How can forces acting on a system be represented both visually and algebraically?

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How can Newton's laws be modelled mathematically?

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Understandings

• Newton's three laws of motion
• forces as interactions between bodies
• that forces acting on a body can be represented in a free-body diagram
• that free-body diagrams can be analysed to find the resultant force on a system
• the nature and use of the following contact forces
• normal force F_N is the component of the contact force acting perpendicular to the surface that counteracts the body
• surface frictional force F_f acting in a direction parallel to the plane of contact between a body and a surface, on a stationary body as given by F_f ≤ μ_sF_N or a body in motion as given by F_f= μ_dF_N where s and d are the coefficients of static and dynamic friction respectively
• tension
• elastic restoring force F_H following Hooke's law as given by F_H=-kx where k is the spring constant
• viscous drag force F_d acting on a small sphere opposing its motion through a fluid as given by F_d= 6πηrv where η is the fluid viscosity, r is the radius of the sphere and v is the velocity of the sphere through the fluid
• buoyancy F_b acting on a body due to the displacement of the fluid as given by F_b= ρVg where V is the volume of fluid displaced
• the nature and use of the following field forces
• gravitational force Fg is the weight of the body and calculated is given by F_g= mg
• electric force Fe
• magnetic force Fm

Guidance:

• Sketches and interpretations of free-body diagrams and a determination of the resultant force are for one and two-dimensional situations only.
• Forces should be labelled using commonly accepted names or symbols.
• Newton's first law will be applied to problems involving translational equilibrium.
• Examples of Newton's third law will include the identification of force pairs in various situations.

How can forces acting on a system be represented both visually and algebraically?

Draw free body diagrams with forces acting on a car, boat and plane when they are in motion

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Reference: Types of Forces

6 min

Free-body diagram

• Draw the object under consideration as a box
• Include all forces that act on the object, representing these forces as vectors of relative size.
• Do not include the forces that the object exerts on its environment.

Medium

Large

Small

Defining force and resultant / net force

A force is a push or pull exerted on an object that causes it to accelerate, decelerate, change direction, or deform. It is a vector quantity, meaning it has both magnitude and direction. The unit of force in the International System (SI) is the newton (N).

The Resultant Force or the Net Force is the overall force acting on an object, or, the vector sum of all forces on an object.

20 N

30 N

10 N

F_net =

10 N

14 N

11 N

11 N

F_net =

To find the net force:

1. Resolve forces into horizontal and vertical components (if necessary).
2. Add all horizontal forces to determine the horizontal component of the net force.
3. Add all vertical forces to determine the vertical component of the net force.
4. Combine the horizontal and vertical components to find the overall net force.

20 N

10 N

30°

50 N

45°

45°

50 N

F_net = 0 N

Vector Review

To find the net force:

• Resolve forces into horizontal and vertical components (if necessary).
• Add all horizontal forces to determine the horizontal component of the net force.
• Add all vertical forces to determine the vertical component of the net force.
• Combine the horizontal and vertical components to find the overall net force.

F_net =

F =

Understanding Newton’s laws with Rocket Sledder

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1. Make the applied force 40 N by changing the slider. What happens to F, v, and a?
2. Now reduce it to 0. What happens to F, v, and a?
3. Turn Friction Force on. What happens to F, v, and a?
4. Make the applied force 40 N by changing the slider. Turn Air Drag on. Turn Friction Off. What happens to F, v, and a? What is causing the drag?

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10 min

Can you predict the motion of the mass if you know the forces acting on it?

Isaac Newton

1642 – 1726

Sir Isaac Newton’s laws of motion explain the relationship between the motion of a physical object and the forces acting upon it. Understanding this information provides us with the basis of modern physics.

Prior Knowledge - State Newton’s Laws of Motion

Defining translational equilibrium

Translational Motion�Translational motion occurs when an object moves in space such that all its points move the same distance in the same direction over a given period of time. In this type of motion, the object undergoes a change in its position without rotating. For example, a car moving straight down a road or a person walking in a straight line are examples of translational motion. The path can be linear or curved, as long as the object's position changes without any rotation.

Translational Equilibrium�An object is in translational equilibrium when the net force acting on it is zero. This means that all the forces acting on the object cancel each other out, resulting in no change in its translational motion. In other words, the object either remains at rest (if initially stationary) or continues to move with a constant velocity (if already in motion). For instance, a book resting on a table is in translational equilibrium because the downward gravitational force is balanced by the upward normal force.

