SME
Grade 11 Science · Problem type
Angular speed and rim speed
Use this for “A wheel of radius 0.30 m turns at 120 rpm. What is its angular speed in rad/s, and how fast does its rim move?”
Diagram
ω
=
2πN/60
=
2π
×
120
60
ω = 12.6 rad/s
T = 2π/ω = 0.5 s
v = rω = 0.3 × 12.6 = 3.77 m/s
Every point turns at the same ω, but the rim moves fastest.
This is an example. Type your own values to start a new problem.
ω (rad/s)
=
2π
×
N (rpm)
60
r
Radius
example
T
Period
calculated
v
Rim speed
calculated
Formulas
ω
=
2π
×
N
60
12.57
=
2π
×
120
60
T
=
2π/
ω
0.5 = 2π/12.57
v
=
r
×
ω
3.77 = 0.3 × 12.57
Assumptions
•
One turn is 2π radians, so N turns a minute is 2πN/60 radians a second.
•
Every point on the wheel has the same ω, but points farther out move faster: v = rω.
•
The rim’s velocity is along the tangent; the angle must be in radians for v = rω.
Step-by-step
WE KNOW
r = 0.3 m, N = 120 rpm
FIND
angular speed (ω), period (T), rim speed (v)
Step 1 · Find angular speed (ω)
Use ω = 2π × N/60
Each turn is 2π radians, and a minute is 60 seconds.
ω
=
2π
×
120
60
ω
=
4π
ω
=
12.57
rad/s
Step 2 · Find period (T)
Use T = 2π/ω
One
turn,
2π
radians,
at
ω
radians
a
second.
T
=
2π/12.57
T
=
0.5
s
Step 3 · Find rim speed (v)
Use v = r × ω
Each radian turned moves the rim one radius along.
v
=
0.3
×
12.57
v
=
3.77
m/s
Check
Put the numbers back in. Both sides should match.
12.57
≈
2π
×
120
60
✓
0.5 ≈ 2π/12.57 ✓
3.77 ≈ 0.3 × 12.57 ✓
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