Statics, Torque, Equilibrium, Levers, and Friction in the Human Body

Statics

Introduction to Statics

Statics is the branch of physics concerned with the analysis of forces on physical systems in static equilibrium, where the system is at rest or moving at constant velocity. In biological contexts, statics is essential for understanding how forces act on the human body and how stability is maintained.

Torque

Definition of Torque

Torque is the tendency of a force to produce rotation about an axis. It is a fundamental concept in rotational dynamics and is crucial for analyzing the mechanics of the human body and machines.

• Symbol: $L$ (sometimes $\tau$ in other texts)

• Formula: The torque produced by a force $F$ at a distance $d$ (lever arm) from the axis of rotation is given by:

$L = F \cos \theta \times d$

• Lever arm (d): The perpendicular distance from the axis of rotation to the line of action of the force. Also called the moment arm.

Example: Opening a door by pushing at the edge (far from the hinge) produces more torque than pushing near the hinge.

Equilibrium and Stability

Conditions for Equilibrium

A body is in static equilibrium if both the vector sum of all forces and the vector sum of all torques acting on it are zero. This ensures the body does not accelerate linearly or rotationally.

• Translational Equilibrium: $\sum \vec{F} = 0$

• Rotational Equilibrium: $\sum \tau = 0$

Centre of Mass: The point at which the total mass of a body can be considered to be concentrated for the purpose of analyzing translational motion.

Centre of Gravity: The point through which the force of gravity acts on a body.

Stability

The stability of a body depends on the position of its centre of mass relative to its base of support.

• If the centre of mass is above the base of support, the body is stable.

• If the centre of mass moves outside the base, the body becomes unstable and may topple.

Example: A triangle with its centre of mass above its base is stable; if the centre of mass is outside the base, it is unstable.

Equilibrium in the Human Body

For an erect person, the centre of gravity is approximately 56% of their height from the soles of the feet. The centre of gravity shifts as the person moves or bends. Maintaining balance requires keeping the centre of gravity above the feet.

• Carrying an uneven load causes the body to adjust limb positions to maintain balance.

Stability Under External Forces

When external forces act on the body (e.g., a push to the shoulder), the body may topple if the applied torque exceeds the restoring torque due to weight.

• Example Calculation: For a person 1.5 m tall (feet to neck), foot width 0.1 m, mass 70 kg:

Restoring torque due to weight: $T_w = W \times 0.1$

Weight: $W = mg = 70 \times 9.8 = 686\ \text{N}$

$T_w = 686 \times 0.1 = 68.6\ \text{N m}$

Applied torque to topple: $T_a = F_a \times 1.5$ Set $T_a = T_w$

for toppling: $F_a = \frac{68.6}{1.5} = 45.7\ \text{N}$

Interpretation: A force of 45.7 N applied at the shoulder is sufficient to topple the person.

Skeletal Muscles and Levers

Skeletal Muscles

Skeletal muscles are composed of thousands of parallel fibres, which contract in response to electrical stimuli from nerves. Muscle contraction produces a pulling force on the bones to which the muscle is attached.

• Muscles may end in one, two, or three tendons (e.g., biceps, triceps).

• Muscles attach to different bones, allowing movement at joints.

• The force a muscle can exert is proportional to its cross-sectional area.

• Maximum muscle force: about $7 \times 10^5$ Pascals.

Example: More muscle fibres contracting simultaneously produce a stronger force.

Levers in the Human Body

Joints in the body can be analyzed as levers, which are rigid bars that rotate about a fixed point called a fulcrum. Levers allow the body to lift loads efficiently and transfer movement.

• Assumptions: Tendons attach at well-defined points; joints are frictionless.

• Three Classes of Levers: (not detailed in the notes, but standard in physics)

Example: The forearm acts as a lever when lifting an object, with the elbow as the fulcrum.

Additional info: The three classes of levers are:

• First class: Fulcrum between effort and load (e.g., neck muscles).

• Second class: Load between fulcrum and effort (e.g., standing on tiptoe).

• Third class: Effort between fulcrum and load (e.g., biceps curl).

Friction

Nature of Friction

Friction is the resistance to motion that occurs when two surfaces are in contact. Even apparently smooth surfaces have microscopic irregularities that interlock, causing friction.

• Types of Friction: Sliding (kinetic), rolling, and fluid (viscous) friction.

• Coefficient of friction ($\mu$): A property of the surfaces in contact.

Friction Force Formula

The frictional force $F_f$ is given by:

$F_f = \mu F_n$

• $F_n$ is the normal (perpendicular) force pressing the surfaces together.

• The rougher the surfaces, the greater the friction force.

Example: Pushing a block across a table requires a force to overcome friction.

Friction in Biological Systems

In biological systems, friction is important in joints and in the flow of fluids (e.g., blood). Sliding friction is independent of velocity, but fluid (viscous) friction increases with velocity.

• Friction dissipates kinetic energy as heat, eventually stopping moving objects.

• Friction is necessary for walking and balance, providing the reaction force needed for movement.

• Friction can cause wear and tear; lubrication (e.g., oil, synovial fluid) reduces friction and wear.

Friction in Joints

Joints experience significant forces during movement. Frictional wear is minimized by smooth cartilage and synovial fluid, which lubricate the contact areas.

• When walking, the force on the joint can be about 2.4 times body weight.

• Each step rotates the joint about 60°, with a radius of about 3 cm, resulting in a sliding distance of about 3 cm per step.

• Work done against friction is the product of friction force and distance moved.

• Without lubrication, joints would wear rapidly and heat would damage tissues.

• Aging reduces cartilage and lubrication efficiency, leading to joint problems.

Example: By age 70, up to two-thirds of people have knee joint problems, and about one-third have hip problems.

Summary Table: Key Concepts

Concept	Definition	Formula	Example/Application
Torque	Tendency of a force to cause rotation	$L = F \cos \theta \times d$	Opening a door, toppling a person
Equilibrium	Sum of forces and torques is zero	$\sum \vec{F} = 0$, $\sum \tau = 0$	Standing upright, balancing
Friction	Resistance to motion between surfaces	$F_f = \mu F_n$	Walking, joint movement
Lever	Rigid bar rotating about a fulcrum	Varies by class	Forearm, neck, foot