Physics of a Spinning Ballerina

By Farah Fouladi

The forces acting on a dancer
- Gravity (downwards)
- Normal Force from the floor (upwards)
- Friction of the floor

Turns in ballet (the physics terms we can use to describe it)
- Angular velocity = how fast the dancer is spinning
- Rotational Moment of Inertia

A fouette is based of the principal of a moment of inertia.
- When the dancer starts turning her arms are brought together
o Small radius
r = 0.2 m
o Small moment of inertia
Treat the dancer’s body as a solid cylinder
Mass of dancer = 55 kg
I = ½ mr2 = ½ (55 kg)(0.2)2 = 1.1 kg m2
o Thus a large angular velocity
Let’s say 1 rad/s
o L = Iw
= (1.1 kg m2)( 1 rad/s) = 1.1 N m s
- The dancer stops for a second and extends her arms and legs
o Larger radius
r = 0.4 m
o Larger moment of inertia
Treat the dancer’s body as a solid cylinder
Mass of dancer = 55 kg
I = ½ mr2 = ½ (55 kg)(0.4)2 = 4.4 kg m2
o Smaller angular velocity
w = L/I
= (1.1 N m s) / (4.4 kg m2)
= 0.25 rad/s
- The dancer continues turning with arms brought together
o Small radius
o Small moment of inertia
o Thus a large angular velocity

How does the dancer stay balanced?
- This is based on canter of gravity!
- Before spin:
o Top half of body is symmetrical
o Vertical center of mass is at center of middle axis
o Legs are not, one leg is straight and the other is in passé

o Xcm = ((xtop)(mtop) + (xrightleg)(mrightleg) + (xleftlef)(mleftleg))/(mtot)
o Xcm = ((0)(29) + (0.2 m)(13 kg) + (- 0.05 m)(13 kg))/(55 kg)
o = 0.04 m shifted to the right!
- After spin:
o Top half of body is symmetrical
o Vertical center of mass is at center of middle axis
o Legs are not, one leg is straight and the other is in fully extended
o Xcm = ((xtop)(mtop) + (xrightleg)(mrightleg) + (xleftlef)(mleftleg))/(mtot)
o Xcm = ((0)(29) + (0.6 m)(13 kg) + (- 0.05 m)(13 kg))/(55 kg)
o = 0.13 m shifted to the right!
So a dancer must constantly adjust to its new center of mass while turning at all
times!!

The effect on a dancer’s body:
- To rotate faster, a dancer must DECREASE her moment of inertia.
- Can do this in 2 ways:
o Decrease mass
o Make sure the bulk of the body is close to the axis of rotation
o See how this correlates to ballerinas needing to very very thin!