Always look out for initial velocity!

TASK 2.1. Skydiving and Newton’s Laws of Motion

Based on the second clip, using forces, explain how to turn a plane

How can Newton's laws be modelled mathematically?

TASK 2.2. F=ma Relationships (Go to File > Make a Copy)

20 min

Interpreting data

An equation is another representation of a proportional relationship.

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Putting it all together,

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F = km

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Where k is the constant of proportionality, a

F = ka

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Where k is the constant of proportionality, m

m = k/a

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Where k is the constant of proportionality, F

Predicting Motion

A parachutist has just jumped off a plane. The mass of the parachutist is 80 kg and her weight is 800 N. A strong wind of 100 N is blowing from the left. At the same time, there is a 200 N draft 30° to the horizontal coming from the left. The drag to the left is 20 N.

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Find the acceleration of the parachutist (magnitude and direction in angles).

We can use Newton’s laws of motion to predict the acceleration of an object. We can also use the same laws to predict the net force on an object if we know the acceleration.

Worked Problem

Step 1. Draw a Freebody Diagram

A free-body diagram is a sketch of only the object in question and the forces acting upon it. While this could still be a sketch, the emphasis is on the forces, so they must be drawn accurately.

• Draw the object under consideration as a box
• Include all forces that act on the object, representing these forces as vectors to relative scale.
• Include both the magnitude and direction of the vectors.
• Do not include the net force on the object, or the forces that the object exerts on its environment.

Free-body diagrams contain the quantitative information needed to solve the problem.

Self-study: Drawing freebody diagrams (video)

Parachutist

Draft = 200 N

Weight = −800 N

Air resistance = −20 N

Strong wind = 100 N

30°

Step 2. Find the Resultant Force

When an object is subject to several forces, the resultant force alone produces the net acceleration.

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We will use Vector Addition

1. Resolve forces into horizontal and vertical components (if necessary).

2. Add all horizontal forces to determine the horizontal component of the net force.

3. Add all vertical forces to determine the vertical component of the net force.

4. Combine the horizontal and vertical components to find the overall net force.

5. Find the angle to the horizontal.

F_Vertical = −700.0 N

F_Horizontall = 253.2 N

Parachutist

F_Net = −744.4 N

Angle of resultant force

tanθ = −700/253.2

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θ = tan⁻¹(−700/253.2) =−70.1°

Self-study: Angles and Unit Circle Exploration

Step 3: Find the acceleration

A nonzero net force indicates that the forces are unbalanced and there is an acceleration.

We can use Newton’s second law of motion to predict the acceleration.

What’s the difference between mass and weight?

Clarifying Newton’s 3rd Law

The pair of forces in Newton’s 3rd Law must always:

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• Act on two separate bodies;
• Both have the same type of force (e.g. gravitational, frictional, etc);
• Be equal in size;
• Be opposite in direction;
• Act along a straight line.

Do the following pair of forces adhere to Newton’s 3rd Law:

• The two bodies are: book and table;
• Force on book is weight BUT force on table is the reaction force;
• Forces on both bodies are equal;
• Forces are opposite;
• Forces act in a straight line.

Defining Normal Force / Reaction Force

The normal force is a contact force exerted by a surface perpendicular (or "normal") to that surface. It prevents objects from passing through each other and is a reaction force between objects based on Newton's third law of motion.

System vs. Object

You can club two objects to make one object.

Construct a force diagram for books A, B, and C, paying attention to the length of the force vectors

5 min

A man of mass 80 kg stands on the floor of an elevator. Find the force of the scales on the man when the elevator:

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1. Is standing still
2. Moves up at constant speed of 3 ms^-1
3. Moves up with acceleration 4 ms^-2
4. Moves down with acceleration 4 ms^-2
5. Moves down slowing down with deceleration 4ms^-2
6. The cable is cut. The elevator now is now in freefall. What is the force of the scales on the man now?

