Question 2.SP.3: Four forces act on bolt A as shown. Determine the resultant ......

Vector Mechanics for Engineers Statics [1686415]

Four forces act on bolt A as shown. Determine the resultant of the forces on the bolt.

STRATEGY: The simplest way to approach a problem of adding four forces is to resolve the forces into components.

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Four forces act on bolt A.

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In this problem, we are given four forces and we need to find the resultant force. To solve this problem, we can use the method of vector addition.

Step 1:

We start by breaking down each force into its x and y components. This can be done using trigonometry. In the table provided, we enter the x and y components of each force.

Step 2:

We then determine the resultant force by adding the x components and the y components separately. In this case, the x component of the resultant force is the sum of the x components of the individual forces, and the y component of the resultant force is the sum of the y components of the individual forces.

Step 3:

Once we have the x and y components of the resultant force, we can represent it as a vector using the notation R = Rx i + Ry j, where i and j are the unit vectors in the x and y directions, respectively. In this case, the x component of the resultant force is 199.1 N and the y component is 14.3 N, so the resultant force can be written as R = (199.1 N) i + (14.3 N) j.

Step 4:

To determine the magnitude and direction of the resultant force, we can use trigonometry. We can find the angle α between the resultant force and the positive x-axis using the equation tan α = Ry/Rx. In this case, Ry is 14.3 N and Rx is 199.1 N, so α is approximately 4.1 degrees.

Step 5:

Finally, we can find the magnitude of the resultant force using the equation R = Ry/sin α. In this case, Ry is 14.3 N and α is 4.1 degrees, so the magnitude of the resultant force is approximately 199.6 N. The direction of the resultant force can be represented as an angle measured counterclockwise from the positive x-axis, so the resultant force can be written as R = 199.6 N ∠ 4.1 degrees.

In summary, by breaking down the forces into their components, adding the components separately, and using trigonometry, we were able to find the magnitude and direction of the resultant force. This method of vector addition is a useful tool in solving problems involving multiple forces.

Final Answer

MODELING: As we mentioned, solving this kind of problem is usually easier if you arrange the components of each force in a table. In the table below, we entered the x and y components of each force as determined by trigonometry (Fig. 1). According to the convention adopted in this section, the scalar number representing a force component is positive if the force component has the same sense as the corresponding coordinate axis. Thus, x components acting to the right and y components acting upward are represented by positive numbers.

Thus, the resultant R of the four forces is

$\mathbf{R}=R_x \mathbf{i}+R_y \mathbf{j} \quad \mathbf{R}=(199.1 \mathrm{~N}) \mathbf{i}+(14.3 \mathrm{~N}) \mathbf{j}$

You can now determine the magnitude and direction of the resultant. From the triangle shown in Fig. 2, you have

$\tan \alpha=\frac{R_y}{R_x}=\frac{14.3 \mathrm{~N}}{199.1 \mathrm{~N}} \quad \alpha=4.1^{\circ}$

$R=\frac{14.3 \mathrm{~N}}{\sin \alpha}=199.6 \mathrm{~N} \quad \mathbf{R}=199.6 \mathrm{~N} \measuredangle 4.1^{\circ}$

REFLECT and THINK: Arranging data in a table not only helps you keep track of the calculations, but also makes things simpler for using a calculator on similar computations.