Objects that rotate in air or water experience a torque due to drag. With quadratic drag, a drag torque that's negligible at low rpm quickly becomes significant as the rpm increases. Consider a square bar with cross section a x a and length L. It is rotating on an axle through its center at angular velocity ω in a fluid of density ρ. Assume that the drag coefficient C_𝒹 is constant along the length of the bar. Find an expression for the magnitude of the drag torque on the bar. Hint: Begin by considering the drag force on a small piece of the bar of length dr at distance r from the axle.

Verified step by step guidance

1. Step 1: Begin by considering the drag force on a small piece of the bar. The drag force on a small segment of length dr at a distance r from the axle is proportional to the square of the tangential velocity of the segment. The tangential velocity is given by v = ωr, where ω is the angular velocity and r is the distance from the axle.
2. Step 2: The drag force on the small segment can be expressed as dF = (1/2) * C𝒹 * p * A * v², where C𝒹 is the drag coefficient, p is the fluid density, A is the cross-sectional area of the bar (a² for a square cross-section), and v² = (ωr)². Substitute v² into the equation to get dF = (1/2) * C𝒹 * p * a² * (ωr)².
3. Step 3: The torque due to this drag force on the small segment is given by dτ = r * dF, where r is the distance from the axle. Substitute dF into this equation to get dτ = r * (1/2) * C𝒹 * p * a² * (ωr)² = (1/2) * C𝒹 * p * a² * ω² * r³ * dr.
4. Step 4: To find the total drag torque on the bar, integrate dτ over the length of the bar. The limits of integration for r are from 0 to L/2, since the bar extends symmetrically on both sides of the axle. The total drag torque is τ = ∫[(1/2) * C𝒹 * p * a² * ω² * r³] dr, with limits 0 to L/2.
5. Step 5: Perform the integration. The integral of r³ with respect to r is (r⁴)/4. Substitute the limits of integration (0 to L/2) into the result to find the expression for the total drag torque. The final expression will be τ = (1/2) * C𝒹 * p * a² * ω² * [(L/2)⁴ / 4].

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, causing it to rotate about an axis. It is calculated as the product of the force applied and the distance from the axis of rotation to the point where the force is applied. In the context of rotating objects, torque is crucial for understanding how forces, such as drag, affect the motion and stability of the object.

Drag Force

Drag force is the resistance experienced by an object moving through a fluid, such as air or water. It is influenced by factors like the object's shape, size, and speed, as well as the fluid's density and viscosity. For rotating objects, the drag force can vary with the square of the velocity, leading to significant effects on torque at higher rotational speeds.

Quadratic Drag

Quadratic drag refers to the drag force that increases with the square of the velocity of the object moving through a fluid. This relationship becomes particularly important at higher speeds, where the drag force can significantly impact the motion of the object. Understanding quadratic drag is essential for calculating the drag torque on rotating objects, as it dictates how the drag force scales with angular velocity.