Problem 27 - An Introduction to the Mechanics of Solids in SI Units

Question

A lightweight rope of area \(A\) and modulus of elasticity \(E\) is hung over a stationary shaft. A weight \(W\) is attached to the longer end, and, at the same time, the rope is forced against the shaft with a horizontal force \(P\) just sufficient to prevent the weight from dropping. Find the value of \(P\) if the static coefficient of friction between the rope and the shaft is \(f\).

Step-by-Step Solution

Answer

The horizontal force \(P\) required to prevent the weight from dropping is calculated by the equation \(P = fW\), where \(f\) is the static friction coefficient between the rope and the shaft, and \(W\) is the weight of the attached object.

Step 1: Calculate Normal Force

Firstly an understanding of what's happening must be achieved. A rope is hung over a shaft, with one end having a weight \(W\) hanging from it. The rope is being pushed against the shaft with a force \(P\), just enough to stop the weight from dropping. This force \(P\) is acting horizontally. Now, the weight \(W\) is acting vertically downwards, and this is countered by the normal force acting upwards from the shaft. The weight and the normal force balance each other out, hence, the normal force is equal to \(W\).

Step 2: Apply Static Friction Equation

Now, it is known that the force \(P\) is just enough to ensure that the weight doesn't drop. This means that \(P\) is countering the force due to friction, which prevents motion. This static frictional force can be calculated by the formula \(fF_N\), where \(f\) is the coefficient of static friction, and \(F_N\) is the normal force. As the normal force \(F_N\) is equal to \(W\), the static frictional force is equal to \(fW\).

Step 3: Evaluate the Result

Given that \(P\) is balancing the frictional force, the value of \(P\) is equal to the frictional force. As a result, \(P = fW\). Therefore, the value of \(P\) is directly dependent on the weight of the object and the static friction between the rope and the shaft.

Key Concepts

Coefficient of Friction

The coefficient of friction (\(f\)) is a measure of how much frictional force exists between two surfaces that are in contact. Friction is what keeps things from sliding on surfaces.

• Static Friction vs. Kinetic Friction: Static friction occurs when surfaces are not moving relative to each other, while kinetic friction happens when they are moving. In this case, we are dealing with static friction because the rope is neither sliding nor dropping.
• Formula: The static frictional force is given by the product of the coefficient of static friction (\(f\)) and the normal force (\(F_N\)). It can be expressed as \( fF_N \).

Here, the coefficient of static friction is the key factor in determining how large force \( P \) needs to be to keep the weight from falling. By understanding this concept, you can predict and control how objects behave when they are on the verge of slipping.

Normal Force

Normal force (\(F_N\)) is the force exerted by a surface in a perpendicular direction to the object resting on it. In the context of statics problems, it is crucial to understand as it directly affects frictional interactions.

• Direction of Normal Force: In our problem, the normal force acts upwards against the weight \(W\) hanging downwards. The shaft exerts this force to balance the rope's weight.
• Equilibrium Condition: In static equilibrium, the normal force should be equal to the applied vertical force; hence \(F_N = W\).

For this exercise, normal force is pivotal in calculating the frictional force that resists slipping, as it is part of the friction formula itself.

Statics Problems

Statics problems involve analyzing forces on objects that are in equilibrium — meaning they are not moving. Understanding how different forces interact and balance one another is at the heart of these problems.

• Equilibrium: When forces are balanced, the object does not move. In this exercise, the objective is to find the horizontal force \(P\) that counteracts weight \(W\), using friction.
• The Role of Forces: Forces such as weight, tension, and friction must be carefully considered to ensure they balance properly. This involves using vector components and often setting up equations to solve for unknowns.

Successfully solving statics problems like this one requires breaking down the forces into understandable parts and using principles such as equilibrium to find the solutions. By focusing on how forces like \(P\), friction, and \(W\) relate, you'll get the full picture of what keeps the rope in place without moving.