EXAM 2 PT 2

Newton's Laws of Motion

Introduction to Forces and Newton's Laws

Newton's laws of motion form the foundation for classical mechanics, describing the relationship between forces and the motion of objects. Understanding these laws allows us to analyze and predict the behavior of objects under various forces.

• Force: A push or pull acting on an object, capable of changing its state of motion.

• Inertia: The tendency of an object to resist changes in its motion, quantified by its mass.

Newton's First Law (Law of Inertia)

• An object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.

• Application: Identifying when net forces act on an object and predicting its motion.

Newton's Second Law

• The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Equation: $F_{\text{net}} = m a$

• Application: Calculating acceleration from known forces and mass.

Newton's Third Law

• For every action, there is an equal and opposite reaction.

• Application: Understanding interactions between objects and the forces they exert on each other.

Free-Body Diagrams

• Visual representations showing all forces acting on a single object.

• Essential for analyzing forces and solving problems involving Newton's laws.

Apparent Weight

• The normal force exerted on an object, which may differ from its actual weight if the object is accelerating.

• Equation: $F_{\text{apparent}} = m (g + a)$ (for vertical acceleration)

• Comparison: Apparent weight vs. mass (mass is invariant; apparent weight depends on acceleration).

Circular Motion and Gravity

Quantities in Circular Motion

Circular motion involves objects moving along a curved path, requiring a continuous change in direction and thus acceleration, even if speed remains constant.

• Centripetal Acceleration: Acceleration directed toward the center of the circle, keeping the object in circular motion.

• Equation: $a_c = \frac{v^2}{r}$

• Tangential Acceleration: Acceleration tangent to the circle, associated with changes in speed.

Forces in Circular Motion

• Net force toward the center (centripetal force) is required for circular motion.

• Equation: $F_c = m a_c = m \frac{v^2}{r}$

• Examples: Tension in a string, friction for cars on a track, normal force on a roller coaster.

Banked and Unbanked Curves

• Banked Curve: A curve inclined at an angle to help provide the necessary centripetal force without relying solely on friction.

• Unbanked Curve: Flat curve where friction provides the centripetal force.

Newton's Law of Universal Gravitation

• Every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Equation: $F_g = G \frac{m_1 m_2}{r^2}$

• Applications: Calculating weight on Earth's surface, orbital speeds, and gravitational attraction between objects.

Orbits and Circular Motion

• Objects in uniform circular orbits experience gravitational force as the centripetal force.

• Equation for orbital speed: $v = \sqrt{\frac{G M}{r}}$

Work and Energy

Work, Kinetic Energy, and Potential Energy

Work and energy concepts are central to understanding how forces cause changes in motion and how energy is transferred or transformed within systems.

• Work: The product of the force applied to an object and the displacement in the direction of the force.

• Equation: $W = \vec{F} \cdot \vec{d} = F d \cos \theta$

• Kinetic Energy (KE): Energy of motion, $KE = \frac{1}{2} m v^2$

• Potential Energy (PE): Stored energy due to position or configuration (e.g., gravitational, elastic).

Work-Energy Theorem

• The net work done on an object equals the change in its kinetic energy.

• Equation: $W_{\text{net}} = \Delta KE$

Types of Potential Energy

• Gravitational PE: $U_g = mgh$ (near Earth's surface)

• Elastic PE (spring): $U_s = \frac{1}{2} k x^2$

Conservative and Non-Conservative Forces

• Conservative Forces: Work done is path-independent (e.g., gravity, springs).

• Non-Conservative Forces: Work done depends on the path (e.g., friction).

Energy Principle

• Accounts for changes in kinetic and potential energy due to internal and external interactions.

• Energy can be transferred as work or transformed between kinetic, potential, and thermal energy.

Power

• The rate at which work is done or energy is transferred.

• Equation: $P = \frac{W}{t}$

Linear Momentum

Momentum, Impulse, and Collisions

Momentum is a measure of an object's motion, and its conservation is a fundamental principle in analyzing collisions and interactions.

• Momentum (p): $\vec{p} = m \vec{v}$

• Impulse (J): The change in momentum, $\vec{J} = \Delta \vec{p} = \vec{F}_{\text{avg}} \Delta t$

Conservation of Momentum

• The total momentum of a system remains constant if no external forces act on it.

• Equation: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

Types of Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not (objects may stick together).

Mechanical Energy in Collisions

• Mechanical energy is conserved only in elastic collisions.

Center of Mass

• The point representing the average position of the mass distribution of a system.

• Equation (discrete masses): $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$

• Important for analyzing motion and predicting tipping or stability.

Rotational Motion

Angular Variables and Rotational Kinematics

Rotational motion describes the movement of objects around an axis, using angular variables analogous to linear motion.

• Angular Displacement (\theta): The angle through which an object rotates.

• Angular Velocity (\omega): Rate of change of angular displacement, $\omega = \frac{d\theta}{dt}$

• Angular Acceleration (\alpha): Rate of change of angular velocity, $\alpha = \frac{d\omega}{dt}$

• Equations of Motion (constant \alpha):

  • $\omega = \omega_0 + \alpha t$

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

  • $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

Rolling Motion

• Objects that roll without slipping have a relationship between linear and angular velocity: $v = r \omega$

Torque and Angular Acceleration

• Torque (\tau): The rotational equivalent of force, $\tau = r F \sin \phi$

• Net torque causes angular acceleration: $\sum \tau = I \alpha$

• Moment of Inertia (I): The rotational equivalent of mass, depends on mass distribution.

Equilibrium & Elasticity

Conditions for Equilibrium

An object is in equilibrium when the net force and net torque acting on it are both zero. This is essential for analyzing structures and predicting stability.

• Translational Equilibrium: $\sum \vec{F} = 0$

• Rotational Equilibrium: $\sum \tau = 0$

• Used to determine unknown forces or predict if an object will tip over.

Center of Mass and Stability

• The position of the center of mass affects whether an object will tip over.

Elasticity and Hooke's Law

• Describes how materials deform under force.

• Hooke's Law: $F = k x$ (force is proportional to displacement for springs and elastic materials)

• Used to calculate stretching or compression of objects.

Topic	Main Equations	Key Concepts
Newton's Laws	$F_{\text{net}} = m a$	Inertia, force, free-body diagrams, apparent weight
Circular Motion & Gravity	$a_c = \frac{v^2}{r}$ $F_g = G \frac{m_1 m_2}{r^2}$	Centripetal force, banked curves, orbits
Work & Energy	$W = F d \cos \theta$ $KE = \frac{1}{2} m v^2$ $U_g = mgh$	Work-energy theorem, potential energy, power
Momentum	$\vec{p} = m \vec{v}$ $\vec{J} = \Delta \vec{p}$	Impulse, conservation, collisions, center of mass
Rotational Motion	$\omega = \omega_0 + \alpha t$ $\tau = I \alpha$	Angular variables, torque, rolling motion
Equilibrium & Elasticity	$\sum \vec{F} = 0$ $\sum \tau = 0$ $F = k x$	Stability, Hooke's Law, tipping