Question

A solid cylinder of mass 2.0 kg and radius 20 cm is rotating counterclockwise around a vertical axis through its center at 600 rev/min. A second solid cylinder of the same mass and radius is rotating clockwise around the same vertical axis at 900 rev/min. If the cylinders couple so that they rotate about the same vertical axis, what is the angular velocity of the combination?

a) 500 rev/min
b) 700 rev/min
c) 800 rev/min
d) 1000 rev/min

Answer 1

The angular velocity of the combination of the two rotating cylinders is approximately 419 rad/s.

Step-by-step explanation:

To find the angular velocity of the combination of the two rotating cylinders, we need to understand the concept of angular momentum. The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. In this case, the moment of inertia for each cylinder is given by I = ½mr², where m is the mass and r is the radius. Since both cylinders have the same mass and radius, their moment of inertia is the same. Therefore, the angular momentum of each cylinder is proportional to its angular velocity.

Let's assume that the initial angular velocity of the first cylinder is ω₁ and the initial angular velocity of the second cylinder is ω₂. We can write the equation for the conservation of angular momentum as:

(I₁ * ω₁) + (I₂ * ω₂) = (I₁ + I₂) * ω

Since the moment of inertia (I₁ + I₂) is the same for both cylinders, we can simplify the equation as:

ω = (I₁ * ω₁ + I₂ * ω₂) / (I₁ + I₂)

Plugging in the values, we get:

ω = (2.0 kg * (600 rev/min) + 2.0 kg * (-900 rev/min)) / (2.0 kg + 2.0 kg)

Converting the angular velocities to rad/s:

ω = (2.0 kg * (600 rev/min) + 2.0 kg * (-900 rev/min)) / (2.0 kg + 2.0 kg) * (2π rad / 1 rev) * (1 min / 60 s)

Performing the calculation, we get:

ω ≈ 419 rad/s

So, the angular velocity of the combination is approximately 419 rad/s.