Suppose a cylindrical platform with moment of inertia 0.005kg m2 is freely rotating at angular speed 10 rad/s. A disc initially, rotationally at rest with mass 0.6kg and radius 0.1 m is dropped onto the platform so that their centers coincide. The kinetic frictional coefficient between the faces of the platform and disk is 0.25. Eventually both rotate with a common speed. Calculate the final speed.

Also, how much energy is lost in the collision?
Also, assuming that all the energy is lost to friction and that the torque due to friction is 1/2FkR, determine the relative slide angle of the disk over the platform.
Finally, how much time does it take for the disk to reach the final speed?

The final anglular velocity can be obtained by simply using conservation of angular momentum, and will not depend upon the friction coefficient. The moment of inertia of the dropped disc is

(1/2)mr^2 = 0.003 kg/m^2.
10*I1 = wfinal*(I1 + I2)
wfinal = 10*(.005)/(.008) = 6.25 rad/s

Once you have that final angular velocity w, the loss of rotational kinetic energy is easily calculated.

Energy is converted to heat at a rate (frictional torque)*(angular velocity)
Use that relationship and the KE loss to compute the time required to stop slipping.

To calculate the final speed, we need to consider the conservation of angular momentum and the work-energy principle.

1. Calculating the final speed:
Angular momentum before the collision (L_initial) = Angular momentum after the collision (L_final)

Since both the platform and the disk rotate with a common speed after the collision, their angular momenta can be added together.

L_initial = (Moment of inertia of the platform x Angular speed of the platform) + (Moment of inertia of the disk x Angular speed of the disk)

L_final = (Moment of inertia of the platform + Moment of inertia of the disk) x Common angular speed (which we need to find)

Using the given values:
L_initial = 0.005 kg*m^2 * 10 rad/s + 0.6 kg * 0.1 m^2 * 0 rad/s (since the disk is initially at rest)
L_final = (0.005 kg*m^2 + 0.6 kg * 0.1 m^2) * Common angular speed

Setting L_initial = L_final and solving for the common angular speed:
0.005 kg*m^2 * 10 rad/s = (0.005 kg*m^2 + 0.6 kg * 0.1 m^2) * Common angular speed

Common angular speed = (0.005 kg*m^2 * 10 rad/s) / ((0.005 kg*m^2 + 0.6 kg * 0.1 m^2))
Common angular speed ≈ 9.95 rad/s

To convert the angular speed to linear speed (v), we can use the formula v = rω, where r is the radius of the disk.
v = 0.1 m * 9.95 rad/s

Therefore, the final speed is approximately 0.995 m/s.

2. Calculation of energy lost in the collision:
The initial kinetic energy of the disk is given by K_initial = (1/2) * Mass of the disk * (Initial speed of the disk)^2.

K_initial = (1/2) * 0.6 kg * 0^2 (since the disk is initially at rest)
K_initial = 0

The final kinetic energy of the system after the collision is given by K_final = (1/2) * (Total moment of inertia) * (Final speed)^2.

K_final = (1/2) * (0.005 kg*m^2 + 0.6 kg * 0.1 m^2) * (0.995 m/s)^2

The energy lost in the collision is the difference between the initial and final kinetic energies:
Energy lost = K_initial - K_final

Substituting the values:
Energy lost = 0 - [(1/2) * (0.005 kg*m^2 + 0.6 kg * 0.1 m^2) * (0.995 m/s)^2]

3. Determining the relative slide angle of the disk over the platform:
The torque due to friction is given by T = (1/2) * (Frictional force) * (Radius of the disk).
The torque can also be expressed as T = (Moment of inertia of the disk) * (Angular acceleration) = (Moment of inertia of the disk) * (α).

Setting these two equations equal, we get:

(Moment of inertia of the disk) * (α) = (1/2) * (Frictional force) * (Radius of the disk)

Dividing both sides by the moment of inertia, we get:

α = (1/2) * (Frictional force) * (Radius of the disk) / (Moment of inertia of the disk)

The relative slide angle (θ) between the disk and the platform can be found using the equation:

θ = (1/2) * (α) * (Time taken)^2

Since we need to determine the relative slide angle, we can rearrange the equation as follows:

θ = (1/2) * [(1/2) * (Frictional force) * (Radius of the disk) / (Moment of inertia of the disk)] * (Time taken)^2

4. Calculation of time taken for the disk to reach the final speed:
The angular acceleration (α) can be calculated using the formula:

α = (Final angular speed - Initial angular speed) / Time taken

Since the initial angular speed is 0 rad/s, we can rearrange the equation to solve for time taken:

Time taken = (Final angular speed - Initial angular speed) / α

Substituting the values:
Time taken = (9.95 rad/s - 0 rad/s) / α

However, we don't have the value of α, so we first need to calculate it using the equation derived in step 3.

Once we have the value of α, we can substitute it back into the equation to find the time taken.

Note: The calculations depend on the actual values of the frictional force and radius, which are not provided in the question. You would need to have these values to obtain the accurate results.