Conservation of Angular Momentum

[lmc489]

Homework Statement

Two disks with moments of inertia I and 31 are mounted on a common shaft with frictionless bearings. They are initially rotating with angular speeds of w and 2w. They are brought together without being disturbed. What is their final KE as a fraction of their initial KE?

The answer is 49/52 but i don't understand the concept of it.
can someone please explain?

Equations:
Angular Momentum: Iw = Iw

 

[hage567]

You will need to use conservation of angular momentum and conservation of rotational energy.

Start by writing out these quantities for each disk and then when the disks are brought together.

 

[thomate1]

Kinetic Energy: KE = 1/2Iw^2

I can explain the concept of conservation of angular momentum in this scenario. Angular momentum is a property of a rotating object and is defined as the product of its moment of inertia (I) and its angular velocity (w). In this situation, the two disks have different moments of inertia, but they are connected by a common shaft and are rotating at different angular velocities.

When the two disks are brought together without being disturbed, the total angular momentum of the system remains constant. This is known as the law of conservation of angular momentum. In other words, the sum of the individual angular momenta of the two disks before and after they are brought together must be the same.

Now, let's consider the kinetic energy of the system. Kinetic energy is defined as the energy an object possesses due to its motion. In this case, the disks are rotating and therefore have kinetic energy. The equation for kinetic energy in rotational motion is KE = 1/2Iw^2, where I is the moment of inertia and w is the angular velocity.

Since the law of conservation of angular momentum states that the total angular momentum of the system remains constant, we can equate the initial angular momenta of the two disks to their final angular momenta. Therefore, Iw + 31(2w) = (I+31)w_f, where w_f is the final angular velocity of the combined disks.

Solving for w_f, we get w_f = (Iw + 62w)/(I+31). Now, using this value of w_f in the equation for kinetic energy, we get the final kinetic energy of the system as KE_f = 1/2(I+31)((Iw + 62w)/(I+31))^2 = 1/2(Iw + 62w)^2/(I+31).

To find the fraction of final KE to initial KE, we can divide the final KE by the initial KE, which gives us (1/2(Iw + 62w)^2/(I+31))/((1/2)Iw^2) = (Iw + 62w)^2/(Iw^2(I+31)).

Substituting the given values of I and w, we get (Iw + 62w)^2/(Iw^2(I+31)) = (Iw +