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!

- 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!

Hi, I am an 8th grader at McNeel Intermediate, I'm doing a project for a STEAM festival we're having and my topic is about the physics of a dancer spinning. I've searched many things in order to try and find a video to represent the physics but I could not find anything. As I came across your website I learned a lot about the spinning physics. I would like to know if you have any video recommendations for my project. If you do have any ideas, please send me the link through my teacher's email; Dolsen@sdb.k12.wi.us

ReplyDeleteThank you!