Question

Two forces whose magnitudes are in the ratio 5:3 are acting at a point at an angle of \( 60^\circ \) simultaneously. If the resultant of the two forces is 35 N, then the magnitudes of two forces respectively are:

• A. 3N, 5N
• B. 25N, 9N
• C. 25N, 15N
• D. 12N, 20N

Hint:

When solving problems involving forces and angles, use vector addition formulas and remember to account for the cosine of the angle between forces.

Answer 1

The Correct Option is C

Let the magnitudes of the two forces be \( F_1 \) and \( F_2 \), where \( F_1 = 5x \) and \( F_2 = 3x \) (since their magnitudes are in the ratio 5:3). The resultant \( R \) of the two forces is given by the formula: \[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} \] where \( \theta = 60^\circ \) is the angle between the two forces. Substituting the known values: \[ 35 = \sqrt{(5x)^2 + (3x)^2 + 2 \times 5x \times 3x \times \cos 60^\circ} \] Simplify: \[ 35 = \sqrt{25x^2 + 9x^2 + 30x^2 \times \frac{1}{2}} \] \[ 35 = \sqrt{25x^2 + 9x^2 + 15x^2} = \sqrt{49x^2} \] \[ 35 = 7x \] \[ x = 5 \] Now, substitute \( x = 5 \) into the expressions for \( F_1 \) and \( F_2 \): \[ F_1 = 5x = 25 \, \text{N}, \quad F_2 = 3x = 15 \, \text{N} \] Thus, the magnitudes of the two forces are \( 25 \, \text{N} \) and \( 15 \, \text{N} \). The correct answer is option (3), \( 25 \, \text{N} \) and \( 15 \, \text{N} \).

Answer 2

Step 1: Write down the given data
Ratio of two forces, \( F_1 : F_2 = 5 : 3 \)
Let the forces be \( 5x \) and \( 3x \)
Angle between the forces, \( \theta = 60^\circ \)
Resultant force, \( R = 35 \, \text{N} \)

Step 2: Use the formula for resultant of two forces
\[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} \]
Substitute \( F_1 = 5x \), \( F_2 = 3x \), and \( \cos 60^\circ = \frac{1}{2} \):
\[ 35 = \sqrt{(5x)^2 + (3x)^2 + 2 \times 5x \times 3x \times \frac{1}{2}} \]

Step 3: Simplify the equation
\[ 35 = \sqrt{25x^2 + 9x^2 + 15x^2} = \sqrt{49x^2} = 7x \]
So,
\[ 7x = 35 \implies x = 5 \]

Step 4: Calculate the magnitudes of forces
\[ F_1 = 5x = 5 \times 5 = 25 \, \text{N} \]
\[ F_2 = 3x = 3 \times 5 = 15 \, \text{N} \]

Final answer: The magnitudes of the two forces are 25 N and 15 N respectively.