Rotating cylinders are not just physical abstractions: for example, they can turn a flow of the air, thus creating lift as the airplane wings. Historically people have even used rotating cylinders to power ships (although with little efficiency). NASA wrote more about this here.
We will now examine how much time we need to stop a ship-engine-like cylinder. Suppose you take a rotating cylinder and let it roll to the right until it reaches a vertical wall. It then spins against the two walls as shown in the picture, slowly slowing down. How much time will it take in s for the cylinder to stop?
To determine the time it takes for the rotating cylinder to stop, we need to consider the factors that affect its deceleration. The key parameters involved in this calculation include the initial angular velocity of the cylinder, the moment of inertia, and the friction between the cylinder and the walls.
To begin, we need more information about the cylinder's dimensions, mass distribution, and surface properties. Specifically, we need to know the mass of the cylinder, the radius, and the initial angular velocity. These details will allow us to calculate the moment of inertia, which quantifies how the mass is distributed around the axis of rotation.
Once we have the moment of inertia (symbolized as 'I'), we can apply the principles of rotational motion to determine the deceleration of the cylinder. By setting up an equation that relates the torque acting on the cylinder to the moment of inertia and its angular acceleration, we can solve for the angular acceleration (symbolized as 'α').
Next, we need to consider the friction between the cylinder and the walls. The frictional force will act opposite to the direction of motion, causing the cylinder to slow down. The magnitude of the frictional force depends on the coefficient of friction between the cylinder and the wall, denoted as 'μ'.
To calculate the angular acceleration caused by friction, we can use the equation τ = I * α, where τ is the torque exerted by the frictional force. The magnitude of the torque can be determined as the product of the frictional force and the cylinder's radius.
Once we have the angular acceleration caused by friction, we can use the equation of motion for rotational motion, which states that the final angular velocity squared is equal to the initial angular velocity squared plus twice the angular acceleration times the angle turned. In this case, the angle turned by the cylinder is 90 degrees (π/2 radians) since it rotates from a horizontal position to a vertical one.
Rearranging the equation, we can solve for the time it takes to stop the cylinder. The equation becomes:
t = (ω_f - ω_i) / α,
where ω_f is the final angular velocity (zero in this case), ω_i is the initial angular velocity, and α is the angular acceleration caused by friction.
By substituting the calculated values for α and ω_i into the equation, we can determine the time it takes for the cylinder to stop.
Keep in mind that the accuracy of the results will depend on the accuracy of the parameters used and any assumptions made.