Figure 6-39 is an overhead view of a 12 kg tire that is to be pulled by three ropes. One force

step 1 So over here, the tire is pulled by three forces, right? So one forces indicated, like this one,

Figure 6-39 is an overhead view of a 12 kg tire that is to...

Figure $6-39$ is an overhead view of a $12 \mathrm{~kg}$ tire that is to be pulled by three ropes. One force $\left(\vec{F}_{A}\right.$, with magnitude $50 \quad \mathrm{~N}$ ) is indicated. Orient the other two forces $\vec{F}_{B}$ and $\vec{F}_{C}$ so that the magnitude of the resulting acceleration of the tire is least, and find that magnitude if (a) $F_{B}=30 \mathrm{~N}, F_{C}=20 \mathrm{~N} ;$ (b) $F_{B}=30 \mathrm{~N}, F_{C}=10 \mathrm{~N} ;$ and $(\mathrm{c})$ $F_{B}=F_{C}=30 \mathrm{~N}$.

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Key Concepts

Optimization of Force Orientation

This concept involves adjusting the directions of individual forces in order to achieve a desired effect on the resultant vector, such as minimizing or maximizing its magnitude. In problems where forces can be reoriented, selecting the optimum angles can control the net force or acceleration experienced by an object.

Vector Addition

This concept involves combining two or more force vectors to determine their overall effect. In mechanics, vectors are added using both magnitude and direction, often by breaking each force into its horizontal and vertical components and then summing these components to obtain a resultant vector.

Newton's Second Law

Newton's Second Law, expressed as F = ma, is fundamental in relating the net force acting on an object to its acceleration, where F is the net force, m is the mass, and a is the acceleration. This principle is critical when determining how different forces combine to produce a specific acceleration.

Force Decomposition

Force decomposition is the process of resolving a force vector into its orthogonal (usually horizontal and vertical) components. This technique simplifies the analysis of vector sums and is essential when dealing with forces that are applied at angles, as it allows the separate consideration of influences in different directions.

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