Conservation of Angular Momentum – Stellar Collapse

Materials:

  • Hoberman Sphere
  • Stand
  • Rotary Motion Sensor (Optional)

Demo:

A Hoberman sphere is attached to a rotary motion sensor and spun around. While the sphere is spinning, it collapses inward, causing the rotational inertia of the sphere to decrease and its angular velocity to increase as a result of the conservation of angular momentum.

Explanation:

The Hoberman sphere is used to demonstrate conservation of angular momentum since it is able to alter its radius while the mass remains constant, as shown in the video above. This quality is crucial to the demonstration since angular momentum is defined as the product of the rotational inertia of an object about its axis of rotation, which is radius-dependent, and its angular velocity,

Given that the rotational inertia of a thin spherical shell is known to be I = ⅔ MR^2, we can see that the angular momentum can be determined with the following equation:

If there is no net external torque acting on the rotating sphere, then the angular momentum of the system will be conserved even under a change in radius. That is, the angular momentum of the rotating sphere will remain the same before and after it has collapsed,

Looking at the equation above, the angular velocity must change in response to how the mass distribution about the sphere’s axis of rotation changes. In this demonstration, the angular velocity of the sphere after collapse, after, must increase from its initial value, before, to keep the value of the angular momentum the same if L before < L after.

This exchange is demonstrated in the video, seeing as the sphere at maximum radius is recorded to have an angular velocity of ω before = 2.623 rad/s right before it collapses. The angular velocity then increases reaching a maximum of ω after = 12.053 rad/s when the radius is at a minimum. We can use these values to determine the angular momentum of the sphere before and after collapse in order to prove conservation. The sphere was measured to have a mass of M = 530 g ±20 g, an initial radius of R before= 40 cm ± 2.5 cm, and a final radius of r after= 17 cm ± 2.5 cm. When applied to the equations above, these measurements along with the recorded angular velocities yielded an initial angular momentum of L before = 0.148 ± 0.019 (kgm^2)/s, and a final angular momentum of L after= 0.123 ± 0.036 (kgm^2)/s. These two values are comparable within the bounds of their error, therefore demonstrating conservation of angular momentum.

It is also worth noting, however, that the Hoberman sphere is not a perfect system, and can experience friction from a few sources, such as the string used to initiate the collapse, and the rotary motion sensor. These forces contribute to a net external torque that would oppose the motion of the sphere and alter the values of angular momentum. In this case, angular momentum was still found to be conserved.

Modeling Stellar Collapse 

Aside from the Hoberman sphere, this physical phenomenon can be demonstrated with stellar collapse. When a star exhausts its supply of hydrogen, the outward pressure created by fusion decreases and disrupts the equilibrium. Consequently, the inward force of gravity is unchallenged, and causes the star to collapse in a matter of seconds. Similarly to the Hoberman sphere, a dramatic decrease in radius results in a greater angular speed, to ensure the angular momentum is conserved. Therefore, the resulting neutron star is spinning much more rapidly than the original star, and is known as a pulsar.

The Crab Pulsar

This image of the Crab Nebula combines data from NASA’s Imaging X-ray Polarimetry Explorer (IXPE) in magenta and NASA’s Chandra X-ray Observatory in dark purple.
Credits: X-ray (IXPE: NASA), (Chandra: NASA/CXC/SAO) Image processing: NASA/CXC/SAO/K. Arcand & L. Frattare

Located 6,500 lightyears away in the Crab Nebula, the Crab Pulsar is a popular example of a rapidly spinning neutron star that can be observed. This relatively young pulsar dates back to 1040 AD when the original stellar collapse was viewed. The conservation of angular momentum during the collapse led to a substantial increase in speed compared to the original star. On top of this, pulsars tend to have a slow rate of deceleration. In the Crab Pulsar’s case, its period is decreasing at a rate of 3 x 10^-8 seconds each day. Given the pulsar’s young age, this means it is still spinning at an incredible rate, releasing massive amounts of radiation as a result. These qualities make the Crab Pulsar an exciting subject to observe, and a real life demonstration of the conservation of angular momentum.