show a qualitative understanding of frictional forces and viscous/drag forces including air resistance (no treatment of the coefficients of friction and viscosity is required, and a simple model of drag force increasing as speed increases is sufficie
Dynamics – Momentum, Newton’s Laws, Friction & Drag (Cambridge 9702)
1. Newton’s Three Laws (statement only)
- Newton I – Law of Inertia: A body remains at rest or moves with constant velocity unless acted on by a net external force.
- Newton II – Law of Motion: The net external force on a body is equal to the mass times its acceleration
\$\mathbf{F}_{\text{net}} = m\mathbf{a}.\$
(The more general form \(\mathbf{F}_{\text{net}} = d\mathbf{p}/dt\) is useful for deeper study but is not required by the syllabus.)
- Newton III – Action – Reaction: For every action force there is an equal and opposite reaction force acting on a different body. Internal forces cancel when analysing the total momentum of an isolated system.
2. Linear Momentum
Momentum is a vector defined by
\$\mathbf{p}=m\mathbf{v}\$
Its direction is the same as the velocity of the object.
2.1 Momentum in two dimensions
- Write the momentum vector in components:
\$\mathbf{p}=m\bigl(vx\hat{\mathbf{i}}+vy\hat{\mathbf{j}}\bigr).\$
- Conservation of momentum applies to each component separately:
\$\$\sum p{x,\text{initial}}=\sum p{x,\text{final}},\qquad
\sum p{y,\text{initial}}=\sum p{y,\text{final}}.\$\$
- Useful for collisions on smooth tables, projectile impacts, etc.
2.2 Impulse
The impulse delivered by a force \(\mathbf{F}(t)\) over a time interval \(\Delta t\) is
\$\mathbf{J}= \int{t1}^{t_2}\mathbf{F}\,dt = \Delta\mathbf{p}.\$
Impulse changes the momentum of a body without needing the detailed force history.
2.3 Collisions – key ideas
- Momentum is always conserved (elastic or inelastic).
\$\mathbf{p}{\text{initial}}=\mathbf{p}{\text{final}}.\$
- Elastic collisions (perfectly elastic) also conserve kinetic energy.
\$\$\tfrac12 m1v1^{2}+\tfrac12 m2v2^{2}=
\tfrac12 m1v1'^{2}+\tfrac12 m2v2'^{2}.\$\$
Note: the syllabus does not require the coefficient of restitution; the kinetic‑energy equation is needed only for perfectly elastic collisions.
- Inelastic (sticky) collisions – the bodies stick together after impact:
\$v'=\frac{m1v1+m2v2}{m1+m2}.\$
2.4 Example – head‑on collision on a friction‑less track
Cart A: \(mA=2.0\;\text{kg},\;vA=+3.0\;\text{m s}^{-1}\) (right).
Cart B: \(mB=3.0\;\text{kg},\;vB=-2.0\;\text{m s}^{-1}\) (left).
Find the velocities after a perfectly elastic collision.
- Conserve momentum (right positive):
\$mAvA+mBvB = mAvA'+mBvB'.\$
- Conserve kinetic energy (elastic case). Solve the two equations simultaneously to obtain
\$\$v_A' = -0.2\;\text{m s}^{-1},\qquad
v_B' = 2.8\;\text{m s}^{-1}.\$\$
The lighter cart rebounds, the heavier cart continues forward faster – a classic elastic‑collision outcome.
3. Frictional Forces – Qualitative Understanding
3.1 What is friction?
- Friction opposes relative motion (or the tendency for motion) between two solid surfaces in contact.
- Originates from microscopic interlocking of surface asperities and intermolecular attractions.
- Direction: always opposite to the direction of relative motion (or impending motion).
3.2 Types of friction
- Static friction – prevents the start of motion up to a maximum value.
- Kinetic (sliding) friction – acts once the surfaces are sliding past each other.
3.3 Qualitative features (syllabus focus)
- Magnitude is roughly proportional to the normal reaction \(N\) (the “weight” of the object on the surface).
- Independent of the apparent area of contact.
- For most macroscopic surfaces the kinetic friction force is essentially independent of speed.
- Friction converts mechanical energy into thermal energy, slowing the object.
3.4 Optional quantitative reminder (not required for 9702)
For completeness, the usual expressions are
\$\$F{\text{static}}^{\max}= \mus N,\qquad
F{\text{kinetic}} = \muk N,\$\$
where \(\mus\) and \(\muk\) are the static and kinetic coefficients of friction. These are useful for extension work but are not examined.
