Physics I: Forces and Newton's Laws

Forces and Newton's Laws

This study guide covers the fundamental concepts of forces and Newton's Laws of Motion, which are foundational topics in introductory college physics. The notes include definitions, types of forces, vector addition, Newton's three laws, applications to inclined planes, and correlated (connected) motion problems.

Definition of Force

• Force is a vector quantity that measures the strength of an interaction between objects. It can cause an object to accelerate, decelerate, remain in place, or change direction.

• Forces are measured in Newtons (N).

• Forces can be classified as either contact forces (e.g., friction, tension, normal force) or action-at-a-distance forces (e.g., gravity, electromagnetic force).

Types of Forces

• Applied Force (F): A push or pull exerted on an object by a person or another object.

• Normal Force (N): The support force exerted upon an object in contact with another stable object (e.g., a book on a table).

• Frictional Force (f): The force that opposes the motion of an object across a surface.

• Gravitational Force (Fg): The force of attraction between two masses, typically the weight of an object near Earth's surface.

• Tension Force (T): The force transmitted through a string, rope, cable, or wire when it is pulled tight by forces acting from opposite ends.

Vector Nature of Force

• Forces are vectors, meaning they have both magnitude and direction.

• To find the net force ($\vec{F}_{\text{net}}$), sum all individual forces acting on an object:

$\vec{F}_{\text{net}} = \sum \vec{F}_i$

• When adding two forces $\vec{F}_1$ and $\vec{F}_2$ at an angle, use vector addition:

$\vec{F} = \vec{F}_1 + \vec{F}_2$ $F = \sqrt{F_1^2 + F_2^2}$ (if perpendicular)

Newton's Laws of Motion

Newton's First Law (Law of Inertia)

Newton's First Law states that 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.

• Objects resist changes to their state of motion (inertia).

• If $\vec{F}_{\text{net}} = 0$, then:

  • The object is at rest, or

  • The object moves at constant velocity.

Newton's Second Law

Newton's Second Law quantifies the relationship between force, mass, and acceleration.

• The net force on an object is equal to the mass of the object multiplied by its acceleration:

$\vec{F}_{\text{net}} = m \vec{a}$

• In component form (e.g., x and y directions):

$F_{\text{net},x} = m a_x$ $F_{\text{net},y} = m a_y$

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction.

• If object A exerts a force $\vec{F}_{AB}$ on object B, then object B exerts a force $\vec{F}_{BA}$ on object A such that:

$\vec{F}_{AB} = -\vec{F}_{BA}$ $|\vec{F}_{AB}| = |\vec{F}_{BA}|$

• Action and reaction forces act on different objects.

Example: Vector Addition of Forces

• Given forces $F_1$ and $F_2$ at an angle, resolve into components:

$F_{\text{net},x} = F_{1x} + F_{2x}$ $F_{\text{net},y} = F_{1y} + F_{2y}$ $F_{\text{net}} = \sqrt{F_{\text{net},x}^2 + F_{\text{net},y}^2}$ $\tan \theta = \frac{F_{\text{net},y}}{F_{\text{net},x}}$

• Example: If $F_1 = 5\,\text{N}$ at $45^\circ$ and $F_2 = 1\,\text{N}$ in the x-direction:

$F_{1x} = 5 \cdot \frac{\sqrt{2}}{2}$ $F_{\text{net},x} = F_{1x} - 1$ $F_{\text{net}} = \sqrt{F_{\text{net},x}^2 + F_{1y}^2}$

Applications of Newton's Laws

Inclined Plane Problems

When analyzing motion on an inclined plane, resolve the gravitational force into components parallel and perpendicular to the surface.

• Let $\theta$ be the angle of the incline.

• Parallel component: $F_{g,x} = mg \sin \theta$

• Perpendicular component: $F_{g,y} = mg \cos \theta$

• Normal force: $N = mg \cos \theta$

• If friction is present, frictional force: $f_r = \mu N$

• Net force along the incline: $F_{\text{net},x} = mg \sin \theta - \mu mg \cos \theta$

• Acceleration: $a = g (\sin \theta - \mu \cos \theta)$

Correlated (Connected) Motion: Atwood Machine

In systems with two masses connected by a string over a pulley (Atwood machine), the acceleration and tension can be found using Newton's Second Law for each mass.

• Let $m_1$ and $m_2$ be the masses ($m_2 > m_1$):

• For $m_1$: $T - m_1g = m_1a$

• For $m_2$: $m_2g - T = m_2a$

• Solving for acceleration:

$a = g \frac{m_2 - m_1}{m_1 + m_2}$

• Tension in the string:

$T = \frac{2 m_1 m_2 g}{m_1 + m_2}$

Summary Table: Types of Forces

Type of Force	Symbol	Description	Formula
Gravitational	$F_g$	Attraction between masses	$F_g = mg$
Normal	$N$	Support force from a surface	$N = mg \cos \theta$ (on incline)
Friction	$f_r$	Opposes motion	$f_r = \mu N$
Tension	$T$	Force in a string/rope	Depends on system
Applied	$F$	External push or pull	Varies

Key Points

• Always draw a free-body diagram to identify all forces acting on an object.

• Choose a coordinate system that simplifies the problem (e.g., align axes with incline).

• Apply Newton's Second Law in each direction separately.

• For connected objects, use the same acceleration for both masses.

Example Problem: Elevator Apparent Weight

• Given: $m = 65\,\text{kg}$, elevator accelerates upward at $a = 2.0\,\text{m/s}^2$.

• Normal force (apparent weight):

$N = mg + ma$ $N = 65 \times (9.8 + 2.0) = 767\,\text{N}$

• If accelerating downward, $N = mg - ma$.

Additional info:

• Some equations and diagrams were inferred and clarified for completeness.

• All vector quantities are denoted with arrows; scalars are not.