Problem 65 - College Physics

Question

Bones and muscles. A patient in therapy has a forearm that weighs 20.5 \(\mathrm{N}\) and lifts a 112.0 \(\mathrm{N}\) weight. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produce a pull of 232 \(\mathrm{N}\) when the forearm is raised \(43^{\circ}\) above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (Hint: The elbow force and the biceps force together must balance the weight of the arm and the weight it is carrying, so their vector sum must be 132.5 \(\mathrm{N}\) upward.)

Step-by-Step Solution

Answer

The elbow exerts a force of magnitude approximately 189.58 N at an angle of 7.93° above the horizontal.

Step 1: Understand the forces involved

There are four main forces at play: the weight of the arm (20.5 N), the weight being lifted (112.0 N), the biceps force (232 N), and the elbow force, which we need to find. The total weight acting downwards is 132.5 N (20.5 N + 112.0 N).

Step 2: Break down bicep force into components

Since the biceps force is acting at a 43° angle to the horizontal, we need to find its vertical and horizontal components. The vertical component is given by: \( F_{b_y} = 232 \cos(43^\circ) \). The horizontal component is given by: \( F_{b_x} = 232 \sin(43^\circ) \).

Step 3: Find vertical equilibrium

The vertical forces must balance out. Therefore, we have: \( F_{e_y} + F_{b_y} = 132.5 \). Substitute \( F_{b_y} \) from Step 2 and solve for \( F_{e_y} \).

Step 4: Find horizontal equilibrium

The horizontal forces must also balance. Thus, we have: \( F_{e_x} = F_{b_x} \). Substitute \( F_{b_x} \) from Step 2 to find \( F_{e_x} \).

Step 5: Calculate the magnitude of the elbow force

With both components \( F_{e_x} \) and \( F_{e_y} \), calculate the total magnitude of the elbow force using the Pythagorean theorem: \( |F_e| = \sqrt{F_{e_x}^2 + F_{e_y}^2} \).

Step 6: Determine the direction of the elbow force

The direction can be found using the arctangent function: \( \theta = \tan^{-1}\left(\frac{F_{e_y}}{F_{e_x}}\right) \). This will give the angle made by the force with respect to the horizontal axis.

Key Concepts

Mechanics

Mechanics is a branch of physics that focuses on the behavior of physical bodies when subjected to forces or displacements. This includes the analysis of forces and torques as well as their effects on motion. In this exercise, the mechanics involve understanding how various forces act upon the forearm as it lifts a weight. Understanding mechanics is crucial because it allows us to predict how objects will move in response to applied forces. The main forces at play here are:

• The weight of the forearm, the object being lifted, and the forces exerted by the muscles.
• Each force acts at specific angles and directions, influencing the net movement of the arm.

Comprehending these interactions makes it possible to solve for unknown forces, such as the force exerted by the elbow, ensuring that the system is in equilibrium. As you explore these exercises, keep in mind that breaking down vectors into components and understanding system equilibration are key mechanics principles.

Elbow Force

The elbow force is crucial in maintaining the stability and balance of the arm as it lifts or supports a weight. In this scenario, it is the reaction force that counteracts other forces acting on the forearm, such as the weight of the arm and the weight being lifted. When analyzing the forces, the elbow must exert a force that keeps the arm in equilibrium. That means:

• It works together with the biceps force to counterbalance the downward forces of both the arm itself and the weight being carried.
• The elbow force must have both a vertical and horizontal component to align with the forces of the biceps.

Working through these equilibria, we can use known forces and their components to calculate the necessary force the elbow must exert. This ensures that upward and downward forces, as well as horizontal forces, are balanced, keeping the system in stasis.

Biceps Force

The biceps play an integral role in lifting and supporting weight through their force, which is applied perpendicularly to the forearm. Through this force, the arm can lift weights and counter the force of gravity. When evaluating the biceps force:

• Understand that it acts at an angle to the horizontal (in this exercise, 43°), influencing its direction and strength.
• By breaking it into horizontal and vertical components, we can accurately assess its contribution to the system's equilibrium.

The biceps' vertical component aids in supporting the arm's weight, while the horizontal component adds stability. Calculating these components allows us to better understand how the muscle contributes to the overall force required to maintain balance and carry additional weight.

Vector Components

In physics, breaking a force into its components helps us analyze its effect in different directions — horizontal and vertical. This is done using trigonometric functions based on the angle of the force.In this exercise:

• We analyzed the biceps force of 232 N acting at 43° to determine its components.
• The vertical component was found using the cosine function: \( F_{b_y} = 232 \cos(43^\circ) \).
• The horizontal component was calculated using the sine function: \( F_{b_x} = 232 \sin(43^\circ) \).

By understanding these components, you know how each part of the force contributes to moving or supporting the object. This also allows us to set up equations to ensure that all forces within the system are balanced, leading us to find the unknown forces effectively.