Maths › Further Mechanics 2 › Angular speed and horizontal circular motion
Angular speed and horizontal circular motion
Something moving in a circle at steady speed is still accelerating, because its direction keeps changing. The acceleration points at the centre, and something has to supply it.
Builds on Forces and Newton's laws and Radians, arcs and small angles.
IN THIS TOPIC
- Convert between linear speed, angular speed and period.
- Identify the force providing the radial acceleration in a given situation.
- Solve conical pendulum and banked track problems.
COMMON MISCONCEPTION
A particle moving in a circle at constant speed has no acceleration, since its speed is not changing.
Turning is accelerating
Angular speed ω is the rate at which the angle at the centre increases, measured in radians per second, and the linear speed is v = rω. Even at constant speed the velocity keeps changing, because its direction does. So there is an acceleration, it points towards the centre, and its magnitude is:
The booklet has it under Motion in a circle, where the rate of turn is written as a theta with a dot over it rather than as ω. It prints the radial acceleration as −v²/r, the minus sign recording that it points inwards. Same result, different notation for the same rate. Constant speed here is not constant velocity, the same slip as thinking a rebounding ball needs no impulse. One period is 2π/ω seconds, and questions often supply revolutions per minute, which needs converting before anything else happens.
Naming the force
Nothing travels in a circle by itself. Some real force must point at the centre to supply the radial acceleration, and naming it is the first line of every solution. A tension does it for a conical pendulum, friction for a car on a flat bend, the horizontal component of a normal reaction for a banked track, and a component of weight for a bead on a smooth wire.
There is no separate outward force to include. Resolve vertically as usual, then resolve horizontally towards the centre and set the total equal to mrω² or mv²/r.
WORKED EXAMPLE
A conical pendulum
A bob of mass 0.2 kg on a string of length 0.5 m moves in a horizontal circle with the string at 30° to the vertical. Find the tension and the angular speed, taking g = 9.8 m/s².
Vertically: T cos30° = 0.2(9.8), so T = 1.96/0.866 = 2.26 N.
The radius is 0.5 sin30° = 0.25 m. Horizontally: T sin30° = 0.2(0.25)ω².
So 1.132 = 0.05ω², giving ω² = 22.63 and ω = 4.76 rad/s. The period is 2π/ω = 1.32 s.
Banked tracks
On a banked track the normal reaction is no longer vertical, so its horizontal component can supply the radial acceleration with no friction at all. Resolve vertically and horizontally, then divide one equation by the other to remove both the mass and the reaction:
The booklet's circular motion entry stops at the radial and transverse components. Banked tracks and vertical circles are not there at all, so learn this or be ready to resolve twice and divide. That gives the design speed for the bend. Go slower and the vehicle tends to slide down the bank, so friction acts up the slope; go faster and the tendency reverses. Any question that mentions friction is asking for the fastest or slowest safe speed, and the friction term has to be added with the right sign for the case being considered.
GUIDED PRACTICE
Designing a bend
A bend of radius 80 m is to be banked so that a car travelling at 20 m/s needs no sideways friction. Find the banking angle, taking g = 9.8 m/s².
Show the working
Resolving vertically: R cos θ = mg. Horizontally: R sin θ = mv²/r.
Dividing: tan θ = v²/(rg) = 400/(80 × 9.8) = 0.510.
So θ = 27.0°.
The mass has cancelled, so the same bank suits a lorry and a motorcycle. Only the speed matters.
ASSESSMENT FOCUS
- Name the force providing the radial acceleration before writing any equation.
- Resolve vertically and horizontally, not along and perpendicular to a string.
- Use rω² when ω is given and v²/r when the speed is. Converting first wastes a line.
- For a banked track, divide the two equations to remove the reaction and the mass.
- Convert revolutions per minute to radians per second before touching the formula.
CHECK YOURSELF
A particle moves in a circle of radius 2 m at 3 rad/s. Find its speed and its acceleration.
Show a hint
One multiplication each.
Show the answer
v = rω = 6 m/s. a = rω² = 2 × 9 = 18 m/s², directed towards the centre.
Circular motion at constant speed still accelerates, towards the centre, with magnitude rω² or v²/r.
Some real force must supply that acceleration, so name it before writing an equation.
On a banked track the reaction alone suffices at the design speed given by tan θ = v²/rg.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the angular speed and horizontal circular motion questions page.
CHECK YOUR PROGRESS
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- Convert between linear speed, angular speed and period.
- Identify the force providing the radial acceleration in a given situation.
- Solve conical pendulum and banked track problems.
Open the full revision checklist to see every objective in the course in one place.