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Take g=10 ms^-2

Elevator Physics

Help is here

Verify with Sim (use 8 instead of 80)

Similar Problem

Let's break down each situation and solve for the force exerted by the scale on the man. The apparent weight of the man is given by

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1. Elevator is standing still (a = 0):

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2. Moves up at constant speed (a=0):

The force remains the same as in the first case because constant speed means no acceleration:

3. Moves up with acceleration a = 4 ms⁻²

The force increases because the elevator is accelerating upward:

4. Moves down with acceleration a = −4 ms⁻²

The force decreases because the elevator is accelerating downward:

5. Moves down, slowing down with deceleration a = 4 ms⁻²

When decelerating downward, the effective acceleration is upward (g+a):

6. Cable is cut (free fall, a = −g = −10 ms⁻²)

In free fall, the acceleration cancels out the effect of gravity, so the apparent weight is 0

Use apparent weight F_scale = m (g + a) for upward motion or m(g−a) for downward motion.

Freebody Diagrams and Newton’s Third Law

Proportional Reasoning Concept Builder

Extension: A peculiar problem for motion under gravity

Extra Practice

Defining Friction

Friction is a force that resists the motion between two objects that are in contact.

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View Sim

F_N

F_g

T

F_f

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What conclusions can we draw?

2 kg

5 kg

10 kg

Initial Phase (0 to ~5s): The applied force starts from zero and gradually increases as the sensor pulls harder. During this time, the force approaches the maximum static friction, but the block remains stationary since the force is not yet sufficient to overcome static friction.

Transition Phase (~5s to ~6s): At this stage, the applied force exceeds the maximum static friction. This initiates the motion of the block, as the force overcomes the resistance of static friction. The transition is marked by a sudden drop from the maximum static friction to the dynamic friction force.

Dynamic Phase (~6s onward): Once the block starts moving, the applied force stabilizes at the dynamic friction level (green dashed line). At this point, the block moves at a constant velocity, as the pulling force balances the dynamic frictional force.

Static friction has a maximum point

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Dynamic friction is constant

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Dynamic friction is smaller than static friction

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What does friction depend on?

Mass?

Surface Area?

Surface Texture?

Angle?

Acceleration due to gravity?

Friction does depend on the mass because it depends on the Normal force which depends on the Weight.

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Friction does depend on the texture because it depends coefficient of friction between two materials

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Friction does not depend on surface area

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Force of friction depends on the normal force and the coefficient of friction (µ)

What does the coefficient of friction depend on?

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What does the Normal force depend on?

Worked Solution

TASK 2.3. Friction on an inclined plane (File > Make a Copy)

10 min

View Sim

How can we calculate / measure the coefficient of friction?

You notice that the scales at the gym consistently measure weight on the low side of what you know to be true. You also notice that gym floor has a suspicious 15° angle… Why does this happen?

TASK 4. Investigating Hooke’s Law and The Spring Constant

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Hooke’s law

Hooke’s law is given by

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Draw a force diagram for the object at the end of the spring.

Discuss in your groups, what is the significance of the negative sign, and therefore what F_H refers to in the data booklet.

Two identical springs each with a spring constant of 220 Nm^-1 are connected to a trolley and fixed support as shown in the figure.

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When the trolley is in equilibrium each spring is extended by 4.0 cm. Calculate the net force on the trolley when it is moved 2.0 cm to the right.

The force that arises in any body when it is stretched or compressed is called tension.

Tension and Pulleys

A body of weight 98.0 N hangs from two strings that are attached to the ceiling as shown in Figure below. Determine the tension in each string.

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Worked Solution

Step 1: Identify the Forces on the blocks�Step 2: Write Equilibrium Conditions�Step 3: Solve for 𝑚�Step 4: Calculate the Ratio

An ice cube floats in water in a glass.

1. What happens to the level of water when the ice cube melts?

1. Rises
2. Falls
3. Stays the same

2. Given our molecular model of solids and liquids, why does ice float?

Discuss in your groups!�

Buoyancy

W

T

F_b = W - T

"Upward buoyancy force on an object (completely or partially submerged in a fluid) is equal to the weight of the fluid displaced by the object."

Archimedes’ Principle

View Sim

Floating Objects: For objects like boats or ice cubes, the buoyant force equals the object's weight, keeping it afloat.

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Sinking Objects: If the object's weight is greater than the buoyant force (e.g., a dense material like metal), it sinks.

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Neutral Buoyancy: If the buoyant force equals the weight of the object (e.g., a submarine adjusting its density), it remains suspended in the fluid.

In a fluid, the pressure at a point depends on the depth of that point below the fluid surface. The deeper you go, the greater the pressure due to the weight of the fluid above.

