Senior 3 practice book · Unit 1 of 10

Uniform Circular Motion

50 questions that complete the Senior 3 quiz for this unit: 27 core and 23 challenge. Easy and hard questions are mixed. Try each question, then tap “Show answer”.

Common misconceptions

If something moves round a circle at a steady speed, it is not accelerating.Velocity includes direction. A body going round a circle changes direction all the time, so it accelerates towards the centre even when its speed is steady.
Centripetal force is an extra force that appears by itself whenever something goes round in a circle.Centripetal force is only the name for the resultant inward force. A real force (tension, friction, gravity, a push from a wall) must do this job; take it away and the circle stops.
Every point on a turning wheel moves at the same speed.Every point has the same angular velocity ω, but the linear speed v = ωr is larger further from the axle, so the rim moves fastest.
At the top of a vertical loop the rider stays in the seat because an outward force balances the weight.At the top the weight (plus any push from the seat) is the centripetal force. If the speed is high enough, mv²/r is at least mg and the rider is kept on the circular path; nothing balances the weight.
A larger circle always needs a larger force.At the same speed F = mv²/r, so a wider circle needs LESS force. Only when ω is kept the same does a larger radius need more force (F = mω²r).

What this unit covers

Topics marked new are not tested much in the quiz, so this book gives them extra questions.

Angular displacement in radians: converting degrees, revolutions and radians
Period, frequency and angular velocity (including rev/min conversions)
Linear speed and the link v = ωr = 2πr/T
Centripetal acceleration: direction and size a = v²/r = ω²r
Centripetal force F = mv²/r = mω²r and how it depends on m, v and r
What provides the centripetal force: tension, friction, gravity, normal force; banked roads
Centrifugal force as an apparent force; motion along the tangent when the force stops
Arc length and angle: s = rθ
Motion in a vertical circle: forces at the top and bottom, minimum speed at the top
Applications: centrifuges, cream separators, leaning cyclists, chair rides, road design
Experiment: investigating how the centripetal force depends on m, v and r (whirled bung)
Satellites and orbits: gravity as the centripetal force, geostationary orbits

Go to the questions

Questions (1–50)

Easier and harder questions are mixed, just like in a real exam. The 23 harder ones are marked ★ Challenge.

1True or false
When a car rounds a bend on a flat road, the friction that keeps it on the curve acts sideways on the tyres, towards the centre of the bend.
Show answer
Answer: True
The tyres push outwards on the road, so the road pushes inwards on the tyres; this sideways friction is the centripetal force.
Common mistakeThinking friction always acts backwards (against the motion) misses the sideways grip that turns the car.
2True or false · ★ Challenge
A satellite in a lower circular orbit must move faster than one in a higher orbit.
Show answer
Answer: True
Closer to the Earth gravity is stronger and the radius smaller; for gravity to equal mv²/r the speed must be larger.
Common mistakeThinking a higher orbit needs more speed confuses the speed in orbit with the energy needed to reach that orbit.
3True or false
On a chair-swing ride, the chairs hang out at an angle because the tension in each chain must have a horizontal part pointing to the centre.
Show answer
Answer: True
The vertical part of the tension holds up the weight; the horizontal part is the centripetal force.
Common mistakeSaying the chairs are flung out by a centrifugal force misses that the tilted chain is what pulls them inwards.
4True or false · ★ Challenge
At the same speed, a car needs a larger centripetal force on a wide bend of large radius than on a sharp bend of small radius.
Show answer
Answer: False
F = mv²/r: at a fixed speed a larger radius needs a smaller force, so sharp bends are the dangerous ones.
Common mistakeThinking 'bigger circle, bigger force' ignores that r is in the denominator of mv²/r.
5True or false
For a body moving round a circle at steady speed, the centripetal acceleration has a constant size but a direction that keeps changing.
Show answer
Answer: True
a = v²/r is constant in size, but it always points to the centre, so its direction turns with the body.
