A neutron star of mass 2.00 times 10 to the thirtieth kilograms and radius 10.0 kilometres rotates with a period of 0.02000 seconds. What is its rotational kinetic energy?
We’re told here are the star’s mass 2.00 times 10 to the thirtieth kilograms, which we’ll call 𝑚, and that it has a radius of 10.0 kilometres; we’ll call that value 𝑟. The star has a period of 0.02000 seconds, which we’ll call capital 𝑇. We want to know the star’s rotational kinetic energy, which we’ll call 𝐾𝐸 sub 𝑟.
Let’s start by recalling the equation for rotational kinetic energy. The rotational kinetic energy of an object is equal to one-half its moment of inertia multiplied by its angular speed 𝜔 squared. In our instance, the object we’re considering is a star that is a sphere rotating about an axis through its centre.
If we look up the moment of inertia of a solid sphere rotating in this way, we find that the moment of inertia of the sphere equals two-fifths the mass of the sphere times its radius squared. If we apply the equation for rotational kinetic energy to our situation, then 𝐾𝐸 sub 𝑟 equals one-half 𝑖 𝜔 squared or one-half times two-fifths 𝑚𝑟 squared times 𝜔 squared.
Now, we know the mass and the radius of the star; those are given in the problem statement. But what about 𝜔? Recall that 𝜔, the angular velocity, has units of radians per second. We know the time it takes that is the period for the star to go through a complete revolution. And since one revolution is equal to two 𝜋 radians, this means that 𝜔, the angular speed of the star, equals two 𝜋 radians or the number of radians in one revolution divided by the period 0.02000 seconds.
We’re now prepared to substitute in for the variables in our equation for rotational kinetic energy. The mass 𝑚 is 2.00 times 10 to the thirtieth kilograms, the radius 𝑟 in units of metres is 10.0 times 10 to the third metres, and 𝜔 is two 𝜋 radians per 0.2000 seconds. When we enter these values into our calculator, we find a result of 3.95 times 10 to the 42 joules. This is the rotational kinetic energy of the neutron star.