Dynamics

Overview

Dynamics is the study of how forces cause changes in motion. While Kinematics describes motion using displacement, velocity and acceleration, Dynamics explains why that motion occurs.

This chapter develops the link between force and motion through Newton’s Laws of Motion, and extends to the ideas of momentum, impulse, and collisions.

Dynamics is foundational for later topics such as:

• Work, Energy and Power
• Circular Motion
• Oscillations
• Fields and interactions

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Core Ideas

What Dynamics Studies

Typical questions in Dynamics ask:

• What is the acceleration of a body under given forces?
• What force is needed to produce a certain motion?
• What happens when two bodies interact?
• How do objects move when connected together?
• What happens during collisions or recoil?

To solve such questions, combine:

• force laws
• vector resolution
• Kinematics
• momentum methods

Newton’s Laws of Motion

Newton’s First Law

A body remains:

• at rest, or
• moving with constant velocity in a straight line

unless acted upon by a resultant external force.

Meaning

If:

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

then acceleration is zero:

$$\vec{a}=0$$

The object may still be moving at constant velocity.

Inertia

Inertia is the tendency of a body to resist changes in motion.

• Larger mass → greater inertia
• Harder to start moving
• Harder to stop
• Harder to change direction

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Newton’s Second Law

The resultant force equals the rate of change of momentum:

$$\vec{F}=\frac{d\vec{p}}{dt}$$

For constant mass:

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

Key meanings

• Force causes acceleration.
• Acceleration is in the same direction as resultant force.
• If resultant force increases, acceleration increases (for fixed mass).

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Newton’s Third Law

If body A exerts a force on body B, then B exerts an equal and opposite force on A.

Properties of action-reaction pairs

They:

• act on different bodies
• are equal in magnitude
• opposite in direction
• same type of force
• occur simultaneously

Example:

• hand pushes wall
• wall pushes hand

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Mass, Inertia and Weight

Mass

Mass measures inertia.

• scalar quantity
• SI unit: kg
• independent of location

Weight

Weight is gravitational force:

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

• vector quantity
• acts vertically downward near Earth
• unit: N

Mass is constant, but weight depends on gravitational field strength.

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Resultant Force and Acceleration

To analyse motion:

1. Identify all external forces.
2. Resolve forces into perpendicular directions.
3. Apply:

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

Usually:

• horizontal direction
• vertical direction

treated separately.

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Free-Body Diagrams Overview

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

Common forces:

• Weight $(W)$
• Normal contact force $(N)$
• Tension $(T)$
• Friction $(f)$
• Resistive force
• Applied force
• Upthrust (if relevant)

See full treatment: Free-Body Diagrams and Force Analysis

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Force Resolution Overview

When forces act at angles, resolve into components.

Example:

For force $F$ at angle $\theta$:

Horizontal:

$$F\cos \theta$$

Vertical:

$$F\sin \theta$$

Use Vectors carefully.

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Standard Dynamics Applications

1. Horizontal Motion

If frictionless:

$$F=ma$$

If friction present:

$$F-f=ma$$

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2. Vertical Motion / Lifts

For lift accelerating upward:

$$R-W=ma$$

For lift accelerating downward:

$$W-R=ma$$

Where $R$ is scale reading / normal reaction.

If $a=g$ downward:

$$R=0$$

(apparent weightlessness)

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3. Connected Bodies

Bodies linked by light inextensible string share the same magnitude of acceleration.

Use:

• separate FBD for each body
• same tension if pulley/string ideal
• same acceleration magnitude

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4. Inclined Planes

Resolve weight:

Parallel to slope:

$$mg\sin \theta$$

Perpendicular to slope:

$$mg\cos \theta$$

Normal force often equals:

$$N=mg\cos \theta$$

(if no additional vertical forces)

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Momentum Overview

Momentum:

$$\vec{p}=m\vec{v}$$

• vector quantity
• same direction as velocity

Large momentum may arise from:

• large mass
• high speed
• both

See: Momentum and Impulse

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Impulse Overview

Impulse is change in momentum:

$$J=\Delta p$$

For constant force:

$$J=F\Delta t$$

Also equal to area under force-time graph.

Longer collision time reduces average force for same momentum change.

Examples:

• airbags
• crumple zones
• bending knees when landing

See: Momentum and Impulse

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Conservation of Momentum Overview

For an isolated system with no resultant external force:

$$\text{Total momentum before}=\text{Total momentum after}$$

For two bodies:

$$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$$

Used in:

• recoil
• explosions
• collisions

See: Momentum Conservation and Collisions

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Collisions Overview

Elastic Collision

Conserved:

• momentum
• kinetic energy

For 1D collisions:

$$u_1-u_2=v_2-v_1$$

(relative speed of approach = relative speed of separation)

Inelastic Collision

Conserved:

• momentum only

Kinetic energy decreases.

Perfectly Inelastic Collision

Bodies stick together after impact.

Common final speed:

$$v_1=v_2$$

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Short Worked Examples

Example 1: Horizontal Force

A 4.0 kg block experiences 10 N resultant force.

$$a=\frac{F}{m}=\frac{10}{4}=2.5{\text{m s}}^{-2}$$

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Example 2: Momentum

A 2.0 kg trolley moves at 3.0 m s${}^{-1}$.

$$p=mv=2(3)=6.0{\text{kg m s}}^{-1}$$

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Example 3: Impulse

A force of 20 N acts for 0.50 s.

$$J=F\Delta t=20(0.50)=10\text{N s}$$

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Exam Relevance

Dynamics questions test whether you can:

• identify forces correctly
• choose a sensible system
• apply Newton’s laws consistently
• use momentum methods when force-based methods are awkward

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Formula Summary

Newtonian Motion

$$\sum \vec{F}=m\vec{a}$$ $$\vec{F}=\frac{d\vec{p}}{dt}$$

Weight

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

Momentum

$$\vec{p}=m\vec{v}$$

Impulse

$$J=\Delta p$$ $$J=F\Delta t$$

Conservation of Momentum

$$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$$

Elastic Collision (1D)

$$u_1-u_2=v_2-v_1$$

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Common Exam Pitfalls

1. Forgetting vector directions

Momentum, force, acceleration are vectors.

2. Mixing action-reaction pair with balanced forces

Balanced forces act on same body.
Third-law pair act on different bodies.

3. Assuming normal force always equals weight

Only true in special cases.

4. Using momentum conservation when external force exists

Must analyse system carefully.

5. Wrong sign convention

Choose positive direction clearly.

6. Confusing mass and weight

Mass in kg, weight in N.

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Revision Strategy

Master this order:

1. Newton’s laws
2. Free-body diagrams
3. Connected-body systems
4. Inclined planes
5. Momentum
6. Impulse
7. Collisions

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Links

Related Links

• Free-Body Diagrams and Force Analysis
• Momentum and Impulse
• Momentum Conservation and Collisions
• Forces
• Kinematics
• Vectors
• Work, Energy and Power
• Circular Motion

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Final Takeaway

Dynamics turns motion description into physical explanation. If you can:

• draw accurate free-body diagrams
• apply $\sum \vec{F}=m\vec{a}$
• use momentum methods wisely
• handle collisions cleanly

you will be strong in one of the most important H2 Physics chapters.

$$\sum F_y=m{a}_y$$

Here $F_x$, $F_y$, $a_x$, and $a_y$ are signed components of vector quantities along chosen axes. In a one-dimensional problem, the same idea is used with one chosen positive direction; signs then represent direction, even though the equation is written with scalar-looking symbols.

If $\sum \vec{F}=\vec{0}$, then $\vec{a}=\vec{0}$. The object may be at rest or moving with constant velocity.

Mass, Weight, and Apparent Weight

Mass is a scalar measure of inertia. Weight is the gravitational force on a body:

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

In magnitude form near Earth’s surface:

$$W=mg$$

Apparent weight is the normal contact force exerted by a support. A weighing scale reads $N$, not necessarily $mg$. In a vertical lift problem, the vector equation is usually reduced to a one-dimensional signed-component equation. Taking upward as positive:

$$N-mg=ma$$

Therefore:

• if the lift accelerates upward, $N>mg$;
• if the lift accelerates downward, $N<mg$;
• if the lift moves at constant velocity, $N=mg$;
• in free fall, $N=0$ even though gravity still acts.