1. Identify the Forces on the Balloon

2. Find the Net Force

3. Use Newton’s Second Law to solve for acceleration

1. Drop balloons of different sizes. What observations can you make about the way the balloons drop?
2. Why do balloons drop so slowly? Use what you know about forces to propose an idea for why this is. Draw a Force diagram for the balloon to help you.
3. What are the factors that affect the speed at which a balloon drops?

Drag Force

1 m / s

0.67 m / s

0.5 m / s

10 cm

20 cm

15 cm

Stokes' Law describes the force experienced by small spherical particles moving through a viscous fluid due to drag.

“Stokes Law state that, “There is a drag force working on the spherical body which is following in a column and the upward drag force on the body ultimately equals the gravitational force then the body drops with a constant velocity.”

The formula for the drag force according to Stokes' Law is:

Stoke’s Law

Extension

Full derivation for reference

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Different from The Drag Equation

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Learn Why

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How can we predict v?

Video of derivation

What are the four fundamental forces in physics?

Strong

Gravitational

Weak nuclear

Electromagnetic

Two observers, one coffee mug

Do Newton’s Laws apply in all frames of reference?

Your friend is in a car. He puts his cup on the dashboard.

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You stand to the side of his car and see him drive off.

The car starts to move.

How does your friend see the cup?

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What does the motion of the cup look like to you?

Forces on the cup

Hmm… it looks like some force diagrams are inconsistent with the rule we developed…

Friend's Perspective (Non-Inertial Frame):

• The red arrow represents the fictitious force that appears to push the cup backward due to the car's acceleration.

Your Perspective (Inertial Frame):

• The orange arrow represents the frictional force opposing the cup's inertia as the car accelerates forward.

Inertial reference frames

If there is no resultant force, the object does not change speed or direction of motion

This is Newton’s first law, expressed as

∑F=0

Newton’s first law of motion tells us that we are concerned with inertial reference frames only.

An accelerating reference frame is called non-inertial. To be clear.

Is it possible to construct a human powered helicopter?

View Formulation

EXTRA

Test Tips

Model

Read the question carefully:

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• Identify what is given (e.g., mass, forces, acceleration, velocity, displacement, time) and what is to be calculated.
• Identify the type of motion: Is it static (no motion), uniform motion (constant speed), or accelerated motion?
• Write down all known values and variables.

Sketch a free-body diagram (FBD):

• Include all the forces acting on the object.
• Label each force (e.g., gravitational force F_g​).
• Indicate the direction of motion or acceleration (if any).
• Clearly mark axes (x- and y-axes) and choose a consistent coordinate system.
• Each object (or system of objects considered as one object) has its own FBD.

Apply Newton’s Laws and identify Equilibrium Conditions

1. First Law (Static or Uniform Motion):
• If the object is stationary or moving at constant velocity, the net force is zero (F_net = 0 N).
2. Second Law (Accelerated Motion):
• Write the equation for net force: F_net = ma, where:
• F_net​: sum of forces in the direction of motion.
• m: mass of the object.
• a: acceleration of the object.
3. Third Law (Interaction Forces):
• If applicable, identify action-reaction pairs (F_action = − F_reaction). They are always the same force from the perspectives of different objects.

Analyze

Resolve Forces

• Break forces into components (if necessary):
• For inclined planes or angles, resolve forces into perpendicular (F⊥​) and parallel (F∥) components using trigonometry:
• F∥ = Fsin⁡θ
• F⊥= Fcos⁡θ
• Add up forces along each axis (x- and y-axes) to find F_net

Use Kinematic Equations (if applicable)

• If the problem involves motion, use kinematic equations to relate displacement (s), velocity (v), acceleration (a), and time (t):

Solve

Solve the Equations

• Combine equations from the analysis stage to solve for the unknown.
• Check units and ensure they are consistent.
• Simplify and calculate the final answer.

Evaluate

Verify the Solution

• Check if the answer is reasonable (e.g., does the value and direction of force or acceleration make sense?).
• Double-check calculations and signs (positive vs. negative directions).

Forces as Vectors

• To add vectors, place them tail-to-tip or break them into components and then add the components.
• The resultant vector is the sum of the components in each direction. Use pythagoras for the magnitude and trigonometry for the direction.
• If three or more vectors combine to close a loop, it implies that the resultant force is zero.
• When breaking vectors into components to add them, use the shorthand for cosθ and sinθ with respect to the correct θ.

