Work/Energy + Torque

Chapter 4: Work, Power, and Energy

Definitions

  • Work: The ability to do work is fundamentally the force applied to an object multiplied by the distance over which that force is applied in the direction of the force.

  • Power: The rate of work done, expressed as the amount of work done per unit of time. Can be mathematically represented as: P = \frac{W}{t} or equivalently P = Fv (where F = Force and v = Velocity).

    • Units of power are Watts (W), where 1 W = 1 J/s.

  • Efficiency: A measure of how effectively energy or work is transformed into useful output. Its formula is:
    \text{Efficiency} = \frac{\text{Mechanical Work Performed}}{\text{Metabolic Energy Consumed}}

Kinetic and Potential Energy

  • Kinetic Energy (KE): Energy possessed by an object due to its motion, expressed as:
    KE = \frac{1}{2} m v^2
    where m is mass and v is velocity.

  • Potential Energy (PE): Stored energy due to an object's position or configuration. The gravitational potential energy is expressed as:
    GPE = mgh
    where g is acceleration due to gravity and h is height above the ground.

  • Strain Potential Energy (SPE): Energy stored due to deformation of materials, given by:
    SPE = \frac{1}{2} k \Delta x^2
    where k is stiffness and \Delta x is displacement.

Conservation of Mechanical Energy

  • Law of Conservation of Mechanical Energy: When gravity is the only external force acting on a body, the mechanical energy (kinetic + potential) of that body remains constant:
    KE + PE = C

  • This principle implies that energy can convert from one form to another (e.g., from potential to kinetic) but the total energy remains unchanged.

Principle of Work and Energy

  • The net work done by all external forces (not just gravity) acting on an object results in a change in the energy of the object:
    Fd = \Delta \text{Total Mechanical Energy} = \Delta KE + \Delta PE

Efficiency and Economy

  • Efficiency Calculation: To determine efficiency, you can express the change in work done relative to the energy consumed:
    \Delta \text{net mechanical work} = \Delta \text{total mechanical energy}

  • Economy: Defined as metabolic energy used per task.

Example: Robert Running

  • When Robert, who weighs 50 kg, runs up a 20-degree hill at 3 m/s:

    • Mechanical Work to Raise CoM: Calculate the work done by Robert over one minute.

    • He expended 100,000 J of metabolic energy. Calculate efficiency based on this.

Uphill vs. Downhill Running

  • When considering uphill vs downhill:

    • Robert performs the same amount of work in absolute terms (magnitude) for both directions.

    • Work done is positive when running uphill and negative when running downhill.

    • Investigate the metabolic energy requirements related to muscle function under different gravitational conditions.

Muscular Contraction Types

  • Concentric Contraction: Muscle shortens while exerting force.

  • Isometric Contraction: Muscle exerts force without changing length.

  • Eccentric Contraction: Muscle lengthens under tension.

Muscles in Action

  • Agonist Muscles: Primary movers that contract to create movement.

  • Antagonist Muscles: Muscles that oppose the motion created by agonists.

  • Example in a push-up: Agonist and antagonist work in dynamic balance to produce effective movement.

Impact Testing with Gymnastics Mat

  • An example of practical application: A 20 kg cylindrical mass dropped from a height of 1m compresses a 10 cm thick gymnastics mat to 4 cm.

Work Calculations from Impact

  • Calculate the kinetic energy at impact:

    • KE = mgh where h is the height of drop (1m).

  • The work done by the mat can be calculated, indicating how much energy was absorbed.

Mechanical Power

  • Power output relates to how quickly work is done.

    • High forces at low velocities or low forces at high velocities can yield similar power outputs if the product of force and velocity is equivalent.

Torque

  • Torque is the measure of the rotational force applied to an object and is defined mathematically as: T = F d_{\perp} where d_{\perp} represents the perpendicular distance from a line of action of the force to the axis of rotation.

    • Measured in Newton-meters (Nm).

Biceps Curl Example

  • In a biceps curl, the torque experienced due to an applied force by a dumbbell requires calculations of various forces, including muscle force, and how they relate to the torque produced:

    • Example: A mass of 10 kg at a distance of 0.5m from the elbow joint can create torque determined by upward flexion or downward extension.

Equilibrium Conditions

  • In static equilibrium:

    • The sum of all external forces and external torques must equal zero:

    • \Sigma F = 0

    • \Sigma T = 0

Muscle-Related Mechanics

  • The moment arm from muscle insertion to joint center significantly affects force exertion.

  • Understanding muscle pennation angle and its implications on force production efficiency.

Summary

  • Focused understanding of energy, calculations of work and torque, dynamics of muscle-tendon interactions, and principles of static equilibrium.

  • Each concept builds towards a comprehensive understanding of biomechanics and energy efficiency in movement.