Physics · Mechanics · Grade 10

Dynamics of a rigid body

Why does a skater spin several times faster when she pulls in her arms?

The situation

In an ice palace a figure skater begins to turn slowly with her arms and one leg stretched wide. Then, as soon as she draws them in to her body, the spin becomes so fast that her face can no longer be made out. Nobody pushed her, and the ice has almost no friction. Where did the rotation speed come from — and where did the extra energy come from?

Topic

Dynamics of a rigid body. Figure skater's spin: conservation of angular momentum

Key ideas

Conservation of angular momentum
Moment of inertia
Rolling

Formulas

L = I ω = const

With no external torque, angular momentum is conserved.

I = \sum_{i} m_{i} r_{i}^{2}, I = I_{c} + m d^{2}

The moment of inertia depends on how far the mass is from the axis. Steiner's theorem: the axis is shifted by $d$ from the centre of mass.

E_{k} = \frac{I ω^{2}}{2} = \frac{L^{2}}{2 I}

If $L$ stays constant, the kinetic energy increases as $I$ decreases.

M = I ε, \frac{d \vec{L}}{d t} = \vec{M}

The basic equation of rotational dynamics and the torque equation.

a = \frac{g \sin θ}{1 + I / (m r^{2})}

Acceleration of rolling without slipping down an incline: it depends only on shape. For Maxwell's pendulum $θ = 90 °$ , and $r$ is the radius of the axle.

Theory

Conservation of angular momentum

Ice and air give the skater almost no torque, so the angular momentum $L = I ω$ stays constant. When the arms and leg move closer to the axis, the moment of inertia $I$ decreases — and $ω$ increases by the same factor. In the scene the length of the $L$ arrow does not change, while the frequency rises up to 5 times. In theoretical mechanics this law is a consequence of the isotropy of space: all directions in space are equivalent, so $\vec{L}$ of a closed system is conserved (Noether's theorem; details in “Hamiltonian mechanics”). The gyroscope also relies on this law — see “Gyroscope and precession”.

Where does the energy come from?

$E_{k} = L^{2} / (2 I)$ : if $I$ decreases, the kinetic energy increases. This energy is supplied by the skater's muscles: to pull a rotating arm in towards the axis a centripetal force is needed, and it does work along the path of the arm: $A_{m} = \frac{L^{2}}{2} (\frac{1}{I_{2}} - \frac{1}{I_{1}})$ . When the arms are spread out, the energy decreases again — the muscles hold the arms and brake them.

Translational, rotational and plane motion

Any motion of a rigid body is made up of translational motion (all points shift in the same way) and rotational motion. The motion of a body is described by the velocity ${\vec{v}}_{c}$ of the centre of mass (inertia) and the angular velocity $\vec{ω}$ : momentum $\vec{p} = m {\vec{v}}_{c}$ , kinetic energy $\frac{m v_{c}^{2}}{2} + \frac{I_{c} ω^{2}}{2}$ . Rolling is plane motion: the centre moves at $v_{c}$ , the body rotates about it at $ω = v_{c} / r$ , and the contact point is at rest. On an incline $m g h = \frac{m v^{2}}{2} (1 + k)$ , $k = I / (m r^{2})$ : a pipe ( $k = 1$ ) spends half of the energy on rotation and comes down slowest, and the result does not depend on mass or radius. From the race time, $k = \frac{g \sin θ t^{2}}{2 L} - 1$ and $I = k m r^{2}$ are found (the result is in the table). Free axes of rotation — in “Gyroscope and precession”.

Dynamics of rotation

The basic equation of rotation about a fixed axis is $M = I ε$ : torque produces angular acceleration, while the moment of inertia acts as a “resistance”, like mass. In the general case it is the torque equation $d \vec{L} / d t = \vec{M}$ : switch on friction for the skater and the external torque reduces $L$ . Some bodies: hoop $m R^{2}$ , disc $\frac{1}{2} m R^{2}$ , solid sphere $\frac{2}{5} m R^{2}$ , rod rotating about its middle $\frac{1}{12} m L^{2}$ , about its end — $\frac{1}{3} m L^{2}$ . Steiner's theorem $I = I_{c} + m d^{2}$ : in Oberbeck's pendulum, if a weight is moved out to $R$ , $m_{0} R^{2}$ is added to $I$ . The work done in turning, $A = M φ$ , becomes rotational kinetic energy.

Lab: measuring I and μ

Oberbeck's pendulum and a wheel (without and with weights): a load falls a height $h$ in time $t$ , and $I = m r^{2} (\frac{g t^{2}}{2 h} - 1)$ — this is just $M = I ε$ . Maxwell's pendulum: $a = g / (1 + I / m r^{2})$ ; more than 95 % of the energy goes into rotation. Trifilar suspension (torsional oscillations): $I = \frac{M g R r}{4 π^{2} l} T^{2}$ ; on a steel wire $T = 2 π \sqrt{I / D}$ . Suspended pendulum: the amplitude of a ball rolling on an inclined plate decreases by the same amount in each oscillation, $μ = \tan β \frac{α_{0} - α_{n}}{4 n}$ . The rolling friction force $F = μ N = f N / R$ is much smaller than in sliding: a steel wheel on a rail experiences about 100 times less friction than in sliding — bearings are based on this.

Model simplifications

The skater is 50 kg and 1.62 m tall; the body consists of 14 segments, with segment masses from Dempster's table (each arm 5 %, each leg 16 %), and each segment is a uniform cylinder. With “Friction” off there is no ice or air resistance. In the race, the bodies roll without slipping ( $μ \geq k \tan θ / (1 + k)$ ), and air and rolling resistance are neglected. In the laboratory the string is inextensible and massless; friction in Oberbeck's bearing is a constant torque; Maxwell's disc loses 5 % of its speed when the string jerks at the bottom; torsional oscillations are of small angle; the rolling-friction coefficients ( $f$ = 0.007 and 0.03 mm) are approximate table values. Static, sliding and viscous friction are covered in the “Car braking” and “Viscosity and flow” experiments.

Did you know?

Pulsars are extremely dense neutron stars. When a stellar core is compressed to a few tens of kilometres, it starts to spin very fast because angular momentum is conserved: the fastest known pulsar, PSR J1748−2446ad, rotates about 716 times per second.
Divers “tuck” in the air, hugging their knees: the moment of inertia falls several times over and they can do more somersaults in one dive. Before entering the water they straighten the body to slow the rotation.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Basketball: the free throw — Grade 10
Momentum and its conservation — Grade 10
Banked turn — Grade 10
Centre of gravity and equilibrium — Grade 10
Aeroplane wing — Grade 10
Non-inertial frames of reference — Grade 10
Inclined plane and Atwood machine — Grade 10
Free fall and air — Grade 10
Jet propulsion and the rocket — Grade 10
Viscosity and flow — Grade 10