Tail to tip

Common Scenarios

Weight, Normal Force, and Friction on an inclined plane

• Include gravitational force F_g​, normal force F_N, and friction F_f​ in the FBD.
• Weight and mass are different things. Weight is a force, F_g. It depends on the Universal gravitational constant (G), the mass of the planet (M), and the distance between the masses.
• F_N is always perpendicular to the point of contact, F_g is always towards the center of the Earth. Friction is always resisting the direction of motion.
• Resolve F_g​ into parallel and perpendicular components to find the forces opposing friction and the normal force.
• Consider static friction F_f ≤ μ_sF_N​ for bodies at rest and dynamic friction F_f = μ_dF_N​ for bodies in motion

Wheelchair ramps reduce the force required to lift a person vertically. The incline spreads the work over a greater distance.

Spring Force & Hooke’s Law

• If a spring is stretched or compressed by a distance x, the restoring force F tries to return it to equilibrium.
• In static equilibrium, F_app = -kx, where the applied force equals the restoring force in magnitude and opposes the direction of motion (significance of the minus sign)

Mechanical scales use springs that stretch in proportion to the weight placed on them.

Tension and Pulleys

• Account for forces in the rope and equilibrium conditions if multiple objects are connected.
• Draw FBD for one object at a time.
• For a single object hanging from a rope, tension equals the object's weight if it's stationary: T = mg
• If the object is accelerating upward or downward: T = m(g+a) (upward acceleration), T=m(g−a) (downward acceleration)
• In an ideal pulley system (no friction), the tension in the rope is the same throughout.
• For two masses connected by a pulley: T = m₁a + m₁g and T=m₂g−m₂a, where a is the acceleration of the system.
• When a mass is connected to a rope on an inclined plane, decompose forces into parallel and perpendicular components.
• In elevators, weight and apparent weight are different things.

Cranes use pulley systems to lift heavy materials to great heights by reducing the force needed.

Buoyancy in a fluid

• Buoyancy and drag both act on fluids.
• For buoyancy, focus on whether the object floats, sinks, or is neutrally buoyant.
• For a fully submerged object the buoyant force is equal to the weight of the displaced fluid, F_b​=ρ_f​Vg
• For an object floating in equilibrium, the buoyant force equals its weight, F_b​= mg (a very small part of the volume is submerged)
• If an object sinks, the net force is the difference between gravitational and buoyant forces, F_net​ = mg − F_b ​= (ρ_o​−ρ_f​)Vg. (o = object, f = fluid)
• Pressure increases with depth
• Use fluid density and object volume relationships to calculate the buoyant force.

Large ships float because their overall density (including the air inside) is less than the water they displace.

Drag in a Fluid

• Drag force increases with velocity until it balances the gravitational force, at which point the object reaches terminal velocity
• Consider drag in trajectory problems to adjust the motion equations. Drag force reduces acceleration and shortens the range of the projectile.
• For objects moving through water, drag force is significant due to higher density

Cars, trucks, and airplanes are designed with streamlined shapes to minimize drag, improving fuel efficiency and speed.

Understandings

• Newton's three laws of motion
• forces as interactions between bodies
• that forces acting on a body can be represented in a free-body diagram
• that free-body diagrams can be analysed to find the resultant force on a system
• the nature and use of the following contact forces
• normal force F_N is the component of the contact force acting perpendicular to the surface that counteracts the body
• surface frictional force F_f acting in a direction parallel to the plane of contact between a body and a surface, on a stationary body as given by F_f ≤ μ_sF_N or a body in motion as given by F_f= μ_dF_N where s and d are the coefficients of static and dynamic friction respectively
• tension
• elastic restoring force F_H following Hooke's law as given by F_H=-kx where k is the spring constant
• viscous drag force F_d acting on a small sphere opposing its motion through a fluid as given by F_d= 6πηrv where η is the fluid viscosity, r is the radius of the sphere and v is the velocity of the sphere through the fluid
• buoyancy F_b acting on a body due to the displacement of the fluid as given by F_b= ρVg where V is the volume of fluid displaced
• the nature and use of the following field forces
• gravitational force Fg is the weight of the body and calculated is given by F_g= mg
• electric force Fe
• magnetic force Fm

Grade Gorilla Practice Set

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Paper Plainz Practice Set (Last year syllabus - ignore question 1 only)

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1. Try first

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• Then ask a buddy

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• Then look at the Answer (not the video solution) to see if you can work it out

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• Then look at the video solution

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• Then ask your teacher