Problem 44 - Trigonometry

Question

Resultant Force Two trucks are trying to pull an auto out of the mud using chains. The magnitudes of the tension forces in the direction of the chains are 556 pounds and 832 pounds. If the angle between the chains is \(38.5^{\circ}\), find the magnitude of the resultant force.

Step-by-Step Solution

Answer

The magnitude of the resultant force is approximately 1313.07 pounds.

Step 1: Identify Forces and Angles

There are two tension forces: \(F_1 = 556\) pounds and \(F_2 = 832\) pounds. The angle between these forces, \(\theta\), is \(38.5^{\circ}\).

Step 2: Formula for Resultant Force

The magnitude of resultant force \( R \) can be found using the formula: \[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \].

Step 3: Plug in Known Values

Substitute the given values into the formula: \[ R = \sqrt{556^2 + 832^2 + 2 \times 556 \times 832 \times \cos(38.5^{\circ})} \].

Step 4: Calculate Cosine of the Angle

Find \(\cos(38.5^{\circ})\). Using a calculator, \(\cos(38.5^{\circ}) \approx 0.7837\).

Step 5: Substitute and Calculate

Substitute \(\cos(38.5^{\circ})\) and calculate: \[ R = \sqrt{556^2 + 832^2 + 2 \times 556 \times 832 \times 0.7837} \].

Step 6: Perform Squaring and Multiplication

Calculate the squares and products: - \(556^2 = 309136\) - \(832^2 = 692224\) - \(2 \times 556 \times 832 \times 0.7837 = 725404.1472\).

Step 7: Sum Up the Values

Add the calculated values: \(309136 + 692224 + 725404.1472 = 1726764.1472\).

Step 8: Calculate the Square Root

Find the square root of the sum: \( R = \sqrt{1726764.1472} \approx 1313.07 \text{ pounds} \).

Key Concepts

Tension Forces

Tension forces come into play when an object is stretched or pulled. Think of it like the force you feel when pulling on both ends of a string. In our exercise, the trucks are applying tension forces through chains to pull an auto out of the mud. These forces are pulling the vehicle with specific magnitudes:

• The first chain exerts a tension force of 556 pounds.
• The second chain applies a force of 832 pounds.

When two separate forces are acting on an object, especially at angles, understanding tension helps us predict the resultant force effectively. It tells us how strong the pull is and from which direction it is coming.

Angle Between Forces

Angles play a crucial role when examining forces acting in different directions. In physics, the angle between such forces affects how they add up. When you have two forces pulling at an angle, the angle can either increase or decrease the resultant force depending on its size.

• In our specific example, the angle between the tension forces is \(38.5^{\circ}\).

Such an angle indicates that the forces are not working in a straight line but at a small degree from one another. Understanding this angle is essential because it influences the calculation of the resultant force using trigonometry.

Cosine Law

The cosine law comes handy for calculating the resultant force when forces pull at an angle. It's very much like the Pythagorean theorem but applicable in situations involving angles other than \(90^{\circ}\). For two forces, the resultant can be calculated using the formula:

• \[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \]

This equation consists of the magnitude of both tension forces and the cosine of the angle between them. The value of \(\cos(38.5^{\circ})\) was found to be approximately 0.7837. Plugging this into the formula allows us to find the total force working on the auto from both trucks.

Vector Addition in Physics

Vector addition is a way to combine forces that don't point exactly in the same or opposite direction. Forces are represented as vectors, which have both magnitude and direction. When adding vectors like the tension forces in our case, the direction and angle between them matter.In our scenario:

• The tension forces are vectors pointing in different directions at an angle of \(38.5^{\circ}\).
• Using vector addition helped us find the magnitude of the resultant force.

Essentially, vector addition allows us to combine these angled forces into a single 'resultant' vector. This gives us both the overall force and its direction — providing a complete picture of the forces acting on the auto stuck in the mud.