Connected Bodies

Connected-body problems involve objects linked by strings, rods, or contact. For an ideal light inextensible string over a smooth pulley:

• the tension has the same magnitude throughout the string;
• connected bodies have the same acceleration magnitude along the string;
• each body usually needs its own free-body diagram.

The equations are written separately and solved simultaneously. The mass in $m\vec{a}$ must match the selected body or system.

Momentum and Impulse

Linear momentum is a vector:

$$\vec{p}=m\vec{v}$$

Impulse is the change in momentum:

$$\vec{J}=\Delta \vec{p}$$

For a resultant force acting over time:

$$\vec{J}=\int {\vec{F}}_{\text{net}}dt$$

On a resultant force-time graph, impulse is the signed area under the graph. This is different from work, which comes from a force-displacement graph.

Conservation of Momentum and Collisions

The total linear momentum of a system remains constant if the resultant external force on the system is zero, or if the external impulse is negligible during a short interaction:

$$\sum {\vec{p}}_{\text{initial}}=\sum {\vec{p}}_{\text{final}}$$

Momentum conservation is useful for collisions, explosions, and recoil. Internal forces can redistribute momentum between bodies, but they cannot change the total momentum of an isolated system.

For collisions:

• elastic collision: total momentum and total kinetic energy are conserved;
• inelastic collision: total momentum is conserved, but total kinetic energy is not;
• perfectly inelastic collision: objects stick together and move with a common final velocity.

For a one-dimensional elastic collision:

$$u_1-u_2=v_2-v_1$$

This relative velocity equation applies only to one-dimensional elastic collisions. The velocities $u_1$, $u_2$, $v_1$, and $v_2$ are signed components along the chosen positive direction.

Non-Constant Forces and Method Choice

When forces vary with time, position, or speed, constant-acceleration equations may not apply directly. Choose the method based on the quantity asked:

• Use Newton’s second law when acceleration or contact forces are required.
• Use impulse and momentum when forces act over time, especially in collisions.
• Use work and energy when forces vary with displacement or when speed/height/losses are the focus.

The common graph areas are:

• resultant force-time graph: impulse, $\Delta \vec{p}$;
• force-displacement graph: work, only when the plotted force is the component along displacement.

Focused Concept Notes

• Newton’s Laws of Motion
• Force Diagrams and Resolution
• Newtonian Dynamics Applications
• Momentum and Impulse
• Momentum Conservation and Collisions
• Dynamics Methods and Non-Constant Forces (optional enrichment)

Exam Relevance

Mastering dynamics requires a strong understanding of vector quantities and consistent application of Newton’s laws. Students must be proficient in drawing accurate free-body diagrams to identify all forces acting on an object. Distinguishing between Newton’s second and third laws, especially regarding action-reaction pairs, is crucial to avoid common errors. For collision problems, correctly identifying the type of collision dictates which conservation laws apply. Pay close attention to the definition of the system when applying conservation of momentum. Problems involving apparent weight in accelerating frames, such as lifts, require careful analysis of the normal force.

Links

• Prerequisite: kinematics
• Prerequisite: forces
• Prerequisite: vectors
• Related: newtons laws of motion
• Related: force diagrams and resolution
• Related: newtonian dynamics applications
• Related: momentum and impulse
• Related: momentum conservation and collisions
• Related: dynamics methods and non constant forces
• Related: work energy and power
• Related: circular motion
• Related: centre of gravity and stability
• Misconception: newtons laws distinction
• Misconception: force identification errors
• Misconception: vector direction consistency
• Misconception: momentum conservation system
• Misconception: inelastic ke conservation
• Misconception: elastic relative speed application
• Misconception: apparent weightlessness gravity
• Misconception: force resolution angles
• Misconception: force time graph area
• Misconception: scalars vs vectors

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