Two thin planks re moving on four identical cylinders. There is no slipping at any contact points. Calculate the ratio of angular speed of upper cylinder to lower cylinder/

To tackle the problem of two thin planks moving on four identical cylinders without slipping, we need to analyze the relationship between the angular speeds of the upper and lower cylinders. This scenario can be visualized as a system where the planks are positioned above and below the cylinders, and we want to determine how the motion of the planks affects the rotation of the cylinders.

Understanding the Motion

When the planks move, they exert a force on the cylinders, causing them to rotate. Since there is no slipping at the contact points, the linear velocity of the planks must match the tangential velocity of the cylinders at the points of contact. This relationship is crucial for calculating the ratio of angular speeds.

Key Concepts

• Angular Velocity (ω): This is the rate of rotation of an object, typically measured in radians per second.
• Linear Velocity (v): This is the speed of an object moving along a path, measured in meters per second.
• Radius (r): The distance from the center of the cylinder to its edge, which is crucial for relating linear and angular velocities.

Relating Linear and Angular Velocities

The relationship between linear velocity and angular velocity is given by the formula:

v = r * ω

Where:

• v is the linear velocity at the edge of the cylinder.
• r is the radius of the cylinder.
• ω is the angular velocity of the cylinder.

Setting Up the Problem

Let’s denote the angular speed of the upper cylinder as ω₁ and that of the lower cylinder as ω₂. Since both cylinders are identical, they have the same radius (r). When the upper plank moves, it causes the upper cylinder to rotate, while the lower plank causes the lower cylinder to rotate.

Calculating the Ratio

Assuming the upper plank moves with a linear velocity v₁ and the lower plank with a linear velocity v₂, we can express the angular speeds as follows:

ω₁ = v₁ / r

ω₂ = v₂ / r

To find the ratio of the angular speeds, we can set up the following equation:

Ratio = ω₁ / ω₂ = (v₁ / r) / (v₂ / r) = v₁ / v₂

Conclusion on the Ratio

Thus, the ratio of the angular speed of the upper cylinder to that of the lower cylinder is simply the ratio of their linear velocities:

ω₁ / ω₂ = v₁ / v₂

This means that if you know the linear velocities of the planks, you can directly calculate the ratio of the angular speeds of the cylinders. If both planks move at the same speed, then the ratio would be 1:1. If one plank moves faster than the other, the ratio will reflect that difference accordingly.