When two equal forces F act at an angle θ, the resultant force is given by which of the following expressions?

Explanation:

When two equal forces F act at an angle θ, the resultant force R is given by the following expression:

The correct option is:

Option 4: \(R=2Fcos(\frac{\theta}{2})\)

Let's derive and explain why this is the correct option:

Derivation:

When two forces of equal magnitude F act at an angle θ, the resultant force can be calculated using the law of cosines in vector addition. The resultant force R can be determined by breaking down the forces into their components and combining them vectorially.

Consider the two forces F acting at an angle θ:

• Force F1 acts along the positive x-axis.
• Force F2 acts at an angle θ relative to F1.

The components of these forces can be resolved as follows:

• The x-component of F1 is \(F_{1x} = F\).
• The y-component of F1 is \(F_{1y} = 0\).
• The x-component of F2 is \(F_{2x} = F \cos(θ)\).
• The y-component of F2 is \(F_{2y} = F \sin(θ)\).

To find the resultant force, we sum the components:

Resultant x-component \(R_x\):

\[ R_x = F_{1x} + F_{2x} = F + F \cos(θ) \]

Resultant y-component \(R_y\):

\[ R_y = F_{1y} + F_{2y} = 0 + F \sin(θ) \]

Now, the magnitude of the resultant force R is given by the Pythagorean theorem:

\[ R = \sqrt{R_x^2 + R_y^2} \]

Substituting the values of \(R_x\) and \(R_y\):

\[ R = \sqrt{(F + F \cos(θ))^2 + (F \sin(θ))^2} \]

We can simplify this equation:

\[ R = \sqrt{F^2 + 2F^2 \cos(θ) + F^2 \cos^2(θ) + F^2 \sin^2(θ)} \]

Using the Pythagorean identity \(\cos^2(θ) + \sin^2(θ) = 1\):

\[ R = \sqrt{F^2 + 2F^2 \cos(θ) + F^2} \]

Combining the like terms:

\[ R = \sqrt{2F^2 + 2F^2 \cos(θ)} \]

Factoring out \(2F^2\):

\[ R = \sqrt{2F^2(1 + \cos(θ))} \]

We know that \(\cos(θ) = 2 \cos^2(\frac{θ}{2}) - 1\), so:

\[ 1 + \cos(θ) = 1 + 2 \cos^2(\frac{θ}{2}) - 1 = 2 \cos^2(\frac{θ}{2}) \]

Substituting this back into the equation:

\[ R = \sqrt{2F^2 \cdot 2 \cos^2(\frac{θ}{2})} \]

\[ R = \sqrt{4F^2 \cos^2(\frac{θ}{2})} \]

Taking the square root of both terms:

\[ R = 2F \cos(\frac{θ}{2}) \]

Therefore, the resultant force R when two equal forces F act at an angle θ is:

\[ R = 2F \cos(\frac{θ}{2}) \]

This confirms that the correct answer is Option 4.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(R=2F\sin(\frac{\theta}{2})\)

This option is incorrect because it describes the resultant force in terms of the sine function. The correct derivation using the law of cosines and vector addition shows that the resultant force involves the cosine function, not the sine function.

Option 2: \(R=F_1+F_2\)

This option is incorrect because it represents the simple scalar addition of forces. When forces act at an angle to each other, their vector nature must be considered. The correct resultant force requires vector addition, not simple scalar addition.

Option 3: \(R=F_1-F_2\)

This option is incorrect because it represents the difference between the magnitudes of the forces. The resultant force of two equal forces acting at an angle requires vector addition, not subtraction.

Conclusion:

Understanding the vector addition of forces is crucial in determining the resultant force when two equal forces act at an angle. The correct expression for the resultant force is \(R = 2F \cos(\frac{θ}{2})\), which takes into account the angle between the forces and their magnitudes. Evaluating the other options helps clarify common misconceptions and emphasizes the importance of proper vector addition in physics.