Forces

Overview

Forces is a core chapter in mechanics. It studies how interactions between objects produce changes in motion, deformation, or turning effects.

This topic supports later chapters such as:

• Dynamics
• Kinematics
• Work, Energy and Power

This page is the main revision hub for the chapter.

It includes:

• what a force is
• vector nature of force
• 1D signed convention
• common forces
• Hooke’s law overview
• pressure / upthrust / drag overview
• moments / equilibrium overview
• centre of gravity / stability overview
• worked examples
• formula summary
• common exam pitfalls
• related links

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Core Ideas

What is a Force?

A force is an interaction that may:

• change speed
• change direction of motion
• produce acceleration
• deform an object
• produce turning effects

The full Newtonian statement is:

$$\sum \vec{F}=m\vec{a}$$

where:

• $\sum \vec{F}$ = resultant force
• $m$ = mass (scalar)
• $\vec{a}$ = acceleration (vector)

Hence, force is a vector quantity.

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Vector Nature of Force

A vector has:

• magnitude
• direction

Examples:

• $6.0\text{N}$ upward
• $12\text{N}$ to the left

See Vectors.

1D Signed Convention

In one-dimensional motion, force is still a vector, but we often represent it by its signed component along a chosen axis.

Choose rightward as positive:

• $+8\text{N}$ means rightward
• $-3\text{N}$ means leftward

Then:

$$\sum F=ma$$

is the component form of:

$$\sum \vec{F}=m\vec{a}$$

A negative sign means the vector points opposite to the chosen positive direction.

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Resultant Force

Figure: The resultant force is the single vector equal to the vector sum of all forces acting on the object.

The resultant force $\vec{F}$ is the vector sum of all forces ${\vec{F}}_i$ acting on an object.

$$\vec{F}=\sum _i{\vec{F}}_i$$

If resultant force is zero:

$$\vec{F}=\sum {\vec{F}}_i=0$$

then the object has no acceleration by Newton’s second law $\vec{F}=m\vec{a}$.

It may be:

• at rest
• moving with constant velocity

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Common Forces

Weight

Figure: Weight is the gravitational force on an object and always acts vertically downward.

Weight is the gravitational force on an object.

$$\vec{W}=m\vec{g}$$

Magnitude:

$$|\vec{W}|=m|\vec{g}|$$

Direction: vertically downward. The magnitude of the gravitational field strength, $|\vec{g}|$ is conventionally denoted by $g$.

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Normal and Tangential Contact Forces

Figure: A contact surface can exert both a normal force perpendicular to the surface and a tangential contact force along the surface.

The forces exerted by a surface on an object.

Symbol: $\vec{N}$ and $\vec{f}$ in the above example.

Direction: $\vec{N}$ perpendicular to the contact surface, and $\vec{f}$ tangential to the contact surface.

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Tension and tensile force

Figure: Tension is the pulling force transmitted through a light taut string or rope, acting along the string.

Force transmitted through a light taut string, rope, or cable. In the above example, a stationary rope is pulled to two opposite directions. At any point on the rope, two tensile forces, ${\vec{f}}_l$ and ${\vec{f}}_r$, act on the point, with equal magnitude and opposite direction. Their magnitude, $|{\vec{f}}_l|=|{\vec{f}}_r|$ is referred to as “tension”, often denoted with

Symbol: $T$

Direction: along the string, away from the object.

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Resistive Force / Drag

Figure: Resistive forces act opposite to the direction of motion through a fluid such as air or water.

Opposes motion through air or fluid.

Examples:

• air resistance
• water resistance

Direction: opposite to velocity.

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Upthrust

Figure: Upthrust is the upward force exerted by a fluid on a partially or fully immersed object.

Upward force exerted by a fluid on an immersed object. For a floating or suspended object, the upthrust (buoyant force) equals the weight. For an object sinking at a constant velocity, there is an drag force opposite to the motion direction. The upthrust equals weight minus the drag force.

Symbol: $U=\text{weight of the displaced liquid or air}=\rho vg$, where $\rho$ is the density of the fluid or air, $v$ is the volume of the immersed object (i.e., the volume of displaced liquid or air), and $g$ is the gravitational field strength.

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Elastic / Spring Force

Restoring force from a stretched or compressed elastic object.

For ideal (harmonic) springs:

$$F=-k\Delta x$$

$k$ is the spring constant reflecting the stiffness of the spring. $\Delta x=x-x_0$ is the displacement from the equilibrium position $x_0$. The minus sign indicates that the spring force (signed vector) points to the opposite direction of the displacement.

