Lesson 9 · 35 min
Choosing a System: Engineering Applications
You now have two complete toolkits. This lesson is about judgement: recognizing which one a problem is asking for, setting it up cleanly, and using the answers to make engineering decisions about roads, rides and aircraft.
Learning objectives
- Choose rectangular or path coordinates from what a problem gives and asks, and justify the choice.
- Follow a consistent procedure for curvilinear-motion problems.
- Apply \(a_n = v^2/\rho\) to design limits: minimum curve radius, maximum speed over a crest, and \(g\)-loads.
- Combine tangential and normal accelerations in realistic braking and cornering situations.
Which system?
Both systems always give the same vectors, so a "wrong" choice is never incorrect, only longer. Let the information in the problem decide:
| The problem gives or asks for… | Use | Because |
|---|---|---|
| \(x(t)\) and \(y(t)\), or \(a_x\) and \(a_y\) separately | \(x\)–\(y\) | differentiate or integrate each coordinate on its own |
| Gravity only (a projectile) | \(x\)–\(y\) | \(a_x = 0\), \(a_y = -g\): two independent, simple motions |
| A path \(y = f(x)\) with \(v_x\) or \(v_y\) known (guides, slots, cams) | \(x\)–\(y\) with the chain rule | \(v_y = f'v_x\), \(a_y = f''v_x^2 + f'a_x\) |
| A known path and the speed along it (vehicles, trains, rides) | \(n\)–\(t\) | \(a_t = \dot v\) and \(a_n = v^2/\rho\) come straight from the data |
| Circular motion with \(\omega\), \(\alpha\) | \(n\)–\(t\) | \(a_t = \alpha r\), \(a_n = \omega^2 r\) |
| "How sharply does the path bend?", "is it speeding up?", "normal force / friction needed" | \(n\)–\(t\), often converted from \(x\)–\(y\) | \(\rho\), \(a_t\) and \(a_n\) are path quantities (Lesson 8) |
A procedure that always works
- Sketch the path, the particle and the axes or the \(\et\), \(\en\) directions at the instant of interest.
- Choose the system with the table above, and write down what is known in that system (with signs).
- Relate the unknowns with the kinematic equations: differentiate, integrate, or use \(a_t\), \(a_n\), \(\rho\).
- Convert if the question asks for the other system (Lesson 8).
- Check: units, signs, \(|\avec|\) the same in both systems, and whether the size makes physical sense (compare with \(g = 9.81\ \text{m/s}^2\)).
Roads: curves, crests and sags
Road designers limit the normal acceleration so that drivers stay comfortable and tyres keep their grip. On a flat curve at steady speed, \(a_n = v^2/\rho\) must stay below a chosen limit \(a_{n,\max}\), which sets the smallest radius allowed for a given design speed:
\[ \rho_\text{min} = \frac{v^2}{a_{n,\max}} \]Example 9.1 — Designing a highway curve
A highway is designed for \(100\ \text{km/h}\) with a comfort limit of \(0.15g\) on the sideways acceleration. (a) Find the minimum curve radius. (b) A car on that curve at the design speed brakes at \(3\ \text{m/s}^2\). What is its total acceleration?
Show solution
(a) \(v = 100/3.6 = 27.78\ \text{m/s}\) and \(a_{n,\max} = 0.15(9.81) = 1.472\ \text{m/s}^2\):
\[ \rho_\text{min} = \frac{27.78^2}{1.472} = 524.4\ \text{m} \](b) \(a_t = -3\ \text{m/s}^2\) and \(a_n = 1.472\ \text{m/s}^2\):
\[ |\avec| = \sqrt{3^2 + 1.472^2} = 3.341\ \text{m/s}^2 \]Real highways also bank (superelevate) their curves so that part of \(m v^2/\rho\) comes from the normal force of the road; that is a Newton's-law refinement of the same kinematics.
Over a crest, \(\en\) points down. Gravity can supply at most \(g\) of downward acceleration, so if \(v^2/\rho\) exceeds \(g\) the road would have to pull the car down, which it cannot do: the wheels lift off. The limiting speed is
Speed limit over a crest of radius \(\rho\)
\[ \frac{v^2}{\rho} \le g \quad\Rightarrow\quad v_\text{max} = \sqrt{g\rho} \]In a sag, \(\en\) points up, the road pushes harder than usual, and occupants feel heavier: there is no speed at which contact is lost, only a comfort limit.
Example 9.2 — Airtime over a hill
A rural road goes over a hump whose crest has a radius of curvature of \(30\ \text{m}\). Above what speed do the wheels of a car leave the road at the crest?
Show solution
Notice that the answer does not depend on the car's mass: it is pure kinematics. This is why "humpback" bridges carry low speed limits.
Rides and aircraft: \(g\)-loads
Pilots and ride designers describe normal accelerations in multiples of \(g\). At the bottom of a pull-out from a dive, the aircraft is on a curve with the center of curvature above it, \(a_n = v^2/\rho\) points up, and the pilot is pressed into the seat.
Example 9.3 — Pulling out of a dive
A jet at \(150\ \text{m/s}\) pulls out of a dive along a vertical arc. The pilot's normal acceleration must not exceed \(5g\). What is the smallest radius of the pull-out?
Show solution
Doubling the speed would need four times the radius: fast aircraft need a lot of sky to turn.
Check your understanding
Key takeaways
- Use \(x\)–\(y\) when the coordinates or their accelerations are known separately (projectiles, \(x(t), y(t)\), guides); use \(n\)–\(t\) when the path and the speed along it are known.
- Design limits come from \(a_n = v^2/\rho\): \(\rho_\text{min} = v^2/a_{n,\max}\) for curves, \(v_\text{max} = \sqrt{g\rho}\) over a crest.
- Braking or accelerating on a curve adds \(a_t\): \(|\avec| = \sqrt{a_t^2 + a_n^2}\).
- You have finished the lessons. Next: the Practice Lab for unlimited problems, the Self-Check Quiz to test yourself, and the Formula Sheet for revision.