PMDC Unit 40 · 4 outcomes

Rotational and Circular Motion

Unit 4 on the PMDC list is the dictionary of turning: angle in radians, angular velocity, and how those link to ordinary linear s, v and a. Stay inside 4.1–4.4.

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4.1

Angular displacement

PMDC: Define angular displacement and express it in radians.

Angular displacement $θ$ is the angle a radius sweeps. In radians it is the arc length divided by the radius. That is why the radian is the natural unit for linking turning to metres.

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θ = 5 7^{\circ} = 0.99 rad

$θ (r a d) = s / r$
$s = r θ (θ in r a d ian s)$

Hold these

$θ$ is a ratio of lengths, so the radian is a ‘bare’ number in formulae.
Use radians in $s = r θ$ , not degrees.

4.2

Revolution, degree and radian

PMDC: Define revolution, degree and radian.

One revolution is one full turn: back to the same spoke. A degree cuts that turn into 360 equal slices. A radian cuts it so that the arc equals the radius — 2 $π$ of those make a full turn.

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θ = 2 π \times 2 = 12.57 rad

1 $r e v = 36 0^{\circ}$ = 2 $π$ rad
$1 rad = 36 0^{\circ} /2 π \approx 57. 3^{\circ}$
$θ_{r} a d = θ_{d} e g \times (π$ / 180)

Hold these

$π r a d = 18 0^{\circ} .$
Half a turn is $π$ rad, not $180 rad$ .

4.3

Angular velocity

PMDC: Describe the term angular velocity.

Angular velocity $ω$ is how fast the angle changes. A steady spinner has constant $ω$ . Direction (for a wheel) is along the axle by the right-hand rule, but MDCAT mostly wants the size in rad s⁻¹.

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ω = 3.1 rad/s

$ω = Δ θ /Δ t (a v er a g e)$
$ω = 2 π / T = 2 π f$
unit: rad s⁻¹

Hold these

f is revolutions per second. $ω$ is radians per second. $ω$ = 2 $π$ f.
Do not mix rpm into $s = r θ$ until you convert.

4.4

Linear and angular quantities

PMDC: Find the relations between linear and angular variables: displacements, velocities, and accelerations.

The same turning looks bigger in metres if you sit farther from the centre. Multiply the angular quantity by r to get the linear one. $θ, ω, α$ become s, v, $a_{t an g e n t i}$ al.

These need $θ, ω, α$ in radians. The radial (centripetal) acceleration v²/r is extra — it points inward and is not r $α$ .

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v = r ω = 0.5 \times 4 = 2 m/s

$s = r θ$
$v = r ω$
$a_{t} = r α (t an g e n t ia l)$
$a_{c}$ = v² / $r = r ω$ ² (centripetal, to the centre)

Hold these

Farther out, same $ω$ , larger v.
r $α$ is along the rim. v²/r is toward the centre. They are different arrows.

Unit 40 formula sheet

Write these until they are cheaper than thinking, then open the quiz.

$θ (r a d) = s / r$
$s = r θ (θ in r a d ian s)$
1 $r e v = 36 0^{\circ}$ = 2 $π$ rad
$1 rad = 36 0^{\circ} /2 π \approx 57. 3^{\circ}$
$θ_{r} a d = θ_{d} e g \times (π$ / 180)
$ω = Δ θ /Δ t (a v er a g e)$
$ω = 2 π / T = 2 π f$
unit: rad s⁻¹
$s = r θ$
$v = r ω$
$a_{t} = r α (t an g e n t ia l)$
$a_{c}$ = v² / $r = r ω$ ² (centripetal, to the centre)

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