Question

In the movie Terminator, Arnold Schwarzenegger lifts someone up by the neck and, with both arms fully extended and horizontal, holds the person off the ground. If the person being held weighs \(700 \mathrm{N},\) is \(60 \mathrm{cm}\) from the shoulder joint, and Arnold has an anatomy analogous to that in Fig. \(8.30,\) what force must each of the deltoid muscles exert to perform this task?

Short answer

Each deltoid muscle exerts 3500 N to hold the person.

Step-by-step solution

Understand the Problem

We need to find the force exerted by Arnold's deltoid muscles to hold a person weighing 700 N at a distance of 60 cm from his shoulder joint. Each of Arnold's arms is held horizontally, which means the muscles must provide the torque required to counteract the torque due to the weight of the held person.

Identify the Forces and Distances

The person is held at a distance of 60 cm (or 0.6 m) from the shoulder joint. The force due to gravity on the person is 700 N. The deltoid muscle applies force at a smaller distance to create the necessary torque to balance the system.

Write the Torque Balance Equation

For equilibrium, the total torque about the shoulder joint must be zero. This means the torque due to the person's weight is equal to the torque produced by the deltoid muscle. The torque equation is: \[ F_d \times d_d = 700 \times 0.6 \]where \( F_d \) is the force exerted by the deltoid muscle, and \( d_d \) is the perpendicular distance from the shoulder joint to where the deltoid force acts.

Solve for the Deltoid Muscle Force

Assuming the deltoid force acts at a typical anatomical angle (e.g., 12 cm or 0.12 m, based on typical human anatomy), substitute values into the torque equation:\[ F_d \times 0.12 = 700 \times 0.6 \]Simplify to find \( F_d \):\[ F_d = \frac{700 \times 0.6}{0.12} = 3500 \, \text{N} \]

Account for Each Arm

Since the force calculated is per deltoid muscle and Arnold uses both arms to hold the person, this force applies to each deltoid muscle independently. Therefore, each deltoid muscle exerts a force of 3500 N.

Key concepts

Understanding Newton's Laws of Motion

Newton's laws of motion are fundamental in understanding how objects interact through forces. They explain that an object will remain in its state of motion unless acted upon by a non-zero net external force. Here, we see this principle at work as Arnold Schwarzenegger lifts someone off the ground. His muscles exert a force that exactly balances the gravitational force pulling the person down. This is an application of the first law, which is also known as the law of inertia.

• Newton's First Law: An object in motion stays in motion unless a force acts on it.
• Newton's Second Law: The force acting on an object is equal to the mass of that object times its acceleration (n=man). Here it helps us calculate how much force the deltoid muscles need to exert.
• Newton's Third Law: For every action, there is an equal and opposite reaction. Arnold's arms exert a force upward to hold the weight, while the earth pulls downward.

This exercise is a classic example where balancing forces and understanding motion laws come into play. By calculating force, we effectively manage to stop any unwanted motion securely.

Principle of Force Equilibrium

Force equilibrium occurs when all the forces acting on an object result in no net force, keeping the object in a stable condition. In this exercise, force equilibrium explains why Arnold can hold the person without them falling. The sum of all vertical forces must be zero because the person is at rest.
The key to achieving equilibrium is having the upward force by Arnold's arms equal to the downward gravitational force of the person. Thus, the torque produced by the deltoid muscle force exactly balances the torque caused by the person's weight.
The equation \[F_d \times d_d = W \times d_w\]helps us ensure this balance. Here:

• \(F_d\) represents the deltoid force.
• \(d_d\) represents the distance from the shoulder joint to where the deltoid muscle acts.
• \(W\) represents the weight of the person.
• \(d_w\) represents the distance from the shoulder joint to where the weight of the person acts.

Ensuring equilibrium is crucial in biomechanics as it prevents unwanted movements and maintains stability.

Fundamentals of Biomechanics

Biomechanics combines principles of physics, like motion and force, with anatomical systems to understand movement in the human body. This field enables a deeper insight into how muscle forces enable complex actions like lifting.
For Arnold to lift the person, biomechanics studies help describe how muscle forces produce torque around joints. The deltoid muscle plays a crucial role in shoulder movement, essential in this scenario.
To perform any movement efficiently:

• Muscle forces must be optimized to produce the necessary torque.
• Acceleration or velocity of limbs can affect force production.
• Understanding lever systems in the body helps in distributing forces effectively.

Biomechanics not only helps athletes like Arnold optimize their strength but also assists in designing supportive equipment to aid movements more safely.

Insights into Anatomy and Physiology

Anatomy and physiology focus on the structure and function of the body, respectively. In this instance, understanding shoulder anatomy and the role of the deltoid muscle is crucial.
The shoulder is a complex joint with the deltoid muscle playing a critical role in arm movement. The deltoid helps lift the arm, which is essential for holding something horizontally. The power of the deltoid muscle allows it to generate the necessary torque for the tasks, as demonstrated in the exercise.

• The shoulder joint functions as a center of rotation.
• The deltoid muscle covers the shoulder and attaches to the collarbone and the shoulder blade.
• Its fibers work together to move the arm in different directions.

Understanding the interplay between anatomy and physiology allows us to grasp how muscles exert force, thus contributing to biomechanics effectively.

The Role of Lever Arm Distance

Lever arm distance is the perpendicular distance from the axis of rotation to the line of action of the force. This concept is pivotal in calculating torque, which is the rotational equivalent of linear force.
In this exercise, the person being lifted acts as a weight, generating torque around Arnold's shoulder joint. The torque can be calculated by multiplying the force by the lever arm distance, in this case, the distance from the shoulder to the position where the weight is held.

Key Elements:

• Torque expression: \( \tau = F \times d \)
• Where \( F \) is the force applied and \( d \) is the lever arm distance.
• The larger the distance, the greater the torque for the same force.
• Positioning of muscles impacts leverage and force efficiency.

Understanding lever arm distances is instrumental in applying the right amount of force efficiently, which ultimately maximizes performance in various physical activities.