Levers in Biological Systems

Introduction to Levers

• Levers are crucial for animal movement.
• Animals use convergent mechanisms: muscles pull on skeletal elements, which then push on the environment.
• Movement includes flying, swimming, and running, facilitated by muscles acting on skeletal elements.
• Skeletal muscular systems function as levers, with a fixed point around which rotation occurs.
• Applied force pulls up, and a load is lifted against gravity.
• Forces are applied at varying distances from the pivot point.

Levers in the Arm

• The biceps muscle attaches to the lower arm bone.
• Muscle contraction causes the arm to rotate around the elbow (pivot point).
• This rotation allows lifting a load against gravity.
• The setup enables work to be done.

Qualitative Definition of Levers

• Forces are input and output around a pivot point.
• Lever systems can either increase force output (mechanical advantage) or increase movement output (velocity advantage).
• Biological levers typically do not provide mechanical advantage; more force is often required as input than is obtained as output.
• Velocity advantage: small muscle movements result in large bone movements, either over a large distance or at high speed.
• Velocity is achieved through large movements over a short period.
• Biological levers commonly utilize velocity advantage.

Simple Lever System: Seesaw

• Pivot point in the middle with two sides that rotate up and down.
• Force applied on one side at a distance from the pivot point lifts a load on the other side, also at a distance.
• If distances are equal, input force equals the lifted load.

Three Types of Lever Systems

Type 1

• Force and load are on opposite sides of the pivot.

Type 2

• Load and force on the same side, with the load closer to the pivot.

Type 3

• Load and force on the same side, with the force closer to the pivot.

Examples

• Type 1: Seesaw, where force and load (kids) are on either side of the pivot point.
• Type 2: Wheelbarrow, where the wheel is the pivot; the load (kids) is closer to the pivot than the applied force (dad).
• Type 3: Pivot point at a joint, load at the end, and force applied in the middle.

Biological Lever Systems

• Commonly of two types (Type 1 and Type 3).

Type 1 in Biological Systems

• Pivot point (ankle) in the middle.
• Achilles tendon applies force on the back of the foot.
• Toes push down, with force and load on opposite sides.
• Example: Wiggling toes, pivoting about the ankle by contracting the Achilles tendon.

Type 3 in Biological Systems

• More common setup.
• Pivot point at a joint (e.g., knee).
• Muscle and tendon cross the pivot point, attaching on the far side (force application).
• Load is farther down (e.g., the foot).
• Applied force is close to the pivot point, and the load is much farther away.

Quantitative Analysis of Lever Systems

• Pivot point with distances measured as $d$.
• $d_{muscle}$: Distance from pivot to muscle force.
• $d_{load}$: Distance from pivot to load force.
• $F_{muscle}$: Muscle force applied.
• $F_{load}$: Load force lifted.
• Variables: $F<em>{muscle}$, $F</em>{load}$, $d<em>{muscle}$, $d</em>{load}$.

Lever Equation

• Torques or angular forces about the pivot point must be equal.
• Torque: Force acting perpendicular to the distance.
• Equation: $F<em>{muscle} \\times d</em>{muscle} = F<em>{load} \times d</em>{load}$.

Rearranging the Lever Equation

• Solving for load: $F<em>{load} = F</em>{muscle} \times \frac{d<em>{muscle}}{d</em>{load}}$.
• Symmetric case: If $d<em>{muscle} = d</em>{load}$, then $F<em>{load} = F</em>{muscle}$.

Mechanical Advantage

• Joint closer to load: $d_{load}$ is smaller.
• Results in more load force out than input force.
• Ratio of load to muscle force is greater than one.
• Typical in engineering applications.
• Example: Crowbar, where a small input force creates a large output force.

Biological Levers and Velocity Advantage

• Muscle acting close to the pivot point or joint.
• $F<em>{load} = F</em>{muscle} \times \frac{d<em>{muscle}}{d</em>{load}}$.
• Load distance is large, so $F<em>{load} < F</em>{muscle}$.
• Not ideal for mechanical advantage.
• Common in biological levers.

Advantages of Biological Lever Systems

• Easy muscle application.
• When a muscle contracts a small distance, the load moves a large distance.
• $x_{muscle}$: Distance muscle moves.
• $x_{load}$: Distance load moves.

Similar Triangles and Velocity

• Relating distances: $\frac{d<em>{load}}{d</em>{muscle}} = \frac{x<em>{load}}{x</em>{muscle}}$.
• Velocities: $\frac{d<em>{load}}{d</em>{muscle}} = \frac{v<em>{load}}{v</em>{muscle}}$.
• The load moves much further in the same amount of time, meaning the load has a bigger velocity compared to what the muscle is doing.
• Velocity advantage: Load velocity is greater than muscle velocity.

Mechanical Ratio vs. Velocity Ratio

• Mechanical Ratio: $\frac{F<em>{load}}{F</em>{muscle}}$.
• Velocity Ratio: $\frac{v<em>{load}}{v</em>{muscle}}$.
• Relationship to distances:
  • $\frac{F<em>{load}}{F</em>{muscle}} = \frac{d<em>{muscle}}{d</em>{load}}$.
  • $\frac{v<em>{load}}{v</em>{muscle}} = \frac{d<em>{load}}{d</em>{muscle}}$.
• The two ratios are inversely proportional.

Implications for Engineering vs. Biological Systems

• Engineering systems: Designed for mechanical ratio, with typically the input distance bigger than the output distance.
• Biological systems: The output distance is almost always much greater than the muscle distance.

Biological Levers - Advantage

• $d<em>{load} > d</em>{muscle}$, so the ratio is less than one.
• Mechanical ratio is less than one.
• Biological systems: High velocity ratio, offering a velocity advantage.
• Increased velocity output compared to muscle input.

Biological Examples in Nature's Design

Bear Leg Example

• The leg of a bear: femur, knee, tibia/fibula, tarsals/metatarsals.
• Muscles attach to the back of the foot, causing the toes to push down when contracted.
  Muscle force input, load force output to potentially dig.
• Has the leg system a mechanical advantage or velocity advantage?
  Calculating mechanical ratio and velocity ratio.

Animal Comparisons - Speed vs. Strength

• Deer example is for speed.
• Dogs are somewhere in between bear and Deer.

Muscle Fiber Types

• Muscle fiber types are important to muscle design.
• Either long thinner muscles or shorter muscles:

*** The long thinner muscles allow for excursions, great changes from contraction.
*** Shorter fatter muscles will have more strands and perform better for force.

Conclusion

• Diverse ecologies lead to evolved biological machines: skeletal systems, muscular systems.
• Levers allow the interplay of systems for running, strength, or compromise.
• Muscles move skeletons, with biological lever systems generally favoring velocity over mechanical advantage for speed ratios.
• Slight systemic changes will allow for less mechanical disadvantage and will get the organism more strength.