Common mistakeSaying the acceleration is 'constant' as a vector forgets that its direction changes all the time.
6True or false
Two wheels of different sizes turn with the same frequency, so the larger wheel has the larger angular velocity.
Show answer
Answer: False
ω = 2πf depends only on the frequency, so both wheels have the same angular velocity; only the rim speed v = ωr is larger for the larger wheel.
Common mistakeConfusing angular velocity with rim speed makes learners think size changes ω.
7True or false · ★ Challenge
When a stone on a string moves round a vertical circle at constant speed, the tension in the string is greatest at the lowest point.
Show answer
Answer: True
At the bottom T = mg + mv²/r; at the top T = mv²/r − mg. The bottom needs the largest tension, so strings break there.
Common mistakeThinking the tension is the same all round forgets that the weight helps at the top and opposes at the bottom.
8Multiple choice · ★ Challenge
Which change doubles the centripetal acceleration of a point on a turning wheel?
1. ADoubling r while ω stays the same
2. BDoubling ω while r stays the same
3. CDoubling the speed while r stays the same
4. DHalving the period while r stays the same
Show answer
Answer: A. Doubling r while ω stays the same
a = ω²r: doubling r at the same ω doubles a. Doubling ω, doubling v or halving T each make a four times larger.
Common mistakeChoosing 'doubling ω' forgets that ω is squared in a = ω²r, so a becomes four times larger, not twice.
9Fill in the blank
When a point at distance r from the axle turns through an angle θ measured in radians, the length of arc it travels is s = ______.
Show answer
Answer: rθ
This is the definition of the radian: θ = s/r.
Common mistakeWriting s = 2πrθ mixes the arc formula with the circumference formula.
10Multiple choice · ★ Challenge
A minibus turns sharply to the left and a passenger is pressed against the right-hand door. Seen from the roadside, what is really happening?
1. AA centrifugal force throws the passenger towards the right
2. BHe tends to go straight on; the door pushes him left to turn him
3. CThe turning engine pushes the passenger towards the right door
4. DThe passenger's weight acts sideways towards the right-hand door
Show answer
Answer: B. He tends to go straight on; the door pushes him left to turn him
From the ground the only sideways force on the passenger is the inward push of the door; without it he would continue in a straight line.
Common mistakeChoosing the centrifugal force describes how it feels inside the bus, not the real forces seen from outside.
11Fill in the blank
If the radius of the circle is doubled while the angular velocity stays the same, the centripetal force becomes ______ times as large.
Show answer
Answer: 2 (two)
F = mω²r, so at fixed ω the force is proportional to r: doubling r doubles F.
Common mistakeAnswering 'half' uses F = mv²/r without noticing that v = ωr also doubles when r doubles at fixed ω.
12Multiple choice
In a dairy cooperative, a cream separator spins milk fast. Where does the denser skimmed milk collect?
1. AAt the axis, where the spinning effect is greatest
2. BEvenly through the whole bowl
3. COn the surface near the middle
4. DNear the outer wall, furthest from the axis
Show answer
Answer: D. Near the outer wall, furthest from the axis
Denser liquid needs a larger inward force to follow the circle; it moves outwards and collects at the wall, while the lighter cream is pushed towards the axis.
Common mistakeChoosing 'at the axis' thinks the spin is strongest there; ω is the same everywhere and the dense part moves outwards.
13Multiple choice · ★ Challenge
A cart on a circular track has its mass doubled and its speed halved; the radius stays the same. The centripetal force needed becomes:
1. Athe same
2. Bhalf as large
3. Ctwice as large
4. Da quarter as large
Show answer
Answer: B. half as large
F = mv²/r: mass ×2 and v² × ¼ give ×2 × ¼ = ×½.
Common mistakeChoosing 'the same' treats v as not squared, so ×2 and ×½ seem to cancel.
14Multiple choice
Clothes in a spinning washing-machine drum move in a circle. What provides the centripetal force on them?
