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A-Level Kinematics of uniform circular motion and centripetal acceleration
What the A-Level syllabus expects for Kinematics of uniform circular motion and centripetal acceleration, and how to practise it.
What the syllabus expects
- State angular displacement using radians
- Understand the idea of angular velocity and work with it
- Apply v = rω in problems from memory
- Understand centripetal acceleration for steady circular travel, and describe in words why moving along an arc needs a net force pointing inward at right angles to the velocity
- Use the centripetal-acceleration expressions a = rω2 and a = v2/r in calculations
- Apply F = mrω2 and F = mv2/r when solving problems
How it's examined
Questions on this topic most often ask you to calculate, determine, explain, find. About 4% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (3 marks)
(b) In a model that treats Earth as a uniform sphere of constant radius 6380 km, the planet rotates once every 24 hours. (i) A person weighs 775 N when measured at the North Pole. Calculate how much this differs from the weight measured for the same person at the equator, the difference arising from Earth's rotation. difference = .............. N
Show the worked answer
At the North Pole the person lies on the rotation axis, so there is no rotational (centripetal) effect and the measured weight equals the gravitational pull: mg = 775 N. At the equator the scale reading (apparent weight) is N = mg - mω²r, because part of gravity provides the centripetal force. The difference is therefore mω²r. m = 775/9.81 = 79.0 kg. ω = 2π/T = 2π/86400 = 7.27x10⁻⁵ rad/s. r = 6.38x10⁶ m. Difference = mω²r = 79.0 x (7.27x10⁻⁵)² x 6.38x10⁶ = 79.0 x 5.29x10⁻⁹ x 6.38x10⁶ = 2.7 N.
Example 2 (4 marks)
(iii) A student observes a chef shaping a pizza base by spinning the dough between his hands, and sees a rounded lump of dough flatten out into a thin disc. The student suggests that the rapidly spinning spherical Earth of part (b)(ii) might grow flatter and flatter until it behaves like a uniform spinning disc of radius r2 and thickness h. Demonstrate that, for angular momentum to be conserved, the thickness of this disc-Earth is h = (5 T1 / 3 T2) r1 where T1 and r1 are the period and radius of the rapidly spinning spherical Earth model and T2 is the period of the disc-Earth model.
Show the worked answer
Angular momentum L = I omega with omega = 2 pi / T. Sphere: I = (2/5) M r1², so L1 = (2/5) M r1² (2 pi / T1). Disc: I = (1/2) M r2², so L2 = (1/2) M r2² (2 pi / T2). Conserving L: (2/5) r1² / T1 = (1/2) r2² / T2, giving r2² = (4/5)(T2/T1) r1². Mass (uniform density) is conserved, so equal volumes: (4/3) pi r1³ = pi r2² h, giving h = 4 r1³ / (3 r2²). Substituting: h = 4 r1³ / [3 (4/5)(T2/T1) r1²] = (5 T1 / 3 T2) r1.
Example 3 (2 marks)
(ii) Work out the period of rotation, expressed in hours, that would make a person at the equator feel 'weightless'. period of rotation = .............. hours
Show the worked answer
'Weightless' means the contact/normal force is zero, so gravity alone supplies the centripetal force: mg = m x omega² x R => omega = sqrt(g/R). Taking g = 9.81 m s⁻² and Earth's equatorial radius R = 6.4 x 10⁶ m: omega = sqrt(9.81 / 6.4 x 10⁶) = 1.24 x 10⁻³ rad s⁻¹. Period T = 2 pi / omega = 6.283 / 1.24 x 10⁻³ = 5.06 x 10³ s. In hours: T = 5060 / 3600 = 1.4 hours.
More worked questions on this topic
- For a car travelling at constant speed around a horizontal circular path: explain why a resulta (2 marks)
- (b) A university researcher plans to reproduce Fizeau's experiment at a reduced size, setting t (2 marks)
- A fairground ride uses a spinning circular platform 10.0 m across. A child in a seat can be tre (2 marks)
- For the same aircraft (radius 12 km, period 250 s, lift L at angle θ to the vertical), determin (3 marks)
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