Angular displacement θ, angular velocity ω, angular acceleration α: Question bank

The five characters (and the sign convention we use)

The picture above is the master diagram: a spinning disc with the radius $r$ , the tangential velocity $v$ , the tangential acceleration $a_{t}$ (along the path), the centripetal acceleration $a_{c}$ (toward the centre), and the axis vector $ω$ (out of the page). Keep it in mind — most traps are just this figure with one arrow changed.

Two derivations the traps lean on

True or false — justify

Two points on the same rigid spinning disc, one at the rim and one near the hub, have the same angular velocity.

True — every point of a rigid body sweeps the same angle per second, so

ω

is shared; only the linear speed

v = r ω

differs because

r

differs.

Two points on the same rigid disc have the same linear speed.

False —

v = r ω

scales with radius, so the rim point (larger

r

) is faster even though

ω

is identical for both.

An object can have angular acceleration

α = 0

while still accelerating.

True — with

α = 0

the spin rate is constant, but the object still turns, so its velocity vector keeps changing direction: that is centripetal acceleration

a_{c} = ω^{2} r \neq = 0

ω = 0

at one instant, then

α

must also be

0

at that instant.

False — at the top of a swing or the moment a decelerating fan stops,

ω = 0

but

α \neq = 0

; just as a ball thrown up has

v = 0

at the peak yet

g

still acts.

A negative

α

always means the object is slowing down.

False — with our anticlockwise-positive convention, a negative

α

slows the body only if

ω

is positive; if

ω

is also negative (spinning clockwise), a negative

α

speeds it up. Slowing =

α

and

ω

having opposite signs.

The equation

s = r θ

works whether

θ

is in degrees or radians.

False — it holds only in radians, because the radian is defined so that arc equals radius at angle 1; degrees carry a hidden factor

π /180

Angular velocity is a vector that points along the direction of motion of the rim.

False —

ω

points along the axis of rotation (right-hand rule, out of the page for anticlockwise), perpendicular to the plane of motion, not tangent to the path.

For constant

α

, the average angular velocity over an interval equals the arithmetic mean

\frac{1}{2} (ω_{0} + ω)

True — constant

α

makes

ω (t)

a straight line in time, so the area under it (the angle) is a trapezium whose average height is the midpoint of the endpoints.

Doubling the radius of a point on a spinning body doubles its centripetal acceleration.

True —

a_{c} = ω^{2} r

is linear in

r

at fixed

ω

, so doubling

r

doubles

a_{c}

(while

a_{c} = v^{2} / r

would mislead you, since

v

also doubles).

Spot the error

"A wheel turns

90°

, so plug

θ = 90

into

s = r θ

Error:

θ

must be in radians.

90° = π /2 \approx 1.571

rad, so using

90

overshoots the arc by a factor of about

57

"The fan slows down, so I'll set

α = + 45

rad/s² in

ω^{2} = ω_{0}^{2} + 2 α θ

Error: with anticlockwise-positive and positive

ω

, slowing requires negative

α

. Forcing a positive value gives an increasing speed, contradicting "slows down."

"At high spin the tangential acceleration

a_{t} = r α

dominates the centripetal

a_{c} = ω^{2} r

Error:

a_{c}

grows as

ω^{2}

while

a_{t}

is independent of

ω

; at high

ω

the centripetal term dominates, not the tangential.

"Since

v = r ω

, a point at the axis (

r = 0

) has undefined speed."

Error:

r = 0

gives

v = 0

, perfectly defined — a point on the axis does not move. Nothing is being divided by zero here.

θ = ω_{0} t + \frac{1}{2} α t^{2}

works for any motion."

Error: this equation assumes constant

α

. If

α

varies with time it is invalid; you must integrate

α (t)

directly.

"Because

a_{t}

and

a_{c}

are both accelerations, the total is

a = a_{t} + a_{c}

Error: they are perpendicular (one along the path, one toward the centre), so they add by Pythagoras:

a = a_{t}^{2} + a_{c}^{2}

, not by simple sum.

