Physics 211: Forces, Newton's Laws, Friction, Work, Energy, and Momentum – Midterm 2 Review

Forces and Motion

Introduction to Forces

In classical mechanics, forces are interactions that cause changes in the motion of objects. Understanding forces is fundamental to analyzing physical systems and predicting their behavior.

• Force is a vector quantity, meaning it has both magnitude and direction.

• Common types of forces include gravitational, frictional, normal, and applied forces.

• Forces can be represented using free-body diagrams to visualize all forces acting on an object.

• Motion results from the net force acting on an object, as described by Newton's laws.

Newton's Laws of Motion

Algebraic Formulation of Newton's Laws

Newton's laws provide the foundation for understanding the relationship between forces and motion.

• Newton's First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: The net force on an object is equal to the mass of the object multiplied by its acceleration.

• Newton's Third Law: For every action, there is an equal and opposite reaction.

Key Equations:

• Net force: $\vec{F}_{\text{net}} = \sum_i \vec{F}_i$

• If $\vec{F}_{\text{net}} = 0$, then $\frac{d\vec{v}}{dt} = 0$ (no acceleration).

• Newton's Second Law: $\vec{F}_{\text{net}} = m\vec{a}$

• Newton's Third Law: $\vec{F}_{A\text{ on }B} = -\vec{F}_{B\text{ on }A}$

Application: Horse and Cart Problem

This classic scenario illustrates Newton's Third Law and the importance of considering all forces acting on a system.

• The horse claims that pulling the cart is futile because the cart exerts an equal and opposite force.

• Key Point: The horse is propelled forward by the force from the ground on the horse, not just the force from the cart.

• The interaction between horse and cart causes them to move together as a single system.

• Example: When the horse pushes against the ground, the ground pushes back, allowing the horse (and cart) to move forward.

Free-Body Diagrams

Constructing Free-Body Diagrams

Free-body diagrams are essential tools for visualizing and analyzing the forces acting on an object.

• Each force acting on the object is represented by an arrow pointing in the direction of the force.

• Common forces include:

  • Normal force (N): Perpendicular contact force from a surface.

  • Friction force (Ffriction): Opposes motion between surfaces.

  • Gravitational force (mg): Downward force due to gravity.

  • Applied force (F): Any external force applied to the object.

• Action-reaction pairs (Newton's Third Law) act on different objects and are not shown together in a single free-body diagram.

Example: A box on a surface experiences normal force upward, gravitational force downward, friction force opposing motion, and an applied force.

Newton's Second Law in Practice

Analyzing Forces and Acceleration

Newton's Second Law allows us to relate the net force on an object to its acceleration.

• For horizontal acceleration: $a_x = \frac{F}{m}$

• For vertical forces: $F_y + N - w = ma_y$

• If $a_y = 0$, then $F_y + N = w$ (no vertical acceleration).

• If $N = 0$, the object lifts off the ground.

Example: Pushing a box against a wall requires balancing forces so the box does not move.

Friction

Static and Kinetic Friction

Friction is a resistive force that opposes the relative motion of two surfaces in contact.

• Static friction (fs): Prevents motion up to a maximum value.

• Kinetic friction (fk): Acts when surfaces are sliding past each other.

• Maximum static friction: $f_s = \mu_s N$ where $\mu_s$ is the coefficient of static friction and $N$ is the normal force.

• To prevent motion, the applied force must be less than or equal to $f_s$.

Example: Pushing a box against a wall requires an applied force $F$ such that $F \geq \frac{mg}{\mu_s}$.

Work and Kinetic Energy

Definition and Calculation of Work

Work is done when a force causes displacement of an object. The amount of work depends on the force, displacement, and the angle between them.

• Work formula: $W = F \cos \theta \Delta x$ where $\theta$ is the angle between force and displacement.

• If $\theta = 90^\circ$, $\cos 90^\circ = 0$, so no work is done.

• Example: Carrying a stone horizontally does no work against gravity.

Kinetic Energy and the Work-Energy Theorem

Kinetic energy (K) is the energy of motion. The work-energy theorem relates the work done on an object to its change in kinetic energy.

• Kinetic energy: $K = \frac{1}{2}mv^2$

• Work-energy theorem: $K_f = K_i + W$

• On a frictionless surface, all work goes into increasing kinetic energy.

• With friction, some work is lost to heat, reducing the net kinetic energy gained.

Momentum

Conservation of Momentum

Momentum is a measure of an object's motion, defined as the product of mass and velocity. In the absence of external forces, total momentum is conserved.

• Momentum: $p = mv$

• Conservation of momentum: $p_{i,\text{total}} = p_{f,\text{total}}$

• Applies to collisions and explosions.

• Example: Rifle and bullet problem – the rifle and bullet move in opposite directions after firing, sharing momentum according to their masses.

Collisions

Collisions can be classified as elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved).

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

• Inelastic collision: Momentum is conserved, but some kinetic energy is transformed (e.g., into heat or deformation).

• For a head-on elastic collision with a massive stationary target: $v_f = \frac{m_1 - m_2}{m_1 + m_2}v_i$ (for the moving object)

• For a perfectly inelastic collision (objects stick together): $v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$

Potential Energy

Gravitational Potential Energy

Potential energy (U) is stored energy due to an object's position or configuration. Gravitational potential energy depends on height above a reference point.

• Gravitational potential energy: $U = mgh$ where $h$ is the height above the ground.

• Mechanical energy: $E = K + U$

• In the absence of non-conservative forces, mechanical energy is conserved.

Example: An object dropped from height $h$ converts potential energy to kinetic energy as it falls.

Energy Conservation and Applications

Energy conservation is a powerful tool for solving problems involving motion and forces.

• At any point: $E = K + U = \text{constant}$

• Final velocity from energy conservation: $v_f = \sqrt{2gh}$ (if initial kinetic energy is zero)

• For systems with springs: $U = \frac{1}{2}kx^2$ where $k$ is the spring constant and $x$ is the displacement.

• Work done by friction reduces mechanical energy.

Example: Roller coaster problem – use energy conservation to find velocity at different points.

Summary Table: Key Concepts

Concept	Equation	Notes
Net Force	$\vec{F}_{\text{net}} = \sum_i \vec{F}_i$	Sum of all forces acting on an object
Newton's Second Law	$\vec{F}_{\text{net}} = m\vec{a}$	Relates force and acceleration
Friction	$f_s = \mu_s N$	Maximum static friction
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion
Work	$W = F \cos \theta \Delta x$	Work done by a force
Momentum	$p = mv$	Product of mass and velocity
Potential Energy	$U = mgh$	Gravitational potential energy
Spring Potential Energy	$U = \frac{1}{2}kx^2$	Energy stored in a spring

Additional info: These notes expand on the provided slides and handwritten content, filling in academic context and definitions for clarity and completeness.