Problem 47 - College Physics

Question

\(\bullet\) You're hanging from a chinning bar, with your arms at right angles to each other. The magnitudes of the forces exerted by both your arms are the same, and together they exert just enough upward force to support your weight, 620 \(\mathrm{N}\) . (a) Sketch the two force vectors for your arms, along with their resultant, and (b) use components to find the magnitude of each of the two "arm" force vectors.

Step-by-Step Solution

Answer

Each "arm" force vector has a magnitude of approximately 438.4 N.

Step 1: Understand the Problem

You're hanging motionless, which means the upward forces from your arms equal your weight (620 N). Each arm exerts an equal force at a 45-degree angle to the vertical direction because your arms are at right angles to each other, forming a 90-degree angle between them.

Step 2: Sketch the Force Vectors

Draw the force vectors for each arm on a coordinate system. The vectors form an angle of 90 degrees between them, and the resultant (net force) is vertically upwards. This resultant force, which is your weight, is 620 N.

Step 3: Set Up the Equations

Let the magnitude of the force exerted by each arm be \( F \). The components of these forces in the vertical direction are added together to equal the total weight. Since the angles are 45 degrees, the vertical component of each arm's force is \( F \cos(45^{\circ}) \). Thus, the equation is: \[ 2F \cos(45^{\circ}) = 620 \].

Step 4: Solve for F

Since \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), substitute this into the equation: \[ 2F \frac{\sqrt{2}}{2} = 620 \]. Simplifying gives \( F \sqrt{2} = 620 \). Dividing both sides by \( \sqrt{2} \) gives \( F = \frac{620}{\sqrt{2}} \). Simplify \( F \) to get approximately 438.4 N.

Key Concepts

Understanding Vector Components

Vector components are crucial in breaking down a vector into parts that are easier to work with. Imagine you are hanging from a bar; the forces your arms exert can be represented by force vectors. These vectors have both horizontal and vertical parts, known as components. By dividing these vectors into horizontal and vertical components, we can simplify complex problems.
In any physics problem involving vectors, each vector can be split into its x (horizontal) and y (vertical) components. For a given vector force, if it forms an angle \( \theta \) with a reference direction, the components can be calculated as follows:

• The vertical component (y-axis) is \( F_y = F \cos(\theta) \).
• The horizontal component (x-axis) is \( F_x = F \sin(\theta) \).

Understanding these components lets us handle each part separately, making difficult physics problems much more manageable.

Equilibrium Conditions in Physics

Equilibrium in physics occurs when all forces acting on an object are balanced, causing no change in motion. In the case of hanging from a bar, you remain stationary as the upward forces equal your weight, ensuring equilibrium. This scenario is a simple demonstration of equilibrium conditions.
For an object to be in equilibrium, the sum of forces in any direction must be zero. In mathematical terms this is expressed as:

• \( \sum F_{x} = 0 \): All horizontal forces must balance.
• \( \sum F_{y} = 0 \): All vertical forces must balance.

In your exercise, your arms provide upward forces that counteract the gravitational force pulling you down. This situation of balanced forces ensures that you don't fall or rise, keeping you in a stable hanging position.

Role of Trigonometric Functions

Trigonometry is a powerful tool in physics, particularly when dealing with force vectors. In your exercise, each of your arm's forces acts at a 45-degree angle to the vertical. Trigonometric functions allow us to break down these angled forces into vertical and horizontal components, crucial for analyzing vectors.
Trigonometric functions like sine, cosine, and tangent relate angles to ratios of sides in right-angled triangles. For a force vector at an angle \( \theta \), the components are determined as:

• Vertical: \( F \cos(\theta) \), which represents the force parallel to the vertical axis.
• Horizontal: \( F \sin(\theta) \), which represents the force parallel to the horizontal axis.

Understanding these relations is key to solving problems involving vectors and equilibrium, allowing for precise calculations in various scientific contexts.