Newton's Laws of Motion

• Newton's Laws of Motion form the foundation of classical mechanics, describing the relationship between a body and the forces acting upon it.
• These laws allow for the prediction of a body's motion under the influence of given forces.

Newton's 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 a force.
• This law defines inertia, the tendency of an object to resist changes in its state of motion.
• Implications include: an object maintains its velocity (constant speed and direction) unless a force changes it.
• In mathematical terms, if the net force on an object is zero ($\sum F = 0$), then the object's velocity is constant.

Newton's Second Law - Law of Acceleration

• The acceleration of an object is directly proportional to the net force acting on the object, is in the same direction as the net force, and is inversely proportional to the mass of the object.
• Expressed as F = ma, where F is the net force, m is the mass, and a is the acceleration.
• Force is a vector quantity, possessing both magnitude and direction.
• Mass is a scalar quantity representing the amount of matter in an object and its resistance to acceleration.
• Acceleration is a vector quantity describing the rate of change of velocity.
• The second law effectively defines force and provides a means to calculate it.
• If multiple forces act on an object, $\sum F = ma$ applies, where $\sum F$ is the vector sum of all forces.

Newton's Third Law - Law of Action-Reaction

• For every action, there is an equal and opposite reaction.
• If object A exerts a force on object B ($F_{AB}$), then object B exerts a force of equal magnitude and opposite direction on object A ($F_{BA}$).
• These forces act on different objects, preventing them from canceling each other out.
• $F_{AB} = -F_{BA}$ is the mathematical representation.
• The action and reaction forces are of the same type (e.g., gravitational, electromagnetic).

Applications of Newton's Laws

• Newton's laws are applied to solve a wide variety of problems in mechanics.
• These include statics (objects at rest) and dynamics (objects in motion), covering linear and rotational motion.

Free-Body Diagrams

• A free-body diagram (FBD) is a visual tool used to analyze forces acting on an object.
• Represent the object as a point mass.
• Draw vectors representing all external forces acting on the object, including their magnitudes and directions.
• Choose a coordinate system to resolve force vectors into components.
• FBDs simplify the application of Newton's Second Law by visually organizing the forces.

Types of Forces

• Gravitational Force: The attractive force between objects with mass, $F_g = mg$, where g is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
• Normal Force: The force exerted by a surface on an object in contact with it, perpendicular to the surface.
• Frictional Force: A force that opposes motion or attempted motion between surfaces in contact.
  • Static friction: Prevents motion from starting.
  • Kinetic friction: Opposes motion when an object is moving.
• Tension: The force transmitted through a string, rope, cable, or wire when it is pulled tight by forces acting from opposite ends.
• Applied Force: A force that is applied to an object by a person or another object.
• Spring Force: The force exerted by a spring, described by Hooke's Law: F = -kx, where k is the spring constant and x is the displacement from equilibrium.

Solving Problems Using Newton's Laws

• Identify the object of interest and draw a free-body diagram.
• Choose a coordinate system and resolve forces into components along the axes.
• Apply Newton's Second Law to each axis: $\sum F_x = ma_x$ and $\sum F_y = ma_y$.
• Solve the resulting equations to find unknowns, such as acceleration, force, or mass.
• Include units in final answers and verify solutions.

Example Application - Object on an Inclined Plane

• Decompose the gravitational force into components parallel and perpendicular to the inclined plane.
• The normal force is equal in magnitude and opposite in direction to the perpendicular component of gravity.
• The component of gravity parallel to the plane causes the object to accelerate down the plane (if friction is negligible).
• If friction is present, the frictional force opposes the motion, reducing the acceleration.

Example Application - Atwood Machine

• An Atwood machine consists of two masses connected by a string over a pulley.
• Draw free-body diagrams for each mass, considering tension and gravitational force.
• Apply Newton's Second Law to each mass, accounting for the direction of acceleration.
• Solve the resulting system of equations to find the tension in the string and the acceleration of the masses.

Example Application - Circular Motion

• For an object moving in a circle, there must be a net force directed towards the center of the circle; this is the centripetal force.
• $F_c = \frac{mv^2}{r}$, where m is the mass, v is the speed, and r is the radius of the circle.
• Examples include: a car turning a corner, a satellite orbiting Earth.
• Centripetal force is not a fundamental force; it is the net force causing circular motion, provided by forces such as tension, friction, or gravity.

Non-Inertial Frames of Reference

• Newton's laws are strictly valid in inertial frames of reference (frames that are not accelerating).
• In non-inertial frames, fictitious forces (also called pseudo-forces) appear.
• Examples of fictitious forces: centrifugal force, Coriolis force.
• These forces are not real forces but are corrections needed to apply Newton's laws in accelerating frames.