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Chapter 4: Problem 11

On her way to visit Grandmother, Red Riding Hood sat down to rest and placed her 1.2 -kg basket of goodies beside her. A wolf came along, spotted the basket, and began to pull on the handle with a force of \(6.4 \mathrm{N}\) at an angle of \(25^{\circ}\) with respect to vertical. Red was not going to let go easily, so she pulled on the handle with a force of \(12 \mathrm{N}\). If the net force on the basket is straight up, at what angle was Red Riding Hood pulling?

Short Answer

Expert verified
Answer: Red Riding Hood was pulling at an angle of approximately 20.69° below the horizontal.

Step by step solution

01

Identify the forces

Let's first visualize the forces. The wolf's force has a magnitude of 6.4 N and is applied at a 25° angle with respect to the vertical. Red's force is along an unknown angle, θ, and has a magnitude of 12 N. Let F₁ be the wolf's force (6.4 N) and F₂ be Red's force (12 N).
02

Decompose the forces into vertical and horizontal components

In order to find the angle θ, we need to first decompose the forces into their vertical and horizontal components. For the wolf's force: - Vertical component: \(F_{1V} = F_1 * cos(25°) = 6.4 * cos(25°)\) - Horizontal component: \(F_{1H} = F_1 * sin(25°) = 6.4 * sin(25°)\) For Red's force: - Vertical component: \(F_{2V} = F_2 * cos(\theta) = 12 * cos(\theta)\) - Horizontal component: \(F_{2H} = F_2 * sin(\theta) = 12 * sin(\theta)\)
03

Apply the net force condition

Since the net force on the basket is straight up, the sum of the vertical components of the forces is equal to the total force up, and the sum of the horizontal components is zero. Therefore: - Vertical force: \(F_{1V} + F_{2V} = F_{Total}\) - Horizontal force: \(F_{1H} + F_{2H} = 0\) Substitute the expressions for vertical and horizontal components calculated earlier: - \((6.4 * cos(25°)) + (12 * cos(\theta)) = F_{Total}\) - \((6.4 * sin(25°)) + (12 * sin(\theta)) = 0\)
04

Solve for the angle θ

From the horizontal force equation, we can solve for the angle θ: \(12 * sin(\theta) = -6.4 * sin(25°)\) \(\theta = arcsin(-\frac{6.4 * sin(25°)}{12})\) Now calculate the angle: \(\theta = arcsin(-\frac{6.4 * sin(25°)}{12}) \approx -20.69°\) Since the angle is negative, it means that Red Riding Hood was pulling at an angle of 20.69° below the horizontal.

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