Question

A horizontal beam of negligible weight is \(18.0 \mathrm{ft}\) long and is supported by columns at either end. A vertical load of 14,500 lb is applied to the beam at a distance \(x\) from the left end. (a) Find \(x\) so that the reaction at the left column is 10,500 lb. (b) Find the reaction at the right column.

Short answer

The distance x from the left end is 13.0 ft, and the reaction at the right column is 4,000 lb.

Step-by-step solution

Understanding the System

The system is a beam on two supports, experiencing a single load. The sum of the reactions at the supports must equal the total load, and the moments around any point must balance.

Setting up the Equilibrium Equations

Let the reaction at the left column be represented by R1 and the right column by R2. According to the equilibrium of forces, we have R1 + R2 = 14,500 lb. Since we want R1 to be 10,500 lb, R2 can be found by subtracting R1 from the total load: R2 = 14,500 lb - R1.

Calculating the Distance x

Set up the moment equation about the left end to find x. The moment due to the load must be equal to the moment due to the left reaction R1. The equation will be 14,500 lb * x = 10,500 lb * 18.0 ft. Now solve for x.

Solving for x

Rearrange the moment equation to isolate x: x = (10,500 lb * 18.0 ft) / 14,500 lb. Now calculate x.

Calculating the Reaction at the Right Column

Use the equilibrium of forces equation R2 = 14,500 lb - 10,500 lb to find R2.

Key concepts

Static Equilibrium

Static equilibrium occurs when an object is at rest or moving at a constant velocity, meaning there's no net force or net moment acting on it. In the context of a beam supported at both ends, like in our exercise, static equilibrium implies two main conditions.

The first is that the algebraic sum of all horizontal and vertical forces must be equal to zero. For a horizontal beam experiencing only vertical loads, only the vertical equilibrium is typically considered. The second condition is that the algebraic sum of moments (turning forces) about any point must also equate to zero, ensuring the beam does not rotate.

Why is Static Equilibrium Important?

Understanding this concept is crucial in structural engineering and physics as it helps in determining the forces and reactions occurring within a structure. Ensuring a structure is in static equilibrium guarantees stability and safety, which is paramount when designing things like buildings, bridges, or in our case, simple beams.

Reaction Forces

Reaction forces are forces exerted by supports or joints in response to loads applied on a structure. These forces keep the structure in equilibrium. There are as many reaction forces as there are supports, and these can be in the form of vertical forces, horizontal forces, or moments depending on the type of support and the loads applied.

The exercise we're discussing involves a beam supported by columns at each end. The reactions at these supports are vertical forces because the columns prevent the beam from moving vertically. In the step-by-step solution, the reaction forces were denoted R1 and R2. To find the value of one reaction force, we must first know either the total load on the beam or one of the other reaction forces, then apply the equilibrium condition to solve for the unknown.

Simplified Calculation for Reaction Forces:

In simpler terms, if you imagine a seesaw with different weights at varying distances from the pivot point, the reaction forces are like the hands that push up on the seesaw from below to balance out the weights, thus maintaining equilibrium.

Moment of Force

The moment of force, often simply called 'moment', is the rotational effect produced by a force. It is calculated by multiplying the force by its perpendicular distance from the point or axis of rotation. The unit typically used in engineering is the foot-pound (ft-lb) in the Imperial system, or newton-meter (Nm) in the Metric system.

Moments are particularly important in understanding the rotational aspects of equilibrium, as they dictate whether an object will spin, rotate, or stay balanced. When solving for the moment about one support, we fix a pivot at that point and examine the tendency of forces to cause rotation about it.

Applying the Moment in Beam Equilibrium:

In our example, the moment created by the vertical load must be balanced by the moment created by the reaction force at the left column in order for the beam to remain in static equilibrium. By setting up an equation that equates these two moments, we can solve for the unknown distance 'x' from the step-by-step solution.