2. Two forces of 7 Newtons each acting at 45 degrees to each other will have a resultant
of approximately
a. 6 Newtons
b. 8 Newtons
c. 10 Newtons
d. 13 Newtons

The correct answer is (d) 13 Newtons because the resultant of two 7 N forces inclined at 45∘45∘ is approximately 13 N when calculated using the standard vector addition formula for forces at an angle.​

Resultant of Two Forces of 7 Newtons Each Acting at 45°

Introduction

In physics and engineering, exam questions on the resultant of two forces of 7 Newtons each acting at 45 degrees
are very common because they test understanding of vector addition and basic trigonometry. Using the
law of cosines or the parallelogram law, the resultant force can be calculated precisely and
matched to the nearest option in multiple‑choice questions.

The Question

Two forces of 7 Newtons each acting at 45 degrees to each other will have a resultant of approximately:

• a. 6 N
• b. 8 N
• c. 10 N
• d. 13 N

Step‑by‑Step Solution

1. Use the Standard Resultant Formula

For two forces F₁ and F₂ acting at an angle θ, the magnitude of the resultant R is:

R = √(F₁² + F₂² + 2F₁F₂ cos θ)

Here:

• F₁ = 7 N
• F₂ = 7 N
• θ = 45°

Substitute the values:

R = √(7² + 7² + 2 × 7 × 7 × cos 45°)

Since cos 45° = 0.707, we get:

R ≈ √(49 + 49 + 98 × 0.707) = √(167.286) ≈ 12.93 N

Explanation of Each Option

Option (a) 6 Newtons

This is far below the correct value (~13 N) and even less than each individual force of 7 N,
which is not possible when the angle between equal forces is less than 180°. For equal forces F,
the minimum possible resultant is 0 (when they are opposite) and the maximum is 2F (when they are in the same direction).
At 45°, the resultant must be greater than 7 N and less than 14 N, so 6 N is impossible.

Option (b) 8 Newtons

Although 8 N is greater than 7 N, it is still much lower than the calculated resultant of about 12.93 N.
Some students get this by incorrectly adding components without considering the cosine factor.

Option (c) 10 Newtons

This may look “reasonable” but results from rough approximations or arithmetic mistakes such as mishandling cos 45°.
The exact formula gives about 13 N, not 10 N.

Option (d) 13 Newtons

Using the correct vector resultant formula for two equal forces at 45°, the computation yields ≈12.93 N, which rounds to 13 N.
This matches the only option close to the computed value.

Key Formula Recap for Exams

R = √(F₁² + F₂² + 2F₁F₂ cos θ)

For two equal forces F:

R = 2F × cos(θ/2)

Applying this formula systematically helps solve many similar multiple-choice questions
involving two forces acting at a given angle.