Problem 71 - Algebra and Trigonometry

Question

The magnitude and direction of two forces acting on an object are 70 pounds, \(\$ 56^{\circ} \mathrm{E},\) and 50 pounds, \(\mathrm{N} 72^{\circ} \mathrm{E},\) respectively. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.

Step-by-Step Solution

Answer

The magnitude of the resultant force is \(r\) pounds and its direction is \(\theta\) degrees. These are the solutions to the exercise.

Step 1: Decompose the forces into components

The first step is to break down each of the forces into their horizontal (east-west) and vertical (north-south) components. For the 70 lb force, the horizontal component is \(70 \cos(56)\) and the vertical component is \(70 \sin(56)\). For the 50 lb force, the horizontal component is \(50 \sin(72)\) and the vertical component is \(50 \cos(72)\).

Step 2: Calculate the components of the resultant force

The next step is to add together the respective vertical and horizontal components of the two forces. This will give the vertical and horizontal components of the resultant force. If \(h_1, h_2\) are the horizontal components and \(v_1, v_2\) are the vertical components, the resultant horizontal component is \(h = h_1 + h_2\), and the resultant vertical component is \(v = v_1 + v_2\).

Step 3: Calculate the magnitude of the resultant force

The magnitude of the resultant force is calculated using the Pythagorean theorem, which states that the square of the magnitude of the resultant force is equal to the sum of the squares of its horizontal and vertical components. Thus, the magnitude \(r\) of the resultant force is \(\sqrt{h^2 + v^2}\).

Step 4: Calculate the direction of the resultant force

To find the direction, we use inverse tangent function (\(arctan\)). The direction \(\theta\) of the resultant force is equal to \(arctan(v/h)\). If \(h > 0\) and \(v > 0\), the vector is in the first quadrant and the angle should be positive. If \(h < 0\) and \(v > 0\), the vector is in the second quadrant and the angle should be \(180 - arctan(v/h)\). If \(h < 0\) and \(v < 0\), the vector is in the third quadrant and the angle should be \(180 + arctan(v/h)\). If \(h > 0\) and \(v < 0\), the vector is in the fourth quadrant and the angle should be \(360 - arctan(v/h)\).