Problem 25 - An Introduction to the Mechanics of Solids in SI Units

Question

A freely pivoted light rod of length \(l\) is pressed against a rotating wheel by a force \(P\) applied to its middle. The friction coefficient between the rod and wheel materials is \(f\). Compute, for both directions of rotation, the friction force \(F\) as a function of the variables \(l, P\), and \(f\), and any others which are relevant. One of these two situations is sometimes referred to as a friction lock. Which one, and why?

Step-by-Step Solution

Answer

The friction force \(F\) between the rod and wheel would be \(fP\) in the situation where the wheel's rotation tends to lift the rod and less than \(fP\) in the other situation. The 'friction lock' refers to the former situation, because the maximum static friction force is how much force it would take to start moving the rod.

Step 1: Identify the Variables

Here the length of the rod is \(l\), the applied force is \(P\), and the friction coefficient is \(f\). The goal is to compute the friction force \(F\). The direction of rotation of the wheel will affect how these forces interact.

Step 2: Visualize the Setup

Visualize a rod of length \(l\) resting against a wheel. The force \(P\) is applied to the middle of the rod, pressing it against the wheel. The frictional force \(F\) acts at the point where the rod contacts the wheel.

Step 3: Calculate Total Force in Each Situation

Consider the two possible directions of rotation of the wheel. If the wheel rotates such that it tends to lift the rod, then the friction force \(F=fP\), because static friction comes into play and the maximum static friction force that can be exerted is equal to the coefficient of static friction multiplied by the normal force. In the other situation, where the wheel's rotation tends to push the rod downwards, kinetic friction comes into play. In this case, the frictional force \(F\) would be less than \(fP\) because kinetic friction is usually less than maximum static friction. This is dictated by the pointing direction of the force \(P\), whether above or below the center of rotation.

Step 4: Identify the Friction Lock

A 'friction lock' refers to the situation in which movement is effectively stopped due to friction. Considering our analysis, the friction lock would refer to the situation in which the wheel's rotation tends to lift the pivoted rod, because in this situation the friction force \(F\), and thus resistance to movement, would be at its maximum of \(fP\).

Key Concepts

Friction Lock

A friction lock occurs when an object is halted due to frictional forces. In the scenario with a rod and a rotating wheel, a friction lock happens when the wheel's rotation attempts to lift the rod. This is because in this situation, the static friction force reaches its maximum potential, effectively locking the rod in place. Static friction is often more potent than kinetic friction, which means that once the rod reaches its limit, it requires a significant force to move again. The force preventing motion is directly tied to the static friction coefficient and the normal force, given by the formula

• Maximum static friction force: \( F_s = fP \)

This situation demonstrates a friction lock, where the static grip between two surfaces prevents motion. Even slight changes in force or direction can alter whether this lock occurs, underscoring its balance dependency on the forces at play.

Static Friction

Static friction is the force that keeps the rod stationary against the wheel when it is not moving. It is the strongest compared to kinetic friction because it is what initially prevents motion. It acts to resist the initiation of motion when a force is applied. The static friction between the rod and the wheel is proportional to the force pressing the rod into the wheel

• The resistance to initial movement: \( F_s = fP \)

Here, \(f\) is the coefficient of static friction, representing how grippy the surfaces are when not moving. Static friction acts to counter any applied force until the threshold for movement is surpassed. In our example, it effectively prevents the rod from moving until the force applied exceeds this frictional limit.

Kinetic Friction

Kinetic friction comes into play once the rod begins to move against the wheel. It's typically less than static friction, which means it is easier to keep an object in motion than to start it moving. In the example of the rotating wheel, if the rotation direction forces the rod down, kinetic friction is involved, offering less resistance than static friction would. This means the frictional force when the rod is sliding is reduced

• Moving friction force: \( F_k < fP \)

Kinetic friction, therefore, facilitates continued motion but does not support preventing motion initiation. Its reduced force compared to static friction makes objects easier to move consistently once started but requires maintaining some force to prevent stopping. This feature explains why it usually takes more energy to initiate movement than to keep it going.