Problem 81 - University Physics with Modern Physics

Question

Bones and Muscles A patient in therapy has a forearm that weighs 20.5 \(\mathrm{N}\) and that lifts a \(112.0-\mathrm{N}\) weight. These two forces have direction vertically downward. 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 produces 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. (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 be 132.5 \(\mathrm{N}\) , upward.)

Step-by-Step Solution

Answer

The elbow exerts a force of approximately 171.3 N at an angle of 8.5° below the horizontal.

Step 1: Determine Forces in Vertical and Horizontal

We begin by analyzing the forces acting on the forearm. There are two main forces acting downward: the force due to the forearm's weight (20.5 N) and the weight that is held (112.0 N), making a total downward force of 132.5 N. The biceps creates a force of 232 N upwards at an angle of 43 degrees from the horizontal.

Step 2: Resolve Biceps Force into Components

We resolve the biceps force into horizontal and vertical components. For the biceps force (232 N) at an angle of 43 degrees, the vertical component is calculated as: \[ F_{b_y} = 232 \cdot \sin(43^\circ) \approx 158.0 \text{ N} \]and the horizontal component is:\[ F_{b_x} = 232 \cdot \cos(43^\circ) \approx 169.4 \text{ N} \]

Step 3: Apply Equilibrium in Vertical Direction

The vertical forces must sum to zero for equilibrium. Thus, the upward force due from the elbow (E_y) and the vertical component of the biceps force must balance the total downward force (132.5 N):\[ E_y + F_{b_y} = 132.5 \]Plugging in the value of \(F_{b_y}\):\[ E_y + 158.0 = 132.5 \]\[ E_y = 132.5 - 158.0 = -25.5 \text{ N}\]The negative sign means that the elbow force is acting downward.

Step 4: Apply Equilibrium in Horizontal Direction

For horizontal force equilibrium, we have:\[ E_x + F_{b_x} = 0 \]Substitute \(F_{b_x}\):\[ E_x + 169.4 = 0 \]\[ E_x = -169.4 \text{ N} \]So, the elbow force is acting in the opposite direction (left) to the biceps force's horizontal component.

Step 5: Calculate Magnitude of Elbow Force

The magnitude of the elbow force is calculated using the Pythagorean theorem, where the components \(E_x\) and \(E_y\) are perpendicular:\[ E = \sqrt{E_x^2 + E_y^2} = \sqrt{(-169.4)^2 + (-25.5)^2} \approx 171.3 \text{ N} \]

Step 6: Determine Direction of Elbow Force

The direction (angle \(\theta\) below the horizontal) is found using:\[ \theta = \tan^{-1}\left(\frac{E_y}{E_x}\right) = \tan^{-1}\left(\frac{-25.5}{-169.4}\right) \approx 8.5^\circ \]The elbow force is at an angle of \(8.5^\circ\) below the horizontal, acting leftward.

Key Concepts

Biceps Force

The biceps muscle plays a crucial part in lifting and holding objects with the arm. This muscle applies force that acts perpendicularly to the forearm. In our exercise, the biceps generates a force of 232 N. To fully understand how this force operates, it is helpful to break it down into components.

One of the components is the vertical component, denoted as \( F_{b_y}\), which helps to lift the arm and the weight being held. The vertical component is calculated by taking the sine of the angle at which the biceps acts, which is 43 degrees in this context. Thus, the formula becomes:

• \( F_{b_y} = 232 \cdot \sin(43^\circ) \approx 158.0 \text{ N} \)

This means that the vertical component acts upwards, opposing the gravity pulling down on both the forearm and the weight.

Understanding these components is essential for solving equilibrium problems as it lets us see how forces balance each other in comprehensive physics scenarios.

Elbow Force

The elbow force is a vital part of maintaining equilibrium in the arm. It counteracts not just the downward pull from the weights but also works in tandem with the biceps. This force can be split into vertical and horizontal components as well.

In the step-by-step problem-solving process, we found that the vertical component \(E_y\) acts in a downward direction, which might seem counterintuitive. After calculations:

• \( E_y = -25.5 \text{ N} \)

The negative sign signifies it acts in the opposite direction of \( F_{b_y} \).

In the horizontal direction, the force \( E_x \) acted to the left, balancing the horizontal component of the biceps force:

• \( E_x = -169.4 \text{ N} \)

The complete elbow force can be found by taking the magnitude through the Pythagorean theorem:

• \( E = \sqrt{(-169.4)^2 + (-25.5)^2} \approx 171.3 \text{ N} \)

This result helps us visualize the elbow force, which opposes the combined effects of the biceps and weights held, maintaining stability.

Vector Components

Breaking forces into vector components is a fundamental skill in understanding physics problems involving equilibrium. Imagine trying to analyze a single, complex force. It is far easier to break it down into simpler perpendicular parts.

In this exercise, the force exerted by the biceps is resolved into horizontal and vertical components. Here’s how it works:

• The horizontal component for the biceps, \( F_{b_x} \), is determined using the cosine of the angle (43 degrees):\( F_{b_x} = 232 \cdot \cos(43^\circ) \approx 169.4 \text{ N} \)

This shows how much of the force pulls sideways, which must be counteracted by the elbow force to keep the arm steady.

When dealing with forces and equilibrium, always break down forces into components to solve for individual impacts horizontally and vertically. This method simplifies complex physics problems and allows you to tackle each force individually, providing clear understanding and solutions.