Intro to Forces (Dynamics)

Concept: Introduction to Forces and Newton’s Second Law

Forces are fundamental interactions that cause changes in an object's motion. In physics, a force is defined as a push or pull that can change an object's velocity. The SI unit of force is the Newton (N), where . Newton’s Second Law of Motion states that if a net force acts on an object, it will accelerate in the direction of the net force. The net force is the vector sum of all forces acting on the object.

• Newton’s Second Law:

• Net Force: The resultant after adding all forces (vector sum).

• Sign Conventions: Choose a positive direction (usually right and up). Forces along the positive direction are positive; those against are negative.

• Solving Steps: 1) Choose direction of +, 2) Write and expand , 3) Solve for the unknown.

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

Concept: Solving for Forces Using Newton’s Second Law

When solving for unknown forces or acceleration, always expand with correct signs. The sign of acceleration indicates its direction, while the magnitude of force is always positive.

• Always write as a variable; plug in the correct sign if known.

• When solving for , the sign gives direction; when solving for forces, report the magnitude.

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 using .

Concept: Newton’s First Law (Law of Inertia)

Newton’s First Law states that an object will remain at rest or in uniform motion unless acted upon by a net force. This property is called inertia. Mass is a measure of inertia; heavier objects resist changes in velocity more than lighter ones.

• Inertia: Resistance to changes in velocity.

• Objects in motion do not require a force to keep moving at constant velocity; only a net force changes velocity.

• For the same net force, a heavier object accelerates less: .

Example: A box is pushed to the right with 20 N and to the left with 20 N. If the mass is 6 kg, the acceleration is zero because the forces cancel.

Concept: Types of Forces

Multiple types of forces can act on an object. Always draw forces as arrows from the object’s center. The most common forces are:

• Applied Force (FA): Direct push or pull.

• Tension (T): Force from a rope or string.

• Normal Force (N): Reaction force from a surface, perpendicular to the surface.

• Friction (f): Opposes motion between rough surfaces.

• Weight (W): Gravitational force, always points toward Earth’s center.

Example: Identify all forces acting on a tire swing (weight and tension) or a couch being pushed across a carpet (weight, applied, normal, friction).

Concept: Free-Body Diagrams (FBDs)

A Free-Body Diagram shows all forces acting on a single object, represented as a dot or box. Draw all forces as arrows from the center, in this order: 1) Weight, 2) Applied Force & Tension, 3) Normal, 4) Friction.

• FBDs help visualize and solve force problems by isolating the object of interest.

Example: Draw the FBD for a block pushed against a rough wall at a 45° angle, or for a box being pulled upward by a rope.

Concept: Solving 1D Motion Problems with Forces

Forces cause acceleration, which changes an object’s speed or direction. To solve problems involving both force and motion variables, use Newton’s Second Law and the Uniformly Accelerated Motion (UAM) equations:

Example: A 20 kg block on a frictionless surface accelerates from rest to 30 m/s in 6 s. Find the applied force using and UAM equations.

Concept: Weight Force and Gravitational Acceleration

All objects near Earth experience gravity, which produces a force (weight) and an acceleration. Weight is the force due to gravity, calculated as . Mass is the amount of matter and does not change with location, while weight does.

• Mass (m): Quantity of matter, constant everywhere.

• Weight (W): Force due to gravity, varies with gravitational acceleration.

• Gravitational Acceleration (g): , .

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

Concept: Vertical Forces and Acceleration in the Y-Axis

Vertical forces can cause acceleration along the y-axis. The net vertical force is the sum of all upward and downward forces, and determines the acceleration using .

• If upward forces exceed downward, acceleration is positive (upward).

• If downward forces exceed upward, acceleration is negative (downward).

• If forces are balanced, acceleration is zero.

Example: A 5.1 kg block is pulled vertically by a string. For different tensions, calculate the acceleration using .

Concept: Equilibrium

An object is in equilibrium if all forces acting on it cancel, resulting in zero acceleration (). Equilibrium does not require the object to be at rest; it can move at constant velocity.

• Static Equilibrium: Object at rest ().

• Dynamic Equilibrium: Object moves at constant velocity ().

Example: A box pulled by two equal forces in opposite directions moves at constant speed; net force is zero, so .

Concept: The Normal Force

The normal force (N) is the reaction force from a surface, always perpendicular to the surface. There is no general equation for N; it must be calculated using in the direction perpendicular to the surface.

• Normal force balances weight and any other vertical forces.

Example: A 2.04 kg book rests on a table. N (approx.).

Concept: 2D Forces in the Horizontal Plane

When forces act in two dimensions, decompose each force into x and y components. The net force in each direction is found by vector addition, and Newton’s Second Law is applied separately in each axis.

• Decompose angled forces: , .

• Apply and .

Example: A 5 kg block is pulled by two forces at different angles. Find the net force and acceleration in both x and y directions.

Concept: Solving an Unknown 2D Force

When a force’s magnitude or direction is unknown, treat its components as variables and solve using the system of equations from and .

Example: Three forces act on a 40 kg block. Given two forces and the desired acceleration, solve for the third force’s magnitude and direction.

Concept: 2D Forces in Horizontal & Vertical Planes

If a force acts at an angle above or below the horizontal, decompose it into horizontal and vertical components. The normal force may change depending on the vertical component of the applied force.

• If , the object is in equilibrium in the y-axis ().

Example: A 5.1 kg block is pulled by a 10 N force at 37° above the horizontal. Find the normal force and acceleration.

Concept: Equilibrium in 2D

In 2D equilibrium, all forces cancel in both x and y axes. Decompose all forces and set and to solve for unknowns.

Example: A box is suspended by two cables at angles. Calculate the tension in each cable using equilibrium equations.

Concept: Newton’s Third Law

Newton’s Third Law states that every action force has an equal and opposite reaction force. These forces act on different objects and are always equal in magnitude and opposite in direction.

• Action-reaction pairs do not result in equal accelerations unless the masses are equal.

• For each object, sum forces acting on it to find its acceleration.

Example: You push an ice block; the block pushes back on you with equal force in the opposite direction.

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

When objects are connected (e.g., by a string), they move together with the same acceleration. To solve, draw FBDs for each object, write for each, and use equation addition or substitution to solve for unknowns.

• For vertical systems, each tension supports the total weight below it.

Example: Two blocks connected by a string are pulled horizontally. Find the acceleration and tension in the string.

Concept: Forces in Systems of Objects with Pulleys

In pulley systems, connected objects have the same acceleration and velocity. For massless pulleys, the tension is the same on both sides. Choose the positive direction as the direction the heavier object will move.

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

Concept: Combining Connected Systems into a Single Object

To find the acceleration of a system, combine all masses into a single object and ignore internal forces (tensions, normals between objects). Use .

Example: Two blocks connected by a string are pulled upward by a rope. Find the acceleration of the system and the tension in the connecting string.

Summary Table: Common Forces