1. AThe weight of the clothes acting downwards
2. BThe friction of the water flowing out through the holes
3. CThe inward push of the drum wall on the clothes
4. DAn outward centrifugal force from the drum
Show answer
Answer: C. The inward push of the drum wall on the clothes
The wall pushes the clothes towards the axis (a normal force); this inward push keeps them moving in a circle.
Common mistakeChoosing the centrifugal force picks an outward force; a centripetal force must point inwards.
15Multiple choice · ★ Challenge
Wheel A has a period of 0.20 s. Wheel B turns at 300 rev/min. Which statement is correct?
1. AA turns faster: 31 rad/s against 5.0 rad/s for B
2. BB turns faster, because 300 rev/min is far more than 5 Hz
3. CB turns faster: 1900 rad/s against 31 rad/s for A
4. DBoth have the same angular velocity, about 31 rad/s
Show answer
Answer: D. Both have the same angular velocity, about 31 rad/s
A: f = 1/0.20 = 5.0 Hz. B: f = 300 ÷ 60 = 5.0 Hz. So ω = 2π × 5.0 ≈ 31 rad/s for both.
Common mistakeChoosing 1900 rad/s multiplies 300 rev/min by 2π without dividing by 60 s.
16Multiple choice
Mud flies off the rim of a turning moto wheel. In which direction does a lump of mud move just after it leaves the tyre?
1. AAlong the tangent at the point where it leaves
2. BStraight out along the radius, away from the axle
3. CRound the wheel for a moment, then straight down
4. DBack towards the axle of the wheel
Show answer
Answer: A. Along the tangent at the point where it leaves
Once the mud is free no inward force acts, so it keeps the velocity it had, which is along the tangent.
Common mistakeChoosing 'along the radius' assumes an outward force throws it; there is no such force, it simply keeps its velocity.
17Multiple choice · ★ Challenge
The Moon orbits the Earth at 3.8 × 10⁸ m from the Earth's centre, once every 27.3 days. What is its orbital speed?
1. A8.7 × 10⁷ m/s
2. B3.6 × 10⁶ m/s
3. C1.0 × 10³ m/s
4. D1.6 × 10² m/s
Show answer
Answer: C. 1.0 × 10³ m/s
T = 27.3 × 24 × 3600 = 2.36 × 10⁶ s; v = 2πr/T = 2π × 3.8 × 10⁸ ÷ 2.36 × 10⁶ ≈ 1.0 × 10³ m/s.
Common mistakeChoosing 8.7 × 10⁷ m/s uses T in days instead of seconds.
18Multiple choice
In the simple model of the hydrogen atom, an electron moves in a circle round the nucleus. What provides the centripetal force?
1. AThe gravitational pull of the nucleus on the electron
2. BThe tension in a bond joining the two particles
3. CAn outward force balancing the motion of the electron
4. DElectric attraction between it and the nucleus
Show answer
Answer: D. Electric attraction between it and the nucleus
Opposite charges attract; this electric force points to the nucleus and keeps the electron on its circle.
Common mistakeChoosing gravity is tempting by analogy with planets, but between such tiny masses gravity is far too weak.
19Fill in the blank
One radian is the angle at the centre of a circle when the length of the arc is equal to the ______ of the circle.
Show answer
Answer: radius
If the arc length equals the radius, θ = s/r = 1 rad.
Common mistakeSaying 'diameter' or 'circumference' gives a different angle; the radian is defined with the radius.
20Short answer · ★ Challenge
A bend on a test track is banked (tilted inwards). Explain how the bank helps a car round the bend, and why at one particular speed a car could take the bend even on ice.
Show answer
Model answer: On a banked road the normal force from the road is tilted towards the centre, so part of it supplies the centripetal force. At one design speed this part is exactly mv²/r, so no sideways friction is needed and the car can go round even on ice.
Common mistakeLearners often say the bank makes the weight point to the centre; the weight is always vertical, it is the normal force that tilts.
