Angular velocity, in , is given by the length traveled divided by the time taken to travel the length:

We are told that the amount of time taken to make one revolution is 3min. One revolution is equal to , and 3 minutes is equal to 180 seconds. Divide the radian value by the seconds value to get the angular velocity.

For this question, the angular velocity  can be given by the equation:

, where  is the angle made and  is the time taken to make this angle. 

In this problem, the wheel makes one full revolution() in  seconds. 

Therefore:

Given initial angular velocity, angular acceleration, and time we can easily solve for final angular velocity with:

If the ferris wheel has height then it must have radius .

The circumference of the ferris wheel, or the distance of one rotation, is then:

Convert the given velocity into meters per minute, or :

Find rotations per minute:

Convert  to :

Convert to

Use the following relationship and plug in known values:

Convert units of time into radians per second:

Convert to linear distance:

Combine equations:

Convert to meters and seconds and plug in values:

We can use the expression for conservation of energy to solve this problem:

Our initial state will be when the rod is horizontal, and our final state will be when mass A is at its highest point. If we assume that point p is at a height of 0 and the system is at rest and mass A is at its highest point, we can eliminate initial potential energy and final kinetic energy to get:

Expanding these terms and applying them to both masses, we get:

We don't need to separate the velocity components for each mass since they are always traveling at the same speed. Since the masses are attached to a rigid rod that spins around its midpoint, we know that the heights of the two masses (with respect to point p) will be equal and opposite. In expression form:

Substituting this into our equation, we get:

Rearranging for final height, we get:

We have values for all of these variables, so time to plug and chug:

Now we can use the sine function to determine what the angle c is at this point:

Where the opposite side is the height we just calculated and the hypotenuse is half the length of the rod. Therefore, we get:

Taking the inverse sine of both sides, we get:

Substituting in our values, we get:

Let's begin with the expression for conservation of energy:

The problem statement tells us that the sphere is initially at rest, so we can eliminate initial kinetic energy. Also, if we assume that the height at the bottom of the ramp is 0, we can eliminate final potential energy. We then have:

Then we can expand both of these terms. We need to remember that kinetic energy will have both a linear and rotational aspect. We then get equation (1):

Moving from left to right, let's begin substituting in expressions for unknown variables. The first term we don't know is height. However, we can use the length of the slope and the its angle to determine the height at the top of the ramp:

Rearranging for height, we get equation (2):

The next term we don't know is final velocity. We can use the relationship between angular and linear velocity:

Rearranging for linear velocity, we get equation (3):

Moving on, the next term we don't know is moment of inertia. We will use the expression for a sphere to get equation (4):

The last term is final rotational velocity. However, this is what we're solving for, so we'll leave it alone. Now let's substitute equations 2, 3, and 4 back into equation 1:

Multiplying each side the equation by , we get:

Factoring the right side of the equation:

Rearranging for final rotational velocity:

We know each of these values, so time to plug and chug: