Problem 17 - An Introduction to the Mechanics of Solids in SI Units

Question

A carpenter with a power saw has a 6 -m plank of uniform weight per unit length \(w_{o}\) and two sawhorses. He wishes to cut a \(1.8-\mathrm{m}\) length from the plank, but in order to minimize splitting of the ends he wants to cut it at a point where the bending moment in the plank is zero. If he places one sawhorse at one end of the plank, where should he put the other so that the bending moment will be zero \(1.8\) \(\mathrm{m}\) from the other end of the plank?

Step-by-Step Solution

Answer

The second sawhorse should be placed \(1.8\, m\) from the cut point to make the bending moment zero at the cut point.

Step 1: Understand the situation

Imagine the plank as a straight line. One end of the plank (say left end) is supported by a sawhorse, another sawhorse is somewhere in between while the carpenter makes a cut at \(1.8\, m\) from the other end (right end). The weight \(w_{o}\) is uniform across the whole length of the plank, meaning each meter weighs the same. So for any distance \(d\), the weight there is \(d \times w_{o}\). We'll consider a moment about the cut point, and place the second sawhorse where the moments on both sides will be equal, causing zero bending.

Step 2: Equate the moments on both sides

Let's denote with \(x\) the distance from the cut point to the second sawhorse. Then the length on the left side of the plank till the cut point is \(6 - 1.8 = 4.2\, m\). The moment on the left is the weight of the plank on this side times \(x\), which is \((4.2 \times w_{o})x\). The moment on the right is the weight of the part with length \(1.8\, m\) times its distance to the cut point, which is \(1.8 \times w_{o} \times (6-x)\).

Step 3: Solve for x

If we are to make the cut so that the bending moment will be zero, then the moments on both sides will be equal. Hence we set the right and left moments equal to each other: \((4.2 \cdot w_{o})x = 1.8 \cdot w_{o}(6-x)\). We can simplify this equation because \(w_{o}\) is on both sides, and then solve for \(x\). After simplifying, we get \(x = \frac{10.8}{6} = 1.8\, m\). This means the other sawhorse should be placed \(1.8\, m\) from the point where the cut is made.

Key Concepts

Uniform Weight Distribution

In statics and beam mechanics, a uniform weight distribution is a crucial concept to understand. It implies that the weight is spread evenly across the entire length of an object, like a plank. Imagine dividing the plank into smaller segments of equal length; each segment would have the same weight, making calculations straightforward.
For the carpenter's plank scenario, the uniform weight per unit length is represented by the variable \( w_o \). This means each meter of the plank has a weight of \( w_o \), making it simple to calculate the total weight over any section of the plank by multiplying the length of that section by \( w_o \).
Using uniform weight distribution makes it easier to predict and calculate reactions and moments in structural components like beams, thus ensuring better design and stability.

Statics

Statics is the branch of mechanics that studies objects at rest. In this context, we are dealing with the plank, where the objective is to maintain equilibrium. In our exercise, to achieve zero bending moment at the cut point, the moments of forces acting on the plank must balance out.
When we refer to moments, we mean the turning effect caused by a force in relation to a point, in this case, the cut point on the plank. Moment is determined by multiplying the force (weight here) by the distance from the point of interest.
The guiding principle in statics for this exercise is to keep the plank in equilibrium by balancing the moments on either side of the cut point. This balance of forces and moments ensures that the carpenter can make a clean cut at the desired point with no stress causing a split.

Beam Mechanics

Beam mechanics involves analyzing the behavior of beams under various forces and conditions. A critical aspect of this is understanding how bending moments occur and affect a beam, such as our wooden plank.
Bending moments arise when a force is applied at a distance from a specific point, causing the beam to bend. In our exercise, we aim to find a configuration where the bending moment at the cut point is zero, meaning the forces are perfectly balanced and no moment is causing bending at that exact spot.
Understanding the beam's behavior in response to applied forces helps engineers and carpenters determine the best support placements. For the carpenter, placing the sawhorses appropriately ensures stability and prevents unwanted deformations or splits, allowing for precise cuts.