21Multiple choice
A laboratory door opens through a quarter of a full turn. Through what angle, in radians, has it turned?
1. A90 rad
2. Bπ/2 rad
3. Cπ/4 rad
4. D2π rad
Show answer
Answer: B. π/2 rad
A full turn is 2π rad, so a quarter turn is 2π ÷ 4 = π/2 rad (about 1.57 rad).
Common mistakeChoosing 90 rad writes the number of degrees with the radian unit; 90° is only π/2 rad.
22Multiple choice · ★ Challenge
A satellite moves at a steady speed in a circular orbit. How much work does the Earth's gravity do on it during one orbit?
1. AA large amount, because gravity acts all the time
2. BNegative work, because gravity slows the satellite down
3. CWork equal to its kinetic energy in each orbit
4. DNone, as the pull is always at 90° to the motion
Show answer
Answer: D. None, as the pull is always at 90° to the motion
Work = force × distance moved in the direction of the force. Gravity points to the centre while the satellite moves along the tangent, so no work is done and the speed stays constant.
Common mistakeChoosing 'a large amount' forgets that work needs motion along the force; a force at right angles does no work.
23Fill in the blank
Centrifugal force is called an apparent (or ______) force because it is only needed to explain motion seen from inside a rotating frame.
Show answer
Answer: fictitious (pseudo)
An observer on the ground sees no outward force; the feeling comes from the body's inertia.
Common mistakeCalling it a real outward force leads to wrong force diagrams with an arrow pointing away from the centre.
24Short answer · ★ Challenge
An athlete whirls a hammer (a heavy ball on a wire) round in a circle and wants to throw it towards a marker on the field. At what point in the circle should she let go? Explain.
Show answer
Model answer: She should let go when the hammer's velocity (the tangent to the circle) points towards the marker, that is when the wire is at right angles to the line towards the marker. After release no inward force acts, so the hammer moves along the tangent.
Common mistakeLetting go when the wire points at the marker is wrong: the hammer leaves along the tangent, not along the wire.
25Multiple choice
Why does a cyclist lean inwards when riding round a bend?
1. ASo that the weight of the cyclist points to the centre
2. BSo that air resistance on the body is reduced
3. CSo the road's total push on the tyres acts through his centre of mass
4. DSo that the speed of the bicycle increases
Show answer
Answer: C. So the road's total push on the tyres acts through his centre of mass
The road's inward friction turns the bike. Leaning makes the total push of the road (normal force + friction) pass through the centre of mass, so it has no turning effect that would topple the cyclist.
Common mistakeChoosing 'the weight points to the centre' is wrong because the weight is always vertical, whatever the tilt.
26Multiple choice · ★ Challenge
A bucket of water is swung in a vertical circle of radius 0.90 m. What is the lowest speed at the top for which the water stays in? (g = 10 N/kg)
1. A3.0 m/s
2. B9.0 m/s
3. C0.30 m/s
4. D4.2 m/s
Show answer
Answer: A. 3.0 m/s
At the lowest speed the weight alone is the centripetal force: mg = mv²/r, so v = √(gr) = √(10 × 0.90) = √9 = 3.0 m/s.
Common mistakeChoosing 9.0 m/s gives gr but forgets to take the square root.
27Fill in the blank
A record turntable turns at 45 revolutions per minute. Its period is ______ s.
Show answer
Answer: 1.33 (4/3)
f = 45 ÷ 60 = 0.75 Hz, so T = 1/f = 1 ÷ 0.75 ≈ 1.33 s.
Common mistakeWriting T = 60 × 45 or T = 45/60 = 0.75 s confuses the period with the frequency.
28Short answer · ★ Challenge
A minibus goes round a roundabout at a steady speed. Describe the directions of its velocity and of its acceleration at any point, and explain why this acceleration does not change its speed.
Show answer
Model answer: Velocity is along the tangent; acceleration points along the radius to the centre. The two are at 90°, so the acceleration has no part along the motion: it only turns the velocity and does not make the bus go faster or slower.
