Force-Length Relationship

Examples of Force-Length Relationship in the following topics:

• Force of Muscle Contraction

  • The force a muscle generates is dependent on its length and shortening velocity.
  • The force a muscle generates is dependent on the length of the muscle and its shortening velocity.
  • If this attachment was removed, for example if the bicep was detached from the scapula or radius, the muscle would shorten in length.
  • In mammals, there is a strong overlap between the optimum and actual resting length of sarcomeres.
  • The force-velocity relationship in muscle relates the speed at which a muscle changes length with the force of this contraction and the resultant power output (force x velocity = power).

• Velocity and Duration of Muscle Contraction

  • The shortening velocity affects the amount of force generated by a muscle.
  • The force-velocity relationship in muscle relates the speed at which a muscle changes length to the force of this contraction and the resultant power output (force x velocity = power).
  • The force generated by a muscle depends on the number of actin and myosin cross-bridges formed; a larger number of cross-bridges results in a larger amount of force.
  • Though they have high velocity, they begin resting before reaching peak force.
  • As velocity increases force and power produced is reduced.

• Fitting a line by eye

  • We want to describe the relationship between the head length and total length variables in the possum data set using a line.
  • In this example, we will use the total length as the predictor variable, x, to predict a possum's head length, y.
  • We could fit the linear relationship by eye, as in Figure 7.7.
  • A scatterplot showing head length against total length for 104 brushtail possums.
  • A point representing a possum with head length 94.1mm and total length 89cm is highlighted.

• Stress and Strain

  • The ratio of force to area $\frac{F}{A}$ is called stress and the ratio of change in length to length $\frac{\Delta L}{L}$ is called the strain.
  • A change in shape due to the application of a force is a deformation.
  • Even very small forces are known to cause some deformation.
  • The ratio of force to area $\frac{F}{A}$ is called stress and the ratio of change in length to length $\frac{\Delta L}{L}$ is called the strain.
  • Tension: The rod is stretched a length $\Delta L$ when a force is applied parallel to its length.

• Elasticity, Stress, and Strain

  • Even very small forces are known to cause some deformation.
  • Strain is the change in length divided by the original length of the object.
  • Experiments have shown that the change in length (ΔL) depends on only a few variables.
  • Additionally, the change in length is proportional to the original length L0 and inversely proportional to the cross-sectional area of the wire or rod.
  • Tension: The rod is stretched a length ΔL when a force is applied parallel to its length.

• Muscles and Joints

  • Viewing them as simple machines, the input force is much greater than the output force, as seen in .
  • Very large forces are also created in the joints.
  • Because muscles can contract but not expand beyond their resting length, joints and muscles often exert forces that act in opposite directions, and thus subtract.
  • Forces in muscles and joints are largest when their load is far from the joint.
  • Training coaches and physical therapists use the knowledge of the relationships between forces and torques in the treatment of muscles and joints.

• Scientific Applications of Quadratic Functions

  • Quadratic relationships between variables are commonly found in physical sciences, engineering, and elsewhere.
  • Quadratic relationships between variables are commonly found in physical sciences, engineering, and elsewhere.
  • Perhaps the most universally used example of quadratic relationships in problem solving concerns right triangles.
  • This says that the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the two legs ($a$ and $b$) of the triangle.
  • The equation relating electrostatic force ($F$) between two particles, the particles' respective charges ($q_1$ and $q_2$), and the distance between them ($r$) is very similar to the aforementioned formula for gravitational force:

• Beginning with straight lines

  • Such plots permit the relationship between the variables to be examined with ease.
  • Figure 7.4 shows a scatterplot for the head length and total length of 104 brushtail possums from Australia.
  • The head and total length variables are associated.
  • Possums with an above average total length also tend to have above average head lengths.
  • Straight lines should only be used when the data appear to have a linear relationship, such as the case shown in the left panel of Figure 7.6.

• Length

  • Length is a physical measurement of distance that is fundamentally measured in the SI unit of a meter.
  • Length can be defined as a measurement of the physical quantity of distance.
  • The distance between objects, the rate at which objects are traveling, and how much force an object exerts are all dependent on length as a variable.
  • Many different units of length are used around the world.
  • The basic unit of length as identified by the International System of Units (SI) is the meter.

• Bond Energy

  • Similarly, the C-H bond length can vary by as much as 4% between different molecules.
  • At internuclear distances in the order of an atomic diameter, attractive forces dominate.
  • At very small distances between the two atoms, the force is repulsive and the energy of the two atom system is very high.
  • The attractive and repulsive forces are balanced at the minimum point in the plot of a Morse curve.
  • Identify the relationship between bond energy and strength of chemical bonds