exists. In a rotating frame, both centripetal and centrifugal forces appear to act.

❓ How does banking of roads help vehicles?

Banking of roads at curves provides several important benefits:

  • Reduces Reliance on Friction: On a banked curve, the horizontal component of the normal force provides part or all of the required centripetal force, reducing the need for friction.
  • Increases Safety: Vehicles can negotiate curves at higher speeds without skidding.
  • Improves Comfort: Reduces the lateral forces felt by passengers.
  • Enhances Stability: Helps prevent vehicles from overturning on sharp curves.

The optimum banking angle is calculated using the formula: \(\tan\theta = \frac{v^2}{rg}\), where v is the design speed, r is the radius of curvature, and g is acceleration due to gravity.

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