Figure: A spring exerts a restoring force opposite to its extension or compression from equilibrium.

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Moments

Figure: The moment about a pivot depends on the force and the perpendicular distance from the pivot to the line of action.

Planar Moment (Torque)

The moment (torque) about a pivot point $O$ describes the turning effect of a force.

Definition

$$|\vec{M}|=|\vec{r}\times \vec{F}|=Fd$$

• $|\vec{M}|$: magnitude of the moment (scalar)
• $F$: magnitude of the force
• $\vec{r}$: position vector from the pivot to a point on the line of action of the force
• $d$: perpendicular distance from the pivot to the line of action of the force

Key Ideas

• The moment depends on both:
  • the magnitude of the force, and
  • the lever arm (perpendicular distance $d$)
• Only the component of force perpendicular to $\vec{r}$ contributes to the moment.

Vector Nature

• The moment is fundamentally a vector:

$$\vec{M}=\vec{r}\times \vec{F}$$

• It is perpendicular to the plane containing $\vec{r}$ and $\vec{F}$.

Direction (Planar Problems)

• Using the right-hand rule of cross-product of vectors:
  • Anticlockwise rotation $\to$ moment out of the page (positive)
  • Clockwise rotation $\to$ moment into the page (negative)
  • Hence, the planar moment vector can be expressed by a signed scalar $M$.

Equilibrium Condition

• For rotational equilibrium about a pivot:

$$\sum M=0$$

i.e., total clockwise moments = total anticlockwise moments.

For a fuller treatment of couples, pivot choice, and rigid-body equilibrium problems, see Equilibrium, Moments, and Couples.

Contact and Non-Contact Forces

Contact Forces

Require physical contact:

• normal force
• tension
• friction
• drag
• upthrust
• spring force

Non-Contact Forces

Act through fields:

• gravitational force
• electric force
• magnetic force

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Free-Body Diagram Overview

A free-body diagram (FBD) shows all external forces acting on one chosen object.

Typical steps:

1. isolate the object
2. draw object as box/dot
3. draw all forces from object
4. label clearly

See Force Diagrams and Resolution.

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Force Resolution Overview

A force at angle $\theta$ may be resolved into perpendicular components.

Horizontal:

$$F_x=F\cos \theta$$

Vertical:

$$F_y=F\sin \theta$$

An angled force can be resolved into horizontal and vertical components using right-triangle geometry.

Useful when analysing slopes, ropes, beams, and equilibrium.

See Force Diagrams and Resolution.

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Hooke’s Law Overview

Within the limit of proportionality:

$$F=kx$$

where:

• $F$ = applied force
• $k$ = spring constant
• $x$ = extension or compression

Force-extension graph is straight line through origin for ideal behaviour.

Elastic potential energy stored:

$$E=\frac{1}{2}k{x}^2$$

Related chapter:

Work, Energy and Power

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Pressure Overview

Pressure is a scalar quantity.

Defined by:

$$p=\frac{F}{A}$$

where:

• $F$ = normal force on surface
• $A$ = area

SI unit: Pa

$$1\text{Pa}=1{\text{N m}}^{-2}$$

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Hydrostatic Pressure Overview

Pressure due to liquid column:

$$p=\rho gh$$

where:

• $\rho$ = fluid density
• $g$ = gravitational field strength
• $h$ = depth

Pressure increases with depth.

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Upthrust Overview

A submerged object experiences greater pressure below than above, producing net upward force.

For displaced fluid volume $V$:

$$U=\rho gV$$

This is Archimedes’ principle.

See Fluid Forces and Resistive Motion.

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Drag and Terminal Velocity Overview

Drag force opposes motion through a fluid.

Usually increases with speed.

When falling:

• weight acts downward
• drag acts upward
• upthrust may act upward

At terminal velocity:

$$\sum \vec{F}=0$$

Acceleration becomes zero and speed is constant.

See Fluid Forces and Resistive Motion.

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Moments Overview

The turning effect of a force about a pivot.

$$M=Fd$$

where:

• $F$ = force magnitude
• $d$ = perpendicular distance from pivot to line of action

Moment direction is associated with clockwise / anticlockwise turning sense.

See Equilibrium, Moments, and Couples.

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Equilibrium Overview

For a rigid body in equilibrium:

Translational Equilibrium

$$\sum \vec{F}=0$$

Rotational Equilibrium

$$\sum M=0$$

So the body has:

• no linear acceleration
• no angular acceleration

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Centre of Gravity Overview

The centre of gravity (C.G.) is the point through which weight may be considered to act.

