Question

Only two forces, \(\vec{F}_{1}\) and \(\vec{F}_{2},\) are acting on a block. Which of the following can be the magnitude of the net force, \(\vec{F}\), acting on the block (indicate all possibilities)? a) \(F>F_{1}+F_{2}\) b) \(F=F_{1}+F_{2}\) c) \(F

Short answer

Answer: The possible magnitudes of the net force acting on the block are: - \(F = F_{1} + F_{2}\) when the forces are acting in the same direction - \(F < F_{1} + F_{2}\) when the forces are acting in opposite directions or at any angle other than 180 degrees and 0 degrees.

Step-by-step solution

Identify the possible cases

Based on the angle between the two given forces, we have three different scenarios that we need to consider: (1) the forces are in the same direction, (2) the forces are in opposite directions, and (3) the forces are at an angle other than 0 and 180 degrees.

Calculate net force in case 1: forces in the same direction

If the forces are in the same direction, they simply add up, and then the net force can be represented as: $$F = F_{1} + F_{2}$$ So, in this case, the answer is option (b).

Calculate net force in case 2: forces in opposite directions

If the forces are in opposite directions, their magnitudes subtract from one another. There are two possibilities: (a) either \(F_{1} > F_{2}\); or (b) \(F_{1} < F_{2}\). In either case, the net force magnitude can be represented as: $$F = |F_{1} - F_{2}|$$ Due to the absolute value, the net force magnitude will always be less than the sum of the magnitudes of the forces, so the answer is option (c).

Calculate net force in case 3: forces at an angle other than 0 or 180 degrees

If the forces are at an angle other than 0 or 180 degrees, the magnitude of the net force can be found using the law of cosines: $$F^{2} = F_{1}^{2} + F_{2}^{2} - 2F_{1}F_{2}\cos(\theta)$$ Since \(0<\theta<180^{\circ}\), we know that the net force \(F\) will satisfy $$F^2 < F_{1}^2 + F_{2}^2$$ Thus, we find that \(F < F_1 + F_2\), and the answer is option (c).

Conclusion

Taking into account all possibilities, we conclude that the magnitudes of the net force acting on the block can only be described by options (b) and (c): - \(F = F_{1} + F_{2}\) when the forces are acting in the same direction - \(F < F_{1} + F_{2}\) when the forces are acting in opposite directions or at any angle other than 180 degrees and 0 degrees. Option (a) is not possible, as the net force magnitude cannot be greater than the sum of the magnitudes of the individual forces.

Key concepts

Vector Forces

When delving into physics problems, especially those involving motion and mechanics, understanding vector forces is essential. Vector forces are not just numbers; they are quantities that have both a magnitude (how much force) and a direction. This is why you can't solely rely on arithmetic to find the sum of two forces; their respective directions have a profound impact on the resultant, or net, force.

For example, if two forces with magnitudes of 3 N and 4 N are acting upon an object in the same direction, they combine linearly to produce a resultant force of 7 N. However, if they are acting in opposite directions, the object will move in the direction of the larger force with a magnitude that is the difference between the two, either 1 N in this case, assuming no other forces are at play. Once angles are involved, the calculation becomes more complex, which leads us to use trigonometric methods like the law of cosines to accurately compute the net force.

Law of Cosines

The law of cosines bridges the gap between trigonometry and physics. It is a formula that relates the lengths of the sides of any triangle with its angles. When working with forces that apply at an angle other than 0 or 180 degrees, we often imagine them at the ends of a vector triangle. In such cases, the law of cosines becomes invaluable.

The law states that for any triangle with sides of lengths a, b, and c, and an angle \(\gamma\) opposite side c, the relationship between them is given by: \[c^2 = a^2 + b^2 - 2ab\cos(\gamma)\].

In terms of vector forces, if \(\vec{F_{1}}\) and \(\vec{F_{2}}\) are represented by 'a' and 'b', and the angle between them by \(\gamma\), the net force 'c' can be found using this important law. Through it, we see that the magnitude of the net force is influenced by the cosine of the angle between the two forces, and thus it could never exceed their simple sum, emphasizing the non-linearity of vector force addition.

Physics Problem Solving

Solving physics problems often requires a clear strategy that considers various principles and laws to find a solution. In dynamics problems, such as finding the magnitude of net force acting on a block, a methodical approach yields clarity and accuracy. This starts with a careful consideration of the scenario and relevant physical laws, then proceeds with a logical sequence of steps designed to isolate and solve for the unknown quantities.

For our net force problem, we examined potential scenarios, applied the appropriate mathematics for each (addition, subtraction, the law of cosines), and logically deduced the possible outcomes. By considering each case—parallel, anti-parallel, and angled forces—we avoided overlooking possibilities which could lead to incorrect conclusions. Critical thinking combined with a comprehensive understanding of physics principles is the key to successful problem solving in physics, making even complex vector problems surmountable.