Physics • Year 12 • Module 5 • Lesson 8
Horizontal Circular Motion
Lock in the key vocabulary, formulas and force-diagram conventions for conical pendulums, banked curves, and flat curves before tackling harder problems.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: conical pendulum, banked curve, design speed, centripetal force, coefficient of static friction, normal force, tension, radius, period, component. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | A bob on a string moving in a horizontal circle, with the string at a constant angle to the vertical. | |
| 1.2 | A road or track inclined at an angle to the horizontal, allowing the normal force to contribute a centripetal component. | |
| 1.3 | The speed at which no friction is required on a banked curve; given by √(rg tan θ). | |
| 1.4 | The net force directed toward the centre of a circular path, equal to mv²/r. | |
| 1.5 | The ratio of the maximum static friction force to the normal force; symbol μs. | |
| 1.6 | The contact force exerted perpendicular to a surface by that surface; greater than mg on a banked curve. | |
| 1.7 | The pulling force in a string or rope, directed along the string toward the attachment point. | |
| 1.8 | The distance from a point on a circular path to the centre of the circle; symbol r. | |
| 1.9 | The time taken for one complete revolution in circular motion; symbol T. | |
| 1.10 | The resolved part of a vector in a specified direction, found by multiplying the vector magnitude by a sine or cosine factor. |
2. True or false — with correction
Circle T or F for each statement. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 At the design speed, friction acts down the slope of a banked curve to provide centripetal force. T / F
2.2 For a conical pendulum, the vertical component of tension balances the weight of the bob. T / F
2.3 The maximum speed on a flat unbanked curve depends on the mass of the vehicle. T / F
2.4 On a banked curve, the normal force is equal to the weight of the vehicle. T / F
2.5 When a car travels faster than the design speed on a banked curve, friction acts up the slope to prevent sliding. T / F
2.6 For a conical pendulum, increasing the speed of the bob causes the angle θ to increase. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word is used once. 8 marks (1 per blank)
Word bank:
centripetal · cos θ · design speed · friction · inward · sin θ · tan θ · vertical
For a conical pendulum, tension T in the string has two components: the ___________ component T ___________ which balances the weight mg of the bob, and the horizontal component T ___________ which provides the ___________ force directed ___________ toward the centre of the circular path. Dividing the horizontal equation by the vertical equation gives ___________ = v²/(rg). On a banked curve, the ___________ is the speed at which no ___________ is required.
4. Formula recall
Complete the table by writing each formula, the variables it contains, and when to apply it. 10 marks (2 each)
| Situation | Formula | Variables and when to apply it |
|---|---|---|
| Conical pendulum — angle vs speed | ||
| Design speed for a banked curve | ||
| Maximum speed on a flat unbanked curve | ||
| Maximum speed on a banked curve with friction | ||
| Period of a conical pendulum |
5. Build a concept map
Draw labelled arrows between the six terms below to show how they relate in horizontal circular motion. Each arrow must carry a linking phrase (e.g. “provides”, “equals”, “depends on”). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: centripetal force · normal force · tension · friction · banking angle · design speed.
Q1 — Term–definition match
1.1 conical pendulum • 1.2 banked curve • 1.3 design speed • 1.4 centripetal force • 1.5 coefficient of static friction • 1.6 normal force • 1.7 tension • 1.8 radius • 1.9 period • 1.10 component.
Q2 — True / false with correction
2.1 False. At the design speed, no friction acts at all. The horizontal component of the normal force alone provides the centripetal force exactly.
2.2 True.
2.3 False. From vmax = √(μsgr), mass cancels from both sides of mv²/r = μsmg. The maximum speed is independent of vehicle mass.
2.4 False. On a banked curve, N cos θ = mg, so N = mg/cos θ > mg. The normal force is greater than the weight.
2.5 False. When v > vdesign, the vehicle tends to slide up the slope, so friction acts down the slope.
2.6 True. From tan θ = v²/(rg), increasing v increases tan θ and therefore θ.
Q3 — Cloze paragraph
In order: vertical / cos θ / sin θ / centripetal / inward / tan θ / design speed / friction.
Q4 — Formula recall
Row 1: tan θ = v²/(rg) — θ = angle string makes with vertical, r = radius of path, g = 9.8 m s−2. Use to find angle or speed of a conical pendulum.
Row 2: vdesign = √(rg tan θ) — r = radius, θ = banking angle. Use to find the speed at which no friction is needed on a banked curve.
Row 3: vmax = √(μsgr) — μs = coefficient of static friction, g = 9.8 m s−2, r = radius. Use when the curve is flat (unbanked).
Row 4: vmax = √{rg( tan θ + μs)/(1 − μstan θ)} — Use when the banked curve also has friction; requires both θ and μs.
Row 5: T = 2π√(L cos θ/g) — L = string length, θ = half-angle, g = 9.8 m s−2. Use to find the period of one revolution.
Q5 — Sample concept map
Accept any valid labelled arrows. Suggested connections (1 mark each, minimum 6 awarded):
- normal force — provides horizontal component of → centripetal force
- tension — provides horizontal component of → centripetal force (conical pendulum)
- friction — can supplement → centripetal force (when v ≠ vdesign)
- banking angle — determines → design speed
- design speed — is the speed at which → friction is zero
- banking angle — increases → centripetal force budget from normal force