Common mistakeLearners often draw the acceleration along the motion; an acceleration along the motion would change the speed.
29Multiple choice
A learner swings a bucket of water in a vertical circle. Why does the water stay in the bucket at the top of the circle?
1. AThe mv²/r it needs is at least its weight, so gravity only bends its path
2. BAn outward centrifugal force exactly balances the weight of the water
3. CGravity stops acting at the very top of the circle
4. DAir pressure on the open top pushes the water in
Show answer
Answer: A. The mv²/r it needs is at least its weight, so gravity only bends its path
At the top the water needs a centripetal force mv²/r downwards. If mv²/r is at least mg, the weight alone is not more than the inward force needed, so it cannot pull the water inside the circle; the bucket base pushes down to supply the rest.
Common mistakeChoosing 'a centrifugal force balances the weight' implies no resultant force, but the water needs a downward resultant to move in a circle.
30Short answer · ★ Challenge
Lorries often skid on a tight bend of a hill road between Kigali and Musanze. Suggest two changes road builders could make to reduce this, and explain the physics of each.
Show answer
Model answer: 1) Make the bend wider (larger r): F = mv²/r, so less centripetal force is needed. 2) Bank the road: part of the normal force then points to the centre. (Also: rougher surface for more friction, or a lower speed limit, since F depends on v².)
Common mistakeOnly asking drivers to 'be careful' gives no physics; the answer must link the change to F = mv²/r or to the force supplying it.
31Multiple choice
The label on the motor of a maize mill says 1440 rev/min. What is the frequency of the motor?
1. A1440 Hz
2. B0.042 Hz
3. C24 Hz
4. D86 400 Hz
Show answer
Answer: C. 24 Hz
f = 1440 ÷ 60 = 24 revolutions per second = 24 Hz.
Common mistakeChoosing 1440 Hz forgets that rev/min counts turns per minute, not per second; divide by 60.
32Short answer · ★ Challenge
A passenger on a big wheel at a fair feels lighter at the top and heavier at the bottom. Explain this using the forces on the passenger.
Show answer
Model answer: The feeling of weight is the push N of the seat. At the top mg − N = mv²/r, so N is less than mg (lighter). At the bottom N − mg = mv²/r, so N is more than mg (heavier). The weight mg itself does not change.
Common mistakeSaying the weight itself changes is wrong; only the seat's push changes because part of the force must be centripetal.
33Multiple choice
In the whirled-bung experiment, why does the learner time 20 revolutions instead of one?
1. ABecause the bung only reaches its full speed after 20 turns
2. BTo make the reaction-time error a small fraction of the total
3. CTo make sure that the radius of the circle stays constant
4. DBecause the stopwatch cannot measure any time shorter than 1 s
Show answer
Answer: B. To make the reaction-time error a small fraction of the total
The reaction-time error (about 0.2 s) is the same for one or twenty turns, so a longer timing makes the percentage error much smaller.
Common mistakeChoosing 'the stopwatch cannot measure under 1 s' is not the reason; the problem is human reaction time, not the watch.
34Multiple choice · ★ Challenge
A 0.50 kg ball on a string 0.80 m long moves in a vertical circle. At the lowest point its speed is 4.0 m/s. What is the tension in the string there? (g = 10 N/kg)
1. A10 N
2. B15 N
3. C5.0 N
4. D2.5 N
Show answer
Answer: B. 15 N
At the bottom T − mg = mv²/r, so T = 0.50 × 10 + 0.50 × 4.0² ÷ 0.80 = 5.0 + 10 = 15 N.
Common mistakeChoosing 10 N forgets that the tension must also hold up the weight at the bottom; T = mg + mv²/r.
35Multiple choice
A 0.30 kg ball on a string moves in a horizontal circle with a centripetal acceleration of 12 m/s². What is the tension in the string? (Ignore gravity.)
