Problem 71 - University Physics with Modern Physics

Question

A physical therapy patient has a forearm that weighs 20.5 N and lifts a 112.0-N weight. These two forces are directed vertically downward. The only other significant forces on this forearm come from the biceps muscle (which acts perpendicular to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised 43.0\(^{\circ}\) above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is carrying, so their vector sum must be132.5 N, upward.)

Step-by-Step Solution

Answer

The elbow exerts a force of 238.7 N at an angle of 29.4° above the horizontal.

Step 1: Analyze the Given Problem

The forearm holds a weight of 112 N and its own weight of 20.5 N, both acting downward. The biceps muscle exerts an upward force of 232 N at a 43.0° angle from the horizontal. We need to calculate the force exerted by the elbow (both magnitude and direction), knowing that the resultant of the biceps and elbow force must be 132.5 N pointing upward.

Step 2: Determine the Components of Biceps Force

The biceps pulls with 232 N at an angle. Calculate the vertical component as \( F_{b_{y}} = 232\sin(43.0^{\circ}) \) and the horizontal component as \( F_{b_{x}} = 232\cos(43.0^{\circ}) \).

Step 3: Apply Equilibrium Conditions

The vertical forces must balance out, so the total upward forces (biceps vertical component and elbow vertical component) must equal the total downward forces (weights of forearm and weight lifted). Calculate the required vertical component of the elbow force, \( F_{e_{y}} = 132.5 - F_{b_{y}} \).

Step 4: Calculate Horizontal Equilibrium

There is no horizontal force from the weight, so the horizontal component of the elbow force must counter the horizontal component of the biceps: \( F_{e_{x}} = F_{b_{x}} \).

Step 5: Solve for Magnitude of Elbow Force

The elbow force can be found using \( F_e = \sqrt{F_{e_{x}}^2 + F_{e_{y}}^2} \). This equation considers both horizontal and vertical components derived in the previous steps.

Step 6: Find Direction of Elbow Force

Calculate the angle \( \theta \) of the elbow force with respect to horizontal using \( \tan(\theta) = \frac{F_{e_{y}}}{F_{e_{x}}} \) and solve for \( \theta \).

Key Concepts

Equilibrium Conditions

In physics, understanding equilibrium conditions is crucial for analyzing systems in balance. When an object is in equilibrium, the sum of all forces and moments acting on it equals zero. This means both forces and torques are balanced. For our problem, the forearm needs to be in equilibrium while supporting its own weight and an additional weight. The total downward forces (weight of the arm and lifted weight) need to be balanced by the upward forces (biceps and elbow reaction).

We apply the principle of equilibrium here by setting the sum of all upward forces equal to the sum of downward forces. Furthermore, in horizontal terms, because no horizontal external forces act on the system, the lateral components must also cancel out. This ensures that the forearm remains stable and doesn't tilt or move horizontally.

Vector Components

Breaking forces into vector components is essential for equilibrium analysis. Vectors can be resolved into two perpendicular directions, often using the sine and cosine functions. The biceps pull is at a 43.0° angle to the horizontal, so it has both a vertical and horizontal component.

We find these components using:

• Vertical component, \( F_{b_{y}} = 232\sin(43.0^{\circ}) \)
• Horizontal component, \( F_{b_{x}} = 232\cos(43.0^{\circ}) \)

This separation into components helps us analyze how each part of the force contributes to keeping the forearm balanced.

Force Analysis

Force analysis involves examining all forces acting on an object to understand their interactions. For the forearm, we're considering:

• The weight of the forearm and additional weight acting downwards
• The biceps force having both vertical and horizontal components
• The force exerted by the elbow also having vertical and horizontal components

The key to solving the problem lies in ensuring that vertical and horizontal forces are both individually balanced. Vertically, the forces from the biceps and the elbow should combine to counteract the downward weight forces. Horizontally, since only the biceps and elbow forces are involved, we make sure their horizontal components are equal and opposite. This is crucial to maintaining the arm's stability.

Trigonometry in Physics

Trigonometry plays a vital role in resolving forces into components and calculating angles within physics problems. For the forearm system, the force exerted by the biceps at an angle requires trigonometric functions to find its effect in horizontal and vertical directions.

We use:

• \( \sin \) for calculating the vertical component
• \( \cos \) for finding the horizontal component
• \( \tan \) to determine the angle of the resultant forces

These functions allow us to derive components and solve for unknowns like the direction of the elbow force, using \( \tan(\theta) = \frac{F_{e_{y}}}{F_{e_{x}}} \). This crucial calculation helps in determining how forces are oriented relative to one another, ensuring equilibrium is achieved.