Question

A baby elephant is stuck in a mud hole. To help pull it out, game keepers use a rope to apply force A, as part

(a) of the drawing shows. By itself, however, force A is insufficient. Therefore, two additional forces B and C are applied, as in part
(b) of the drawing. Each of these additional forces has the same magnitude F and is oriented at θ = 19.5° away from A. The magnitude of the resultant force acting on the elephant in part (b) of the drawing is three times that in part a. Find the ratio F / FA.

Answer 1

The ratio F/FA is approximately 2.83.

Step-by-step explanation:

To find the ratio F / FA, we can use the concept of vector addition. In part (a), force A alone is not enough to pull the baby elephant out of the mud, so two additional forces B and C are applied. These additional forces have the same magnitude F and are oriented at θ = 19.5° away from force A.

Given that the magnitude of the resultant force in part (b) is three times that in part (a), we can set up the equation:

3FA = √((FA + Fcosθ)² + (Fsinθ)²)

Simplifying, we get:

9FA² = FA² + 2FAFcosθ + F²

Using the trigonometric identity cosθ = cos²θ - sin²θ, we can rewrite the equation as:

9FA² = FA² + 2FAF(cos²θ - sin²θ) + F²

Simplifying further, we obtain:

9FA² = FA² + F²

Dividing both sides of the equation by FA², we get:

9 = 1 + (F/FA)²

Subtracting 1 from both sides of the equation, we have:

(F/FA)² = 8

Taking the square root of both sides of the equation, we find:

F/FA = √8 ≈ 2.83

Therefore, the ratio F/FA is approximately 2.83.