Newton's Laws of Motion: Principles, Applications, and Problem Solving

Newton's Laws of Motion

Lecture Outline

• Newton's First Law (Inertia)

• Newton's Second Law (F=ma)

• Newton's Third Law (Action-Reaction)

• Forces of Friction

• Force of Static Friction, $F_s$

• Force of Kinetic Friction, $F_k$

The Concept of Force

Why Does the Motion of an Object Change?

Motion changes due to the action of forces, as described by Newton's laws of motion. These laws are based on experimental observations and apply to objects with velocities much less than the speed of light ($v \ll c$).

• Kinematics: Describes motion without considering its causes.

• Dynamics: Explains motion by considering the forces that cause it.

Fundamental Forces in Physics:

• Gravitational

• Electromagnetic

• Strong Nuclear

• Weak Nuclear

Types of Forces:

• Contact Forces: Tension, friction, applied, normal

• Field Forces: Gravitational, electromagnetic

Force causes a change in the velocity of a particle. The main reason for acceleration is the net force acting on a particle.

Newton's First Law (Law of Inertia)

Statement and Implications

If the net force acting on an object is zero, the object maintains its original motion.

• If an object moves with a velocity and net force is zero, it continues with constant velocity.

• If an object is at rest and net force is zero, it remains at rest.

Mathematical Form:

• $\sum \vec{F}_{\text{NET}} = 0$

• $\vec{a} = 0$

• $\Delta \vec{v} = 0$

Inertia is the resistance of any physical object to a change in its state of motion (speed, direction, or rest).

• Mass is a measure of inertia.

First Law Validity: Applies to inertial frames of reference (moving at constant velocity or at rest).

Newton's Second Law (Fundamental Law; F=ma)

Statement and Mathematical Formulation

Describes what happens when one or more forces act on an object. The acceleration of an object is directly proportional to the net force and inversely proportional to its mass.

• $\sum \vec{F}_{\text{NET}} = m \vec{a}$

• $m = \frac{\sum \vec{F}_{\text{NET}}}{\vec{a}}$

Units of Force:

• 1 Newton ($N$) = $1\,\frac{kg\cdot m}{s^2}$

• 1 Dyne = $1\,\frac{g\cdot cm}{s^2}$

In Three Dimensions:

• $\sum F_{\text{NET},x} = m a_x$

• $\sum F_{\text{NET},y} = m a_y$

• $\sum F_{\text{NET},z} = m a_z$

Mass vs. Weight

Definitions and Differences

• Mass: Inherent property, independent of surroundings, scalar quantity, does not change with location.

• Weight: Not inherent, depends on surroundings (gravitational field), vector quantity, measure of gravitational force.

Mathematical Form:

• $\vec{F}_g = \vec{G} = m \vec{g}$

The mass of an object remains the same on Earth and Moon, but its weight changes due to different gravitational accelerations.

Location	Gravitational Acceleration (m/s2)
Earth	9.81
Moon	1.63
Earth (various altitudes)	9.75 - 9.83

Weight depends on $g$ and varies with geographic location and altitude.

Newton's Third Law (Action-Reaction)

Statement and Examples

Forces result from interactions. Whenever two objects interact, they exert equal and opposite forces on each other.

• If you sit on a chair, your body exerts a downward force (action) and the chair exerts an upward force (reaction).

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

Mathematical Form:

• $\vec{n} = -\vec{F}_g$

• $|\vec{n}| = |\vec{F}_g|$

• $\vec{F}_{12} = -\vec{F}_{21}$

Action and reaction forces always come in pairs and act on different objects.

Applications and Problem Solving

Free Body Diagrams

Free body diagrams are used to represent all forces acting on an object. Each object is treated as a point particle, and forces are shown as arrows.

Example 1: Tension in Cables

An object weighing 125 N hangs from a cable tied to two other cables at angles $\theta_1 = 37^\circ$ and $\theta_2 = 53^\circ$ with the horizontal. Find the tensions $T_1$ and $T_2$.

• System is at rest ($a = 0$), so $\sum F_x = 0$ and $\sum F_y = 0$.

• $T_3 = m g = 125\,N$

• $T_2 \cos 53^\circ - T_1 \cos 37^\circ = 0$

• $T_2 \sin 53^\circ + T_1 \sin 37^\circ - T_3 = 0$

• Solving yields $T_1 = 75.1\,N$, $T_2 = 99.9\,N$

Example 2: Two Blocks in Contact

Two blocks of masses $m_1$ and $m_2$ are in contact on a frictionless surface. A force $F$ is applied to $m_1$.

• Both blocks move with the same acceleration $a_x$.

• For $m_1$: $F - P = m_1 a_x$

• For $m_2$: $P = m_2 a_x$

• $a_x = \frac{F}{m_1 + m_2}$

• $P = m_2 a_x = \frac{m_2 F}{m_1 + m_2}$

Example 3: Atwood Machine

Two masses $m_1$ and $m_2$ ($m_2 > m_1$) are connected by a string over a frictionless pulley. Find the acceleration and tension.

• For $m_1$: $T - m_1 g = m_1 a_y$

• For $m_2$: $m_2 g - T = m_2 a_y$

• $a_y = \frac{(m_2 - m_1)}{(m_1 + m_2)} g$

• $T = \frac{2 m_1 m_2}{m_1 + m_2} g$

Example 4: Masses on an Inclined Plane

Mass $m_1$ and $m_2$ are attached by a cord over a pulley. $m_2$ lies on a frictionless incline of angle $\theta$.

• For $m_1$: $T - m_1 g = m_1 a$

• For $m_2$: $m_2 g \sin \theta - T = m_2 a$

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

• $T = m_1 m_2 \frac{(\sin \theta + 1)}{m_1 + m_2} g$

Friction Forces

Static and Kinetic Friction

• Static Friction ($F_s$): The force that resists the initiation of sliding motion between two surfaces.

• Kinetic Friction ($F_k$): The force that opposes the motion of two surfaces sliding past each other.

Formulas:

• $F_s \leq \mu_s N$

• $F_k = \mu_k N$

Where $\mu_s$ and $\mu_k$ are the coefficients of static and kinetic friction, and $N$ is the normal force.

Summary Table: Mass vs. Weight

Property	Mass	Weight
Nature	Inherent property	Not inherent
Dependence	Independent of surroundings	Depends on surroundings
Measurement	Scalar	Vector
Physical Meaning	Amount of matter	Gravitational force

Additional info: These notes include expanded explanations, formulas, and problem-solving strategies for Newton's Laws of Motion, suitable for college-level physics students.