3.5 Illustrative diagram
4. Viscous/Drag Forces – Qualitative Understanding of Air Resistance
4.1 What is drag?
- When an object moves through a fluid (air, water, oil) it experiences a resistive force called drag.
- Drag always acts opposite to the direction of motion.
- Two main mechanisms:
- Viscous (frictional) drag – due to the fluid’s internal viscosity; dominant at low speeds and for smooth, small objects.
- Pressure (form) drag – caused by pressure differences between the front and rear of the object; becomes important at higher speeds and for bluff bodies.
4.2 Qualitative behaviour of drag
- Low‑speed (laminar) regime: drag increases roughly in proportion to speed.
Conceptual form: \(F_{\text{drag}}\propto v\).
- High‑speed (turbulent) regime: drag increases roughly with the square of speed.
Conceptual form: \(F_{\text{drag}}\propto v^{2}\).
- The transition occurs when the flow changes from laminar to turbulent (characterised by a Reynolds number of order \(10^{3}\!-\!10^{4}\); the exact value is not required for the exam).
- Because drag grows with speed, an object subject to a constant applied force eventually reaches a terminal speed \(v_t\) where the applied force balances the drag, giving zero net acceleration.
Beyond the syllabus – useful for extension work
For students who wish to explore further, the drag force can be written in a simple piece‑wise form:
\$\$%
F_{\text{drag}}(v)=%
\begin{cases}
k{1}\,v, & v\lesssim v{c}\\[4pt]
k{2}\,v^{2}, & v\gtrsim v{c}
\end{cases}%
\$\$
where \(k{1},k{2}>0\) depend on the object's size, shape and the fluid’s density, and \(v_{c}\) is the characteristic speed at which the flow changes regime.
4.3 Example – terminal speed of a falling sky‑diver
Assume a sky‑diver of mass \(m=80\;\text{kg}\) reaches a terminal speed when the weight \(mg\) is balanced by a quadratic drag force \(F{\text{drag}}=k v^{2}\). If the measured terminal speed is \(vt=55\;\text{m s}^{-1}\), the drag constant is
\$k = \frac{mg}{v_t^{2}} = \frac{80\times9.8}{55^{2}}\approx 0.026\;\text{kg m}^{-1}.\$
This illustrates how the qualitative idea of “drag ∝ \(v^{2}\) at high speed” leads directly to a terminal‑velocity calculation.
4.5 Illustrative diagram – drag on a sphere
5. Comparison of Friction and Drag
| Aspect | Friction (solid–solid) | Viscous/Drag (solid–fluid) |
|---|---|---|
| Typical speed dependence | Nearly independent of speed (static/kinetic friction) | Increases with speed – roughly linear at low \(v\), quadratic at high \(v\) |
| Direction of force | Opposite to relative motion (or impending motion) | Opposite to the object's velocity |
| Energy dissipation | Converted to heat at the contact surface | Converted to heat in the fluid and, at high speed, to turbulent kinetic energy |
| Dependence on normal reaction | Approximately proportional to the normal reaction \(N\) | Independent of \(N\); depends on fluid density, object shape and speed |
| Typical syllabus‑level formula | \(F_f \approx \mu N\) (qualitative) | \(F_{\text{drag}}\) ∝ \(v\) (low speed) or ∝ \(v^{2}\) (high speed) |
6. Summary – What You Must Remember for Cambridge 9702
- Newton’s three laws are the foundation of dynamics; for the exam you need the simple form \(\mathbf{F}=m\mathbf{a}\).
- Linear momentum \(\mathbf{p}=m\mathbf{v}\) is conserved in isolated systems; treat each component separately in two‑dimensional problems.
- Impulse \(\mathbf{J}=\Delta\mathbf{p}\) is a convenient way to handle forces acting over short time intervals.
- Collisions:
- Momentum is always conserved.
- For perfectly elastic collisions kinetic energy is also conserved (optional depth).
- For perfectly inelastic (sticky) collisions use the combined‑mass formula.
- Friction (solid–solid):
- Static and kinetic types; magnitude roughly \(\propto N\).
- Direction opposite to relative motion; essentially speed‑independent.
- Energy is dissipated as heat.
- Viscous/drag forces (solid–fluid):
- Oppose motion through a fluid.
- Linear with speed at low \(v\) (laminar regime), quadratic at high \(v\) (turbulent regime).
- Lead to a terminal speed when drag balances the applied force.
- Understanding these resistive forces lets you apply Newton’s laws to real‑world situations such as:
- Blocks sliding on surfaces,
- Cars braking on wet roads,
- Sky‑divers reaching terminal velocity,
- Objects falling through air or water.