The mass of a star is 1.49·1031 kg and it performs one rotation in 26.1 days. Find its new period (in /days) if the diameter suddenly shrinks to 0.59 times its present size. Assume a uniform mass distribution before and after.

To find the new period of rotation for the star after its diameter shrinks, we need to use the principle of conservation of angular momentum. The angular momentum of an object remains constant unless acted upon by an external torque.

The angular momentum of a rotating object can be calculated using the formula:

L = Iω

Where:
L is the angular momentum,
I is the moment of inertia, and
ω (omega) is the angular velocity.

In this case, we are given the mass and period (rotation time) of the star and we need to find the new period after the diameter shrinks.

First, we need to calculate the initial moment of inertia (I) of the star. Since the mass distribution is assumed to be uniform, we can use the formula for the moment of inertia of a sphere:

I = (2/5) * mr^2

Where:
m is the mass of the star, and
r is the radius (half of the diameter).

Since the diameter suddenly shrinks to 0.59 times its present size, the radius becomes 0.295 times its present size.

Now, let's calculate the initial moment of inertia (I_initial):

I_initial = (2/5) * m * (r_initial)^2

Where:
m = 1.49 * 10^31 kg (Given mass of the star)
r_initial = radius of the star before shrinking

Next, we need to calculate the initial angular velocity (ω_initial) using the given period (T_initial) of rotation:

ω_initial = (2π) / T_initial

Where:
T_initial = 26.1 days (Given period of rotation)

Now, we can calculate the initial angular momentum (L_initial):

L_initial = I_initial * ω_initial

After the diameter shrinks, the new radius becomes 0.59 times the initial radius.

Let's calculate the new moment of inertia (I_new):

I_new = (2/5) * m * (r_new)^2

Where:
r_new = 0.59 * r_initial (New radius after shrinking)

Finally, let's calculate the new angular velocity (ω_new) using the principle of conservation of angular momentum:

L_initial = L_new

I_initial * ω_initial = I_new * ω_new

Since the angular momentum is conserved, we can equate the two expressions and solve for ω_new. Then, we can find the new period of rotation (T_new) using the formula:

T_new = (2π) / ω_new

By following these steps, you will be able to find the new period of rotation (T_new) for the star after its diameter shrinks.