SME
University Physics I: Mechanics · Problem type
A spinning skater pulls in her arms
Use this for “A skater spinning at 3 rad/s with her arms out (I = 5 kg·m²) pulls them in to I = 2 kg·m². Find her new spin and the work her arms do.”
Chart
ω(1.6) = 5
ω(I) = ω₁(I)
The curves cross at I = 4.
This is an example. Type your own values to start a new problem.
I₁ (kg·m²)
×
ω₁ (rad/s)
=
I₂ (kg·m²)
×
ω₂ (rad/s)
L
Angular momentum
worked out (not typed) · kg·m²/s
K₁
Kinetic energy before
worked out (not typed) · J
K₂
Kinetic energy after
worked out (not typed) · J
W
Work done by the arms
worked out (not typed) · J
Formulas
L
=
I₁
×
ω₁
8 = 4 × 2
L
=
I₂
×
ω₂
8 = 1.6 × 5
K₁
=
1
2
×
I₁
×
ω₁
2
8
=
1
2
×
4
×
2
2
K₂
=
1
2
×
I₂
×
ω₂
2
20
=
1
2
×
1.6
×
5
2
W
=
K₂
−
K₁
12 = 20 − 8
Assumptions
•
No outside torque acts about the spin axis (the ice’s friction is left out), so the angular momentum L = Iω is kept.
•
Pulling the arms in moves mass closer to the axis: I falls, and ω rises by the same factor.
•
The kinetic energy ½Iω² is not kept: the arms do work W = K₂ − K₁ pulling in, and W is negative letting them out.
Step-by-step
WE KNOW
I₁ = 4 kg·m², ω₁ = 2 rad/s, I₂ = 1.6 kg·m²
FIND
angular momentum (L), spin after (ω₂), kinetic energy before (K₁), kinetic energy after (K₂), work done by the arms (W)
Step 1 · Find angular momentum (L)
Use L = I₁ × ω₁
The angular momentum is the moment of inertia times the spin.
L
=
4
×
2
L
=
8
kg·
m
2
/s
Step 2 · Find spin after (ω₂)
Use L = I₂ × ω₂
No
outside
torque,
so
L
is
the
same
after:
the
new
spin
is
L
divided
by
the
new
I.
ω₂
=
L
I₂
ω₂
=
8
1.6
ω₂
=
5
rad/s
Step 3 · Find kinetic energy before (K₁)
Use K₁ = ½ × I₁ × ω₁²
The energy of spinning is half the moment of inertia times the spin squared.
K₁
=
1
2
×
4
×
2
2
K₁
=
1
2
×
4
×
4
K₁
=
2
×
4
K₁
=
8
J
Step 4 · Find kinetic energy after (K₂)
Use K₂ = ½ × I₂ × ω₂²
The
energy
of
spinning
after:
half
of
I₂
times
ω₂
squared.
K₂
=
1
2
×
1.6
×
5
2
K₂
=
1
2
×
1.6
×
25
K₂
=
0.8
×
25
K₂
=
20
J
Step 5 · Find work done by the arms (W)
Use W = K₂ − K₁
The work-energy theorem: the arms’ work is the change in kinetic energy.
W
=
20
−
8
W
=
12
J
Check
Put the numbers back in. Both sides should match.
8 = 4 × 2 ✓
8 = 1.6 × 5 ✓
8
=
1
2
×
4
×
2
2
✓
20
=
1
2
×
1.6
×
5
2
✓
12 = 20 − 8 ✓
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