A wheel is rotating freely with an angular speed of 4840rad/s on a shaft whose moment of inertia is negligible. A second wheel, initially at rest and with 8 times the rotational inertia of the first is suddenly coupled to the same shaft.

a) What is the angular speed of the resultant combination of the shaft and two wheels? answer in units of rad/s

b) What is the ratio of the final kinetic energy to the initial kinetic energy of the system?

To solve this problem, we can use the law of conservation of angular momentum and the law of conservation of kinetic energy.

a) First, let's calculate the initial angular momentum of the system. The angular momentum (L) of an object can be calculated by multiplying its moment of inertia (I) by its angular speed (ω) using the formula L = Iω. Since the moment of inertia of the first wheel is negligible, the initial angular momentum (L1) is simply equal to the product of the moment of inertia of the second wheel (I2) and its initial angular speed.

L1 = I2 * ω1 (Equation 1)

Next, we need to calculate the final angular momentum of the system. When the second wheel is coupled to the shaft, the total angular momentum (L2) is the sum of the initial angular momentum of the first wheel and the initial angular momentum of the second wheel.

L2 = L1 + L2

Since the second wheel is initially at rest (ω2 = 0), its initial angular momentum (L2) is zero.

L2 = L1 + 0
L2 = L1 (Equation 2)

Now, let's substitute Equation 1 into Equation 2:

L2 = I2 * ω1

The final angular speed (ωf) of the resultant combination of the shaft and two wheels is given by rearranging the equation L2 = I2 * ωf:

L2 = I2 * ωf

Since L2 = L1, we get:

I2 * ωf = I2 * ω1

Dividing both sides of the equation by I2, we find:

ωf = ω1

Therefore, the angular speed of the resultant combination of the shaft and two wheels is 4840 rad/s.

b) The ratio of the final kinetic energy (Kf) to the initial kinetic energy (Ki) of the system can be calculated using the formula:

Kf / Ki = (1/2) * I2 * ωf^2 / (1/2) * I2 * ω1^2

The (1/2) terms cancel out, and since ωf = ω1, we can simplify the equation to:

Kf / Ki = ωf^2 / ω1^2 = (4840 rad/s)^2 / (4840 rad/s)^2 = 1.

Therefore, the ratio of the final kinetic energy to the initial kinetic energy of the system is 1.