1. A40 N
2. B3.6 N
3. C0.025 N
4. D36 N
Show answer
Answer: B. 3.6 N
The tension is the centripetal force: F = ma = 0.30 × 12 = 3.6 N.
Common mistakeChoosing 40 N divides a by m; Newton's second law multiplies them, F = ma.
36Multiple choice
An athlete runs round a curve of radius 32 m at a steady 8.0 m/s. What is her centripetal acceleration?
1. A0.25 m/s²
2. B64 m/s²
3. C2.0 m/s²
4. D256 m/s²
Show answer
Answer: C. 2.0 m/s²
a = v²/r = 8.0² ÷ 32 = 64 ÷ 32 = 2.0 m/s².
Common mistakeChoosing 0.25 m/s² uses v/r and forgets to square the speed.
37Short answer · ★ Challenge
A wheel turns through 2.5 revolutions. Express this angle in degrees and in radians, and explain why angles must be in radians when you use v = ωr.
Show answer
Model answer: 2.5 × 360° = 900°; 2.5 × 2π = 5π ≈ 15.7 rad. The formula v = ωr comes from s = rθ, which is only true when θ is measured in radians, so ω must be in rad/s.
Common mistakeUsing degrees per second in v = ωr gives a speed about 57 times too large, because 1 rad ≈ 57°.
38Short answer
Give two uses of geostationary satellites and explain why a TV dish on a house in Kigali can stay fixed in one position.
Show answer
Model answer: Uses: TV broadcasting, telephone/internet links, weather pictures. A geostationary satellite orbits once in 24 h above the equator in the same direction as the Earth turns, so it stays above the same point and the dish always points at it.
Common mistakeSaying the satellite 'does not move' is wrong; it moves fast but keeps pace with the turning Earth.
39Multiple choice · ★ Challenge
A windscreen wiper blade is 0.45 m long and sweeps through 120°. How long is the arc traced by its tip?
1. A54 m
2. B2.8 m
3. C0.94 m
4. D0.24 m
Show answer
Answer: C. 0.94 m
120° = 120 × π/180 = 2.09 rad; s = rθ = 0.45 × 2.09 ≈ 0.94 m.
Common mistakeChoosing 54 m multiplies 0.45 by 120 degrees; s = rθ only works with θ in radians.
40Short answer
A hospital laboratory spins tubes of blood in a centrifuge. Explain why the red blood cells collect at the bottom of each tube (the end furthest from the axis).
Show answer
Model answer: The tubes swing out so their bottoms are furthest from the axis. Every part needs an inward force mω²r; the plasma cannot supply enough to the denser cells, so they move outwards (towards the tube bottom) and the lighter plasma is pushed towards the axis.
Common mistakeSaying a centrifugal force flings the cells out is the rotating-frame view; from outside, the cells simply are not given enough inward force to stay on the smaller circle.
41Short answer · ★ Challenge
The minute hand of a school clock is 12 cm long. How far does its tip travel in 20 minutes? Show your working.
Show answer
Model answer: 20 min is 1/3 of a turn, so θ = 2π/3 ≈ 2.09 rad. s = rθ = 0.12 × 2.09 ≈ 0.25 m (25 cm).
Common mistakeUsing θ = 20 (minutes) or 120 (degrees) in s = rθ gives a huge distance; the angle must be in radians.
42Short answer
A loaded lorry and an identical empty lorry take the same bend at the same speed. Which needs the larger centripetal force? Explain.
Show answer
Model answer: The loaded lorry. F = mv²/r and v and r are the same, so the force is proportional to the mass; the larger mass needs the larger inward force from friction.
Common mistakeSaying both need the same force because the speed and bend are the same forgets that F is proportional to m.
43Short answer · ★ Challenge
Give two sources of error in the whirled-bung experiment and suggest how to reduce each.
