Motion in a circle

• Constant speed is not constant velocity. The centripetal force is the resultant of the real forces, pointing to the centre; there is no outward force.
• Arc equals radius: one radian. Constant speed, changing velocity. The centripetal force is the resultant of real forces, towards the centre, and there is no outward force.

• 12.1.1 I can define — define the radian and express angular displacement in radians
• 12.1.2 I can understand — understand and use the concept of angular speed
• 12.1.3 I can recall — recall and use ω = 2π / T and v = rω
• 12.2.1 I can understand — understand that a force of constant magnitude that is always perpendicular to the direction of motion causes centripetal acceleration
• 12.2.2 I can understand — understand that centripetal acceleration causes circular motion with a constant angular speed
• 12.2.3 I can recall — recall and use a = rω² and a = v²/r
• 12.2.4 I can recall — recall and use F = mrω² and F = mv²/r

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “It moves round at a constant speed, so it is not accelerating and the resultant force is zero.” Correct Constant speed is not constant velocity. The direction of motion changes continuously, so the velocity changes and there is an acceleration \(v^2/r\) towards the centre. By Newton’s second law there must be a resultant force \(mv^2/r\) towards the centre.
• “The forces on the car are its weight, the normal contact force, friction and the centripetal force.” Correct The centripetal force is not an extra force. It is the name for the resultant of the real forces along the radius — here, the friction of the road on the tyres. Draw only the real forces, then set their resultant towards the centre equal to \(mv^2/r\).
• “When the string breaks, the ball flies outwards.” Correct With the string gone there is no resultant horizontal force, so the ball keeps the velocity it had: it moves in a straight line along the tangent at the release point (Newton’s first law), not along the radius.
• “\(s = r\theta\), so a 30° arc on a 2.0 m circle is 60 m long.” Correct \(s = r\theta\), \(v = r\omega\) and \(a = r\omega^2\) hold only with \(\theta\) in radians and \(\omega\) in rad s−1. 30° = 0.524 rad, so \(s = 2.0 \times 0.524 = 1.05\) m. Put the calculator in radian mode whenever you take a sine or cosine of an angle in radians.
• “At the top of a vertical circle the tension balances the weight.” Correct Nothing balances: at the top the tension and the weight both point down, towards the centre, and together they provide \(mv^2/r\), so \(T + mg = mv^2/r\). State “towards the centre is positive” before you write any vertical-circle equation.
• “\(\omega = 2\pi/T\) is on the formula sheet.” Correct No equation in Topic 12 is printed on the Data and formulas sheet. \(\omega = 2\pi/T\), \(v = r\omega\), \(a = r\omega^2 = v^2/r\) and \(F = mr\omega^2 = mv^2/r\) must all be recalled.
• “The object moves at a constant speed, so it is not accelerating.” Repair Its velocity changes direction continuously, so it has an acceleration \(v^2/r\) towards the centre. Constant speed is not constant velocity.
• “Centripetal force is an extra force acting on the object.” Repair It is the name for the resultant of the real forces (tension, gravity, friction, normal contact) along the radius. Never add it to a free-body diagram.
• “When the string breaks, the ball flies outwards along the radius.” Repair It moves off in a straight line along the tangent at the release point, by Newton’s first law, because it keeps the velocity it had.
• “The rider on a fairground ride is pushed outwards against the wall by a centrifugal force.” Repair There is no outward force on the rider, and “centrifugal force” is not part of this syllabus. The rider’s body tends to continue in a straight line; the wall gets in the way and pushes the rider inwards. That inward normal contact force is the resultant towards the centre.
• “\(s = r\theta\) with \(\theta = 30\).” Repair \(\theta\) must be in radians: 30° = 0.524 rad.
• “All points on a spinning disc have the same speed.” Repair They have the same angular speed \(\omega\); the speed \(v = r\omega\) increases with distance from the axis.
• “\(\omega = 2\pi/T\) with T = 24 for the Earth.” Repair T must be in seconds, 8.64 × 104 s, to give \(\omega\) in rad s−1. Likewise a rate in rev min−1 is multiplied by 2π and divided by 60.
• “The tension in a vertical circle is the same all the way round.” Repair At the top \(T = mv^2/r - mg\); at the bottom \(T = mv^2/r + mg\), with a larger v as well. It is greatest at the bottom.
• “At the top of a vertical loop the weight is balanced by the tension.” Repair At the top both act towards the centre, and together they provide \(mv^2/r\): \(T + mg = mv^2/r\). Nothing is balanced.
• “The centripetal force does work, so the object speeds up.” Repair It is perpendicular to the velocity, so it does no work; in uniform circular motion the speed is constant.
• “A car goes round a level bend because of its engine force.” Repair The engine drives the car forwards, along the tangent. The sideways friction of the road on the tyres provides the force towards the centre.
• “If I double r, F doubles, because \(F = mr\omega^2\); or it halves, because \(F = mv^2/r\).” Repair Both are right, for different conditions. At constant \(\omega\), F doubles; at constant v, F halves. Decide first what the question holds constant.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Units first. Before using \(\omega = 2\pi/T\), put T in seconds (24 h = 86 400 s, not 24). Before using \(v = r\omega\), put r in metres and \(\omega\) in rad s−1 (not rev min−1, not degrees per second).
• Writing the explanation. An explain answer for this outcome needs three linked points: the force is perpendicular to the velocity, so it does no work and the speed is constant; the force is not zero, so there is an acceleration in its direction; that acceleration is perpendicular to the velocity, so it changes only the direction, towards the centre.
• Two conditions to write every time. Every equation containing \(\theta\) or \(\omega\) needs radians. And \(F = mv^2/r\) is about the resultant towards the centre: write “resultant force towards the centre = \(mv^2/r\)”, then say which real forces make it up.
• Name the physical cause. A useful source of error says what went wrong in the apparatus and what it did to the reading: the clip drifting so that L changes during the timing; friction at the tube so that the tension at the bung is not Mg; the plane of the circle not horizontal. A vague source such as “mistakes in the timing” names no cause, so it points to no improvement. An improvement must also be something the method did not already do: “take repeats” adds nothing when repeats were already taken.
• Interleave with the chapters that use this one. When you reach Topic 13, re-answer: which force provides the centripetal force of an orbit, and why does the satellite’s mass cancel? At Topic 17, re-answer: what are ω and the radian in an oscillation? At Topic 20, re-answer: why does a charged particle in a uniform magnetic field move in a circle at constant speed? Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 12 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• There is no multiple-choice paper on A Level content, so Topic 12 is met in Paper 4 structured questions. They ask you to define the radian or angular speed, to explain why an object moving at constant speed in a circle is accelerating, or why a force at 90° to the motion does not change the speed, and to identify the real force that provides the resultant towards the centre.
• Conversions between degrees, radians and revolutions; \(\omega\), \(v\) and \(a\) from a period or a rotation rate; a tension, a friction force or a normal contact force from \(F = mv^2/r\) with the forces resolved along the radius; the minimum speed at the top of a vertical circle. No equation of this topic is on the Data and formulas sheet: \(\omega = 2\pi/T\), \(v = r\omega\), \(a = r\omega^2 = v^2/r\) and \(F = mr\omega^2 = mv^2/r\) are all recalled. Only g = 9.81 m s−2 is given.
• A bung whirled on a thread through a tube, the thread held taut by a hanging mass. Vary the length L of thread (metre rule, to 1 mm) or the hanging mass M; time 20 revolutions with a stopwatch; plot T² against L or against 1/M to find the bung’s mass. The largest uncertainty is keeping L constant while whirling. Paper 5 may set it as a plan or as an analysis with error bars.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.