SME
Grade 11 Science · Problem type
A wheel speeding up
Use this for “A fan turning at 3 rad/s speeds up at 2 rad/s² for 4 s. How fast is it turning, and how many turns does it make?”
Diagram
ω = ω₀ + αt = 3 + 2 × 4 = 11 rad/s
Δθ = ω₀t + ½αt² = 28 rad, the area under the ω–t line
n = Δθ/2π = 28/2π = 4.46 turns
This is an example. Type your own values to start a new problem.
ω (rad/s)
=
ω₀ (rad/s)
+
α (rad/s²)
×
t (s)
Δθ
Angle turned
calculated · rad
n
Net turns
calculated
Formulas
ω
=
ω₀
+
α
×
t
11 = 3 + 2 × 4
Δθ
=
ω₀
×
t
+
1
2
×
α
×
t
2
28
=
3
×
4
+
1
2
×
2
×
4
2
n
=
Δθ/(2π)
4.46 = 28/(2π)
Assumptions
•
The angular acceleration α is constant.
•
These are the straight-line motion equations with θ, ω and α in place of x, v and a.
•
The area under the ω–t line is the angle turned; one turn is 2π radians.
Step-by-step
WE KNOW
ω₀ = 3 rad/s, α = 2 rad/s², t = 4 s
FIND
angular speed (ω), angle turned (Δθ), net turns (n)
Step 1 · Find angular speed (ω)
Use ω = ω₀ + α × t
Start
at
ω₀
and
add
α
for
every
second.
ω
=
3
+
2
×
4
ω
=
3
+
8
ω
=
11
rad/s
Step 2 · Find angle turned (Δθ)
Use Δθ = ω₀ × t + ½ × α × t²
The
angle
at
the
starting
speed,
plus
the
extra
the
steady
α
adds.
Δθ
=
3
×
4
+
1
2
×
2
×
4
2
Δθ
=
3
×
4
+
1
2
×
2
×
16
Δθ = 12 + 16
Δθ = 28 rad
Step 3 · Find net turns (n)
Use n = Δθ/(2π)
Each turn is 2π radians.
n
=
28/(2π)
n
=
4.46
Check
Put the numbers back in. Both sides should match.
11 = 3 + 2 × 4 ✓
28
=
3
×
4
+
1
2
×
2
×
4
2
✓
4.46 ≈ 28/(2π) ✓
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