91Ó°ÊÓ

BIO Forearm. In the human arm, the forearm and hand pivot about the elbow joint. Consider a simplified model in which the biceps muscle is attached to the forearm 3.80 \(\mathrm{cm}\) from the elbow joint. Assume that the person's hand and forearm together weigh 15.0 \(\mathrm{N}\) and that their center of gravity is 15.0 \(\mathrm{cm}\) from the elbow (not quite halfway to the hand). The forearm is held horizontally at a right angle to the upper arm, with the biceps muscle exerting its force perpendicular to the forearm. (a) Draw a free-body diagram for the forearm, and find the force exerted by the biceps when the hand is empty. (b) Now the person holds a 80.0 -N weight in his hand, with the forearm still horizontal. Assume that the center of gravity of this weight is 33.0 \(\mathrm{cm}\) from the elbow. Construct a free-body diagram for the forearm, and find the force now exerted by the biceps. Explain why the biceps muscle needs to be very strong. (c) Under the conditions of part (b), find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) While holding the \(80.0-\mathrm{N}\) weight, the person raises his forearm until it is at an angle of \(53.0^{\circ}\) above the horizontal. If the biceps muscle continues to exert its force perpendicular to the forearm, what is this force when the forearm is in this position? Has the force increased or decreased from its value in part (b)? Explain why this is so, and test your answer by actually doing this with your own arm.

Step by step solution

01

Analyze the Free-Body Diagram for Part (a)

To solve Part (a), start by identifying all forces acting on the forearm. The forces include the weight of the forearm and hand (15.0 N acting downward at 15.0 cm from the elbow) and the force exerted by the biceps at 3.8 cm from the elbow. We also consider the reaction force at the elbow, though it won't affect our torque calculations about the elbow.

02

Compute the Biceps Force in Part (a) Using Torque Balance

To find the force exerted by the biceps, use the principle of torque equilibrium about the elbow:\( T_{biceps} + T_{weight} = 0 \)Thus,\(-F_{biceps} \cdot 0.038 + 15.0 \cdot 0.15 = 0 \).Solving for the biceps force:\( F_{biceps} = \frac{15.0 \times 0.15}{0.038} \approx 59.2 \, \text{N} \).

03

Update the Free-Body Diagram for Part (b)

When an 80.0 N weight is added, it acts downward at 33.0 cm from the elbow. Include this weight in the free-body diagram along with the previously considered forces.

04

Calculate the Biceps Force for Part (b)

Use torque balance about the elbow again:\(-F_{biceps} \cdot 0.038 + 15.0 \cdot 0.15 + 80.0 \cdot 0.33 = 0 \).Solving for \( F_{biceps} \):\( F_{biceps} = \frac{15.0 \times 0.15 + 80.0 \times 0.33}{0.038} \approx 731.58 \, \text{N} \).This large force demonstrates the requirement for the biceps muscle to be very strong due to the mechanical disadvantage of its leverage.

05

Determine the Elbow Joint Force in Part (c)

With all forces and distances established, set the sum of vertical forces to zero:\( F_{biceps} - 15.0 - 80.0 + F_{elbow} = 0 \).Solve for the elbow force:\( F_{elbow} = 15.0 + 80.0 - 731.58 \, \text{N} \approx 636.58 \, \text{N} \), upward.

06

Calculate New Biceps Force for Part (d)

When the forearm is at a 53° angle, we consider both components of each force. For simplicity, only recalculate torques about the elbow considering the perpendicular component of the weight and arm forces.\[ \text{Torque due to 15.0 N} = 15.0 \times 0.15 \times \cos(53°) \]\[ \text{Torque due to 80.0 N} = 80.0 \times 0.33 \times \cos(53°) \] Set total torque to be balanced by the biceps:\[ F_{biceps} \cdot 0.038 = 15.0 \times 0.15 \times \cos(53°) + 80.0 \times 0.33 \times \cos(53°) \]Solving for \( F_{biceps} \) gives:\[ F_{biceps} = \frac{15.0 \times 0.15 \times \cos(53°) + 80.0 \times 0.33 \times \cos(53°)}{0.038} \approx 440.4 \, \text{N} \]. The force has decreased because the angle reduces the component of gravitational force requiring compensation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Biceps Muscle Force

The biceps muscle plays a crucial role in lifting your forearm and any weight it might carry. When you're lifting your forearm, the biceps are essentially pulling up to balance the forces acting on your forearm.

In the exercise, we considered two main scenarios: holding the hand empty and holding an 80 N weight.

• In the first scenario, the force exerted by the biceps was calculated by balancing the torque due to the weight of the forearm and hand.
• In the second scenario, when an additional weight is added, the force required from the biceps increases substantially.

Torque equilibrium tells us that the further a force is from the pivot (the elbow), the more leverage it exerts. But because the biceps attachment is very close to the elbow, it needs to exert a much larger force to achieve equilibrium.

This is why the biceps have to be incredibly strong. Even a small change in the weight or the position of the forearm significantly affects the force the biceps must generate, demonstrating the significant role of mechanical leverage in physical tasks.

Free-Body Diagram

A free-body diagram is a helpful tool in physics for visualizing the forces acting upon an object. It simplifies complex interactions into clear, understandable arrows.

For our forearm exercise, the free-body diagrams showed:

• The weight of the forearm and hand, acting downwards, was labeled as 15 N at a distance of 15 cm from the elbow.
• The force from the biceps, labeled as acting upwards at a much shorter distance of 3.8 cm from the elbow.
• When an additional weight was added, another force arrow was introduced for the 80 N weight at 33 cm from the elbow.

These diagrams help us set up the torque balance equations by clearly laying out where each force acts and in which direction. They are fundamental to solving mechanical problems and understanding how different forces interact with each other.

Mechanical Advantage

Mechanical advantage refers to using a tool or a system to make a task easier. In biomechanics, it's about how the body's structure makes certain tasks more or less difficult.

In the context of the forearm and the biceps:

• The biceps have a mechanical disadvantage because they are attached close to the elbow, requiring them to generate a high force to lift heavy objects or extend the arm.
• This setup is effective for speed and range of motion, allowing the arm to move quickly and rotate considerably. However, it means the biceps muscle must be strong to handle the leverage.

Understanding mechanical advantage explains why some tasks feel easier or require less effort when optimizing the body's positions or using tools. In sports and rehabilitation, enhancing mechanical advantage can improve performance and reduce fatigue for tasks involving force balance.