Show answer
Model answer: 1) The radius changes as the bung speeds up or slows: use a marker (or clip) on the string and keep it just below the tube. 2) The circle is not horizontal: whirl above the head and practise first. (Also: reaction time — time 20 turns; friction at the tube — use a smooth tube.)
Common mistakeWriting 'human error' is too vague; each error must be named with a matching improvement.
44Short answer
Explain the difference between the frequency and the angular velocity of a turning wheel. Give the unit of each and the equation that links them.
Show answer
Model answer: Frequency f is the number of complete turns per second (unit Hz). Angular velocity ω is the angle turned per second (unit rad/s). Since one turn is 2π rad, ω = 2πf.
Common mistakeMany learners think f and ω are the same number; they differ by the factor 2π because one turn is 2π rad.
45Fill in the blank
If F is plotted against v² for a bung of fixed mass on a circle of fixed radius, the graph is a straight line through the origin, so F is directly ______ to v².
Show answer
Answer: proportional
A straight line through the origin means F = constant × v², the constant being m/r.
Common mistakeSaying 'F increases with v' is true but not enough: a straight line through the origin shows direct proportion.
46Multiple choice · ★ Challenge
On a bicycle, the pedal sprocket (radius 0.10 m) turns at 1.5 rev/s. The chain drives a rear sprocket of radius 0.04 m fixed to the rear wheel of radius 0.35 m. How fast does the bicycle move?
1. A3.3 m/s
2. B0.94 m/s
3. C24 m/s
4. D8.2 m/s
Show answer
Answer: D. 8.2 m/s
Chain speed = 2π × 0.10 × 1.5 = 0.94 m/s. Rear ω = 0.94 ÷ 0.04 = 23.6 rad/s. v = 23.6 × 0.35 ≈ 8.2 m/s.
Common mistakeChoosing 3.3 m/s assumes the rear wheel turns at the pedal rate; the chain has one speed, so the small sprocket turns faster.
47Multiple choice · ★ Challenge
A learner keeps the speed of a whirled bung at 2.0 m/s, changes the radius and plots F against 1/r. The straight line has a gradient of 0.20 N m. What is the mass of the bung?
1. A0.050 kg
2. B0.10 kg
3. C0.80 kg
4. D0.025 kg
Show answer
Answer: A. 0.050 kg
F = mv² × (1/r), so gradient = mv². m = 0.20 ÷ 2.0² = 0.20 ÷ 4.0 = 0.050 kg.
Common mistakeChoosing 0.10 kg divides by v instead of v²; the gradient is mv².
48Multiple choice
A roller in a tea factory has a radius of 0.12 m and turns at 25 rad/s. How fast does its surface move?
1. A3.0 m/s
2. B208 m/s
3. C0.0048 m/s
4. D6.0 m/s
Show answer
Answer: A. 3.0 m/s
v = ωr = 25 × 0.12 = 3.0 m/s.
Common mistakeChoosing 6.0 m/s uses the diameter (0.24 m) instead of the radius in v = ωr.
49Multiple choice
A wheel of radius 0.30 m turns through an angle of 2.0 rad as it rolls without slipping. How far does it roll?
1. A0.15 m
2. B6.7 m
3. C3.8 m
4. D0.60 m
Show answer
Answer: D. 0.60 m
s = rθ = 0.30 × 2.0 = 0.60 m.
Common mistakeChoosing 3.8 m treats 2.0 rad as 2 revolutions (2 × 2π × 0.30); radians are not turns.
50Short answer · ★ Challenge
The blades of a ceiling fan are 0.60 m long and turn at 10 rad/s. Compare the speed of the blade tip with the speed of a point 0.20 m from the axle, and explain why the tips wear out first.
Show answer
Model answer: Tip: v = 10 × 0.60 = 6.0 m/s; inner point: v = 10 × 0.20 = 2.0 m/s. Both have the same ω, but v = ωr, so the tip moves three times faster, meets more air and dust each second and wears first.
Common mistakeThinking the whole blade has one speed ignores v = ωr: same ω, different radius, different speed.