Newton's Laws of Motion

Introduction to Forces and Newton’s Second Law

Forces are fundamental to understanding motion in physics. A force is a push or pull that changes an object's velocity, and is represented as a vector (arrow). The unit of force is the Newton (N), defined as .

• Newton’s Second Law (Law of Acceleration): If a net force acts on an object, it accelerates in the direction of the net force.

• Net Force (): The vector sum of all forces acting on an object.

• Equation:

• Sign Convention: Choose a positive direction (e.g., right or up). Forces along the positive direction are written with a plus sign, and those against with a minus sign.

• Example: A 10 kg block is pulled by multiple horizontal forces. Calculate the block’s acceleration using .

Solving for Forces Using Newton’s Second Law

When solving force problems, always expand with correct signs. The acceleration () can be positive or negative, indicating direction, but force magnitudes are always positive.

• Steps:

  1. Choose direction of positive.

  2. Write and expand .

  3. Solve for unknowns.

• Example: A 10 kg box accelerates to the right at , pushed by two forces. If one force is 30 N to the left, calculate the other force.

Newton’s First Law (Law of Inertia)

Newton’s First Law states that objects resist changes in velocity unless acted upon by a net force. This property is called inertia.

• Inertia: The tendency of an object to resist changes in its state of motion.

• Mass: The quantity of inertia; heavier objects accelerate less for the same net force.

• Example: A box is pushed to the right with 20 N and another force of 20 N to the left. If the box has a mass of 6 kg, its acceleration is zero.

Types of Forces

Multiple types of forces can act on an object. Always draw forces as arrows from the object’s center.

• Applied Force: Direct push or pull.

• Tension: Pull from a rope or string.

• Normal Force: Reaction from a surface, perpendicular to contact.

• Friction: Opposes motion, acts parallel to surfaces in contact.

• Weight: Gravitational pull by Earth, always towards Earth’s center.

Free-Body Diagrams (FBDs)

A Free-Body Diagram shows only the forces acting on a single object, drawn as a dot or box. Draw all forces as arrows from the object’s center in the following order:

• Weight (always, unless otherwise stated)

• Applied Force & Tension (if present)

• Normal (if surfaces are in contact)

• Friction (if surfaces are rough)

Solving 1D Motion Problems with Forces

Forces cause objects to accelerate, changing their speed or direction. To solve problems combining force and motion variables, use Newton’s Second Law and kinematic equations.

• Acceleration (): The link between force and motion problems.

• Kinematic Equations (UAM):

• Example: A 20 kg block on a frictionless surface is pushed and accelerates to 30 m/s from rest in 6 s. Calculate the magnitude of the applied force.

Weight Force and Gravitational Acceleration

All objects near Earth are affected by gravity, which produces a force and an acceleration. The term “weight” refers to the force due to gravity, not mass.

• Mass: Quantity of matter, does not change with location.

• Weight: Force due to gravity, changes with location.

• Equation:

• Gravitational Acceleration: ,

• Example: If an object has mass 10 kg on Earth, its weight is ; on the Moon, its weight is .

Vertical Forces and Acceleration in the Y-Axis

Vertical forces cause objects to accelerate along the Y-axis. The net force is the sum of upward and downward forces.

• Equation:

• Example: A 5.1 kg block is pulled vertically by a string. Find the block’s acceleration for different tension forces.

Newton’s Third Law (Law of Action-Reaction)

Newton’s Third Law states that every action (force) results in an equal and opposite reaction. Forces always exist in pairs, acting on different objects.

• Action-Reaction Pair: Equal in magnitude, opposite in direction, act on different objects.

• Equation:

• Example: If you push an ice block with 20 N, the block pushes back on you with 20 N.

Force Problems in Connected Systems of Objects (X-Axis)

When objects are connected, they move together with the same acceleration and velocity. To solve these problems, use equation addition or substitution.

• Steps:

  1. Draw FBDs for all objects.

  2. Choose direction of positive.

  3. Write for each object.

  4. Solve for acceleration using addition or substitution.

  5. Plug acceleration into equations to solve for other targets.

• Example: Two blocks of mass 3 kg and 5 kg are connected by a string. If the 5 kg block is pulled with 30 N, calculate the acceleration and tension.

Forces in Systems of Objects with Pulleys

In systems with pulleys, connected objects have the same acceleration and velocity. For massless pulleys, tension is equal in magnitude but points in different directions.

• Example: A 4 kg block on a table is connected to a 2 kg block hanging off the table by a cable and pulley. Calculate the acceleration and tension.

Combining Connected Systems into a Single Object

When solving for acceleration, you can combine all masses into a single object and ignore internal connecting forces. If asked for an internal force, draw a FBD and write for the relevant object.

• Example: Two blocks connected by a string are pulled vertically upwards by a rope with 100 N. If the blocks are 3 kg and 2 kg, calculate the acceleration and tension in the string.