Kinematics of circular motion — OCR A-Level Physics
Test yourself on Kinematics of circular motion with OCR A-Level practice questions.
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Kinematics of circular motion explained
The radian is the SI unit of angle, defined so that one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
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Because the circumference is 2πr, a full turn is 2π radians, so 360° = 2π rad. To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. For example, 90° = π/2 rad ≈ 1.57 rad. Using radians makes arc length s = rθ and angular speed ω = Δθ/Δt valid directly, without conversion factors, which is why circular-motion equations assume angles in radians.
(b) period and frequency of an object in circular motion
For an object moving in a circular path, the period T is the time taken to complete one full revolution, measured in seconds (s). The frequency f is the number of complete revolutions per second, measured in hertz (Hz), where 1 Hz = 1 s⁻¹. Period and frequency are reciprocals: T = 1/f and f = 1/T. For example, a satellite completing an orbit in T = 120 s has f = 1/120 ≈ 8.3 × 10⁻³ Hz. If a wheel spins at f = 5.0 Hz, each revolution takes T = 1/5.0 = 0.20 s. Period and frequency describe the timing of the motion. They are related to speed v and radius r by the equation v = 2πr/T; therefore, knowing only the period or frequency does not uniquely determine the radius or speed, as a larger radius requires a proportionally greater speed to maintain the same period.
(c) angular velocity ω; ω = 2π/T or ω = 2πf
Angular velocity ω is the rate of change of angle for an object moving in a circle, measured in radians per second (rad s⁻¹). One complete revolution corresponds to an angle of 2π radians, so ω = 2π/T, where T is the period in seconds. Because f = 1/T, this is equivalent to ω = 2πf, where f is the frequency in hertz. For example, a wheel with T = 0.50 s has ω = 2π/0.50 ≈ 12.6 rad s⁻¹; the same wheel has f = 2.0 Hz and ω = 2π × 2.0 ≈ 12.6 rad s⁻¹. Angular velocity describes how fast the angle changes and is the same for all points on a rigid rotating body, even though points further from the axis move faster in a straight-line sense.
Your focus
- Define the radian in terms of arc length and radius.
- Convert angles between degrees and radians accurately.
- Use radian measure in equations such as s = rθ and ω = Δθ/Δt.
Show all 9 objectives
- Define period and frequency for an object in circular motion and state their units.
- Apply the reciprocal relationships T = 1/f and f = 1/T to convert between period and frequency.
- Interpret a value of period or frequency in the context of a rotating or orbiting object.
- Define angular velocity and state its unit as rad s⁻¹.
- Use ω = 2π/T and ω = 2πf to calculate angular velocity from period or frequency.
- Explain why angular velocity is the same for all points on a rigid rotating body.
Kinematics of circular motion exam tips
Marking Points
- Defines the radian as the angle subtended by an arc equal in length to the radius.
- States that a full circle is 2π radians, so 360° = 2π rad.
- Converts between degrees and radians using π/180 and 180/π respectively.
- Explains that arc length s = rθ requires θ in radians.
- Applies radian measure to angular quantities such as angular speed ω = Δθ/Δt.
- Period T is the time for one complete revolution, measured in seconds (s).
- Frequency f is the number of complete revolutions per second, measured in hertz (Hz), where 1 Hz = 1 s⁻¹.
- Period and frequency are reciprocals: T = 1/f and f = 1/T.
- A larger period corresponds to a smaller frequency, and vice versa.
- Period and frequency alone do not determine radius or speed, as they are related by v = 2πr/T.
- Angular velocity ω is the angle swept per unit time, measured in radians per second (rad s⁻¹).
- One complete revolution is 2π radians, giving ω = 2π/T.
- Since f = 1/T, the equivalent form is ω = 2πf.
- Both forms give the same value of ω when T and f are consistent reciprocals.
- Angular velocity is the same for all points on a rigid body rotating about a fixed axis.
Examiner Tips
- 💡Write the conversion factor π/180 explicitly before substituting an angle in degrees.
- 💡Check whether a calculator is in radian or degree mode before evaluating trigonometric functions in circular-motion work.
- 💡Use s = rθ only after confirming θ is in radians.
- 💡Quote angles in radians to an appropriate number of significant figures, often as multiples of π.
- 💡Check the unit: if the answer is in seconds it is a period; if in hertz or s⁻¹ it is a frequency.
- 💡When a question gives revolutions per minute, convert to hertz by dividing by 60 before using T = 1/f.
- 💡Sketch one full revolution and label the start and end of one period to avoid counting part-revolutions as whole ones.
- 💡Convert revolutions per minute to hertz first, then use ω = 2πf.
- 💡Keep π in your working until the final step to reduce rounding error, then round to the required significant figures.
- 💡Check that your answer is in rad s⁻¹; if it is in s⁻¹ you have probably calculated frequency instead.
Common Mistakes
- Treating a full circle as π radians; the correction is that a full turn is 2π radians.
- Substituting degrees into s = rθ; the correction is to convert the angle to radians first.
- Multiplying by 180/π when converting degrees to radians; the correction is to multiply degrees by π/180.
- Assuming the radian is a dimensional unit like the metre; the correction is that it is a dimensionless ratio of two lengths.
- Confusing period with frequency: the error is treating T and f as the same quantity; the correction is that T is time per revolution in seconds while f is revolutions per second in hertz.
- Inverting the reciprocal the wrong way: the error is writing T = f; the correction is T = 1/f, so a frequency of 4.0 Hz gives T = 0.25 s.
- Using degrees or radians per second for frequency: the error is giving f in rad s⁻¹; the correction is that frequency is measured in hertz (s⁻¹), while angular velocity in rad s⁻¹ is a different quantity.
- Assuming period is independent of speed and radius: the error is ignoring the relationship v = 2πr/T; the correction is that for a fixed period, speed must increase proportionally with radius.
- Using degrees instead of radians: the error is substituting 360 for 2π; the correction is that one revolution is 2π rad, so ω = 2π/T.
- Mixing up T and f in the formula: the error is writing ω = 2πT; the correction is ω = 2π/T, so a longer period gives a smaller angular velocity.
- Forgetting the unit: the error is quoting ω in Hz or s; the correction is that angular velocity is measured in rad s⁻¹.
- Treating ω as a linear speed: the error is giving ω in m s⁻¹; the correction is that ω is an angular quantity in rad s⁻¹, while linear speed v = ωr is in m s⁻¹.