Useful in:

• balance problems
• moments
• stability

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Stability Overview

An object is more stable when:

• base is wider
• centre of gravity is lower

Toppling occurs when the line of action of weight falls outside the base.

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Conservative Force Bridge

A conservative force does path-independent work.

Examples:

• gravitational force
• ideal spring force

Non-conservative forces:

• friction
• drag

These often dissipate mechanical energy.

See Work, Energy and Power.

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Short Worked Examples

Example 1: Resultant Force in 1D

Right positive.

Forces on object:

• $+10\text{N}$
• $-4\text{N}$

$$\sum F=10-4=6\text{N}$$

Mass $=2.0\text{kg}$

$$a=\frac{6}{2}=3.0{\text{m s}}^{-2}$$

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Example 2: Spring Extension

Spring constant:

$$k=200{\text{N m}}^{-1}$$

Applied force:

$$F=10\text{N}$$

Then:

$$x=\frac{F}{k}=\frac{10}{200}=0.050\text{m}$$

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Example 3: Moment

Force $15\text{N}$ applied $0.40\text{m}$ from pivot.

$$M=Fd=15(0.40)=6.0\text{N m}$$

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Exam Relevance

Force questions test whether you can identify the physical interaction correctly before choosing the right method, such as force balance, resolution, or moments.

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Formula Summary

Newton’s Second Law

$$\sum \vec{F}=m\vec{a}$$

Weight

$$\vec{W}=m\vec{g}$$

Hooke’s Law

$$F=-k\Delta x$$

The equilibrium position is often set to be zero ($\Delta x=x-x_0=x$), hence $F=-kx$.

Elastic Potential Energy (scalar)

$$E=\frac{1}{2}k(\Delta x{)}^2$$

Pressure (scalar)

$$p=\frac{F}{A}$$

Hydrostatic Pressure (scalar)

$$p=\rho gh$$

Upthrust

$$U=\rho gV$$

Moment (vector)

$$M=Fd$$

Here, $d$ is the perpendicular distance between the pivot and the line of action of the force.
The moment has a direction, often described as clockwise or anticlockwise.

In planar problems, it is often convenient to treat moment as a signed scalar:

• Choose a convention (e.g., anticlockwise positive).
• Clockwise moments are then negative, and anticlockwise moments are positive.

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Common Exam Pitfalls

1. Treating force as scalar always

Force is a vector. Use direction/sign properly.

2. Confusing mass and weight

• mass in kg
• weight in N

3. Wrong trig component

Check angle carefully before using $\sin$ or $\cos$.

4. Using non-perpendicular distance for moment

Use perpendicular distance to line of action.

5. Forgetting drag direction

Drag always opposes motion relative to fluid.

6. Assuming zero force means zero velocity

Zero resultant force means zero acceleration, not necessarily zero velocity.

7. Forgetting 1D sign convention

Define positive direction first.

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Links

Related Links

• Force Diagrams and Resolution
• Fluid Forces and Resistive Motion
• Equilibrium, Moments, and Couples
• Dynamics
• Kinematics
• Vectors
• Work, Energy and Power

Good practice:

1. Choose the object or system.
2. Draw only forces acting on that object.
3. Do not draw forces exerted by the object on something else.
4. Do not add “resultant force” or “centripetal force” as an extra force in a free-body diagram.
5. Label every force clearly.
6. Resolve forces only after the real forces have been identified.

Free-body diagrams are the starting point for later dynamics, equilibrium, circular motion, and field-force problems.

For more detail, see Force Diagrams and Resolution.

Resolving Forces

Forces often need to be resolved into perpendicular components. If a force $F$ makes an angle $\theta$ with the chosen $x$-axis:

$$F_x=F\cos \theta$$ $$F_y=F\sin \theta$$

The component formula depends on where the angle is measured from. The component adjacent to the angle uses cosine; the component opposite the angle uses sine.

When an object is on an inclined plane, axes are often chosen parallel and perpendicular to the plane. This usually simplifies the normal contact force and friction.