"A body moving in a circle at constant speed has zero acceleration."

Error: constant speed still means changing direction, so

a_{c} = ω^{2} r \neq = 0

. Only

a_{t}

is zero here.

"Frequency

f

and angular velocity

ω

are the same thing in different units."

Error: they are related by

ω = 2 π f

, not equal. One full turn is

2 π

radians, so

ω

carries the extra

2 π

factor.

Why questions

Why do we invent angular quantities instead of just tracking metres for a spinning body?

Different points of a rigid body travel different arc lengths, so no single "distance" describes the whole body — but every point shares one angle

θ

, giving a single clean description.

Why is the radian the "natural" unit for rotational formulas?

It is defined so arc length equals radius at angle 1, which makes

s = r θ

exact with no conversion constant; degrees would inject

π /180

into every equation.

Why does the rim of a wheel whizz while the hub barely moves, if both share the same

ω

Linear speed is

v = r ω

; the shared

ω

is multiplied by a larger

r

at the rim, producing a larger

v

despite identical angular motion.

Why can the "time-free" equation

ω^{2} = ω_{0}^{2} + 2 α θ

be more useful than the others?

When you know

ω

ω_{0}

, and

θ

but not the time

t

, this is the only constant-

α

equation that never mentions

t

, so no unknown is left dangling.

Why is

a_{c} = ω^{2} r

called an acceleration if the speed never changes?

Acceleration is the rate of change of the velocity vector, which includes direction; turning changes the vector's direction, so a real acceleration is required even at constant speed.

Why does differentiating

s = r θ

give

v = r ω

only when

r

is constant?

r

changed, the product rule would add a term

θ \frac{d r}{d t}

; holding

r

fixed kills that term, leaving the clean

v = r ω

Why does the linear–angular dictionary

x \leftrightarrow θ

v \leftrightarrow ω

a \leftrightarrow α

work at all?

Both sets obey the same defining relations (

ω = d θ / d t

mirrors

v = d x / d t

), so the same calculus produces the same-shaped equations with symbols swapped.

Edge cases

What are

ω

and

α

for Uniform Circular Motion?

ω

is constant (nonzero) and

α = 0

; the body turns steadily with no change in spin-rate, so only centripetal acceleration remains.

At the instant a decelerating fan momentarily stops, what are

ω

α

, and

a_{c}

ω = 0

, so

a_{c} = ω^{2} r = 0

, but

α \neq = 0

(still decelerating/reversing tendency), giving a purely tangential acceleration at that instant.

For a point exactly on the rotation axis (

r = 0

), what happens to

v

a_{t}

, and

a_{c}

All three vanish:

v = r ω = 0

a_{t} = r α = 0

a_{c} = ω^{2} r = 0

— the axis point is stationary no matter how fast the body spins.

What is the angular displacement after exactly one full revolution, in radians and degrees?

θ = 2 π

rad

= 360°

, since the circumference

s = 2 π r

gives

θ = s / r = 2 π

A wheel spins at constant

ω

but starts to slip and slow — is the constant-

α

equation still valid during the slow-down?

Only if the slow-down happens at constant

α

; if the braking force (hence

α

) varies, the standard kinematic equations no longer apply and you must integrate

α (t)

ω

is negative and

α

is negative, is the body speeding up or slowing down?

Speeding up — same-sign

ω

and

α

means the spin magnitude grows; "slowing down" needs opposite signs.

Can angular displacement be negative, and what does that mean physically?

Yes — a negative

θ

just means rotation in the opposite (clockwise) sense under our anticlockwise-positive convention; the magnitude is still a genuine swept angle.

Connections

Uniform Circular Motion — the $α = 0$ edge case above.
Centripetal Acceleration and Force — the $a_{c} = ω^{2} r$ traps live here.
Linear Kinematics Equations — the dictionary these traps exploit.
Torque and Angular Acceleration — what causes a nonzero $α$ .