Fluid Pressure and Upthrust

Pressure is force per unit area:

$$p=\frac{F}{A}$$

The SI unit of pressure is the pascal:

$$1\mathrm{Pa}=1\mathrm{N}{\mathrm{m}}^{-2}$$

In a fluid at rest, gauge pressure due to depth is:

$$p=\rho gh$$

Total pressure may include atmospheric pressure:

$$p_{\text{total}}=p_{\text{atm}}+\rho gh$$

Upthrust is the upward resultant force exerted by a fluid on an immersed object. By Archimedes’ principle, the upthrust equals the weight of fluid displaced:

$$U={\rho }_f{V}_{\text{disp}}g$$

where ${\rho }_f$ is the density of the fluid and $V_{\text{disp}}$ is the volume of fluid displaced.

An object floats when its weight is balanced by upthrust. At floating equilibrium:

$$U=W$$

For more detail, see Fluid Forces and Resistive Motion.

Drag and Terminal Velocity

Drag is a resistive force that acts opposite the relative motion between an object and the fluid around it. It usually increases with speed.

Terminal velocity occurs when the resultant force becomes zero because the driving force is balanced by resistive forces. For a falling object:

• initially, weight is larger than drag, so the object accelerates downward;
• as speed increases, drag increases;
• at terminal velocity, drag plus upthrust balances weight;
• acceleration is then zero, but velocity is not zero.

Moments and Couples

This hub has already introduced the core moment idea above. The main exam reminders are:

• use the perpendicular distance to the line of action;
• keep a consistent clockwise / anticlockwise sign convention in planar problems;
• remember that a couple gives zero resultant force but a non-zero turning effect.

For a couple:

$$|{\vec{\tau }}_{\text{couple}}|=Fd$$

where $F$ is the magnitude of either force and $d$ is the perpendicular distance between the two force lines of action.

For fuller discussion and worked rigid-body equilibrium problems, see Equilibrium, Moments, and Couples. For the cross-product vector form, see Vectors.

Static Equilibrium

For an object to be in static equilibrium, both translational and rotational equilibrium must hold:

$$\sum \vec{F}=\vec{0}$$ $$\sum \vec{\tau }=\vec{0}$$

The first condition means there is no resultant force. The second condition means there is no resultant moment about any point.

Useful strategy:

1. Draw a free-body diagram.
2. Choose convenient axes.
3. Resolve forces if needed.
4. Write force-balance equations.
5. Take moments about a pivot that removes one or more unknown forces.

For extended bodies, the weight is usually treated as acting through the centre of gravity. The line of action of this weight is central to stability and toppling analysis; see Centre of Gravity and Stability.

Conservative and Non-Conservative Forces

Some forces can be linked to potential energy. Gravitational, elastic, and electrostatic forces are common conservative forces. Work done by a conservative force can be stored or recovered as potential energy.

Friction and drag are non-conservative forces. They dissipate mechanical energy, usually as thermal energy.

This distinction becomes important in Work, Energy and Power.

For more detail, see Potential Energy and Conservative Forces.

Learning Path

Use these deeper notes when more detail is needed:

• Force Types and Interactions
• Force Diagrams and Resolution
• Elastic Forces and Hooke’s Law
• Fluid Forces and Resistive Motion
• Equilibrium, Moments, and Couples
• Centre of Gravity and Stability
• Potential Energy and Conservative Forces

Exam Relevance

Drawing accurate free-body diagrams is the most critical first step in solving force problems. Students must master calculating moments using the perpendicular distance and strategically choosing pivot points to simplify calculations. Applying both translational and rotational equilibrium conditions correctly is essential for solving problems involving multiple unknown forces or moments. Vector resolution skills are frequently tested.

Common exam traps include:

• drawing action-reaction pairs on the same free-body diagram;
• adding “centripetal force” as an extra force instead of identifying the real inward forces;
• forgetting that normal contact force is perpendicular to the surface;
• assuming friction always acts opposite the object’s motion rather than opposite relative motion or tendency of relative motion at the contact surface;
• using the distance to the pivot instead of the perpendicular distance for moments;
• applying only $\sum \vec{F}=\vec{0}$ and forgetting $\sum \vec{\tau }=\vec{0}$ for rigid-body equilibrium;
• confusing pressure with force;
• thinking terminal velocity means the object has stopped.

Links

• Prerequisite: kinematics
• Prerequisite: measurement
• Related: dynamics
• Related: work energy and power
• Related: circular motion
• Related: simple harmonic motion
• Related: oscillations
• Related: vectors
• Related: centre of gravity and stability
• Related: electric fields
• Related: force types and interactions
• Related: force diagrams and resolution
• Related: elastic forces and hookes law
• Related: fluid forces and resistive motion
• Related: equilibrium moments and couples
• Related: potential energy and conservative forces

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