Identifying the limit of proportionality

• What does the equation $F =$$kx$ represent?


  Force equals spring constant times extension
• What occurs beyond the limit of proportionality in materials?

  Elastic deformation followed by plastic deformation
• What is elastic deformation?

  Material returns to original shape after force removal
• What is the elastic limit of the material shown in the graph?

  1%
• What is the relationship between force and extension according to Hooke's Law?

  They are directly proportional to each other
• If the material is loaded beyond the elastic limit, what type of behavior will it exhibit?

  Plastic behavior
• According to the graph, what is the force when the extension is 0.08 m?

  4 N
• What is the title of the graph shown in the image?

  Fluorescence Calibration Graph
• What is the equation of the linear regression line shown in the graph?

  y = 1.92440x + 1.54167
• What happens during plastic deformation?

  Material undergoes permanent changes in shape
• What is the dependent variable shown on the y-axis of the graph?

  Intensity (arbitrary units)
• What is the force-extension relationship beyond the limit of Hooke's Law?

  Non-linear
• What are the different regions of the stress-strain curve?

  • Elastic behavior
  • Plastic behavior
  • Fracture point
• What are the key differences between elastic and plastic behavior in the stress-strain curve?

  • Elastic behavior: Material deforms reversibly and returns to original shape/size when load is removed
  • Plastic behavior: Material deforms irreversibly and does not return to original shape/size when load is removed
• What are the key features of the fluorescence calibration graph that indicate it is a reliable and accurate method for quantifying unknown concentrations?

  • Linear relationship between concentration and fluorescence intensity
  • High coefficient of determination (R^2 = 0.99848) indicating excellent fit of the linear model
  • Wide range of concentrations tested (0 to 25 µg/mL)
  • Consistent, reproducible measurements as indicated by the tight clustering of data points
• What is the coefficient of determination (R^2) of the linear regression line?

  0.99848
• What does the high R^2 value indicate about the linear relationship between concentration and fluorescence intensity?

  • The linear regression line provides an excellent fit to the data
  • There is a strong linear correlation between concentration and fluorescence intensity
  • The model explains a very high proportion (99.848%) of the variability in the data
• What does the linear region of the graph represent?

  Where force and extension are directly proportional
• What does the coefficient in the equation $y =$$1.9244x +$$1.54167$ represent?


  The spring constant
• What is the independent variable shown on the x-axis of the graph?

  Concentration (µg/mL)
• What is the linear region of a force-extension graph?

  It is where the graph is a straight line.
• What is the equation that describes the linear region?

  $F =$$kx$
• What is the term for the end of the linear region?

  Limit of proportionality.
• What does the limit of proportionality signify in relation to Hooke's Law?

  It indicates the end of the linear force-extension relationship
• If the fluorescence intensity of an unknown sample is measured to be 25 arbitrary units, what would be its estimated concentration based on the linear regression equation?

  Using the equation y = 1.92440x + 1.54167:

  Rearranging to solve for x:

  x = (y - 1.54167) / 1.92440

  x = (25 - 1.54167) / 1.92440

  x = 12.2 µg/mL
• How does the linear region appear on the graph?

  As a straight line relationship between variables
• What does the linear region indicate about force and extension?

  They are directly proportional.
• What is the point on the graph where the linear relationship between force and extension ends called?

  Limit of proportionality
• What is the limit of proportionality?

  The point where Hooke's Law stops working
• What happens to extension when force exceeds the limit of proportionality?

  Extension won't increase proportionally
• What happens at the point where the straight line starts to curve?

  The linear region ends.
• What does the equation $y =$$1.9244x +$$1.54167$ indicate about the relationship between force and extension?


  It shows a linear relationship with a specific slope
• Why is the end of the linear region significant?

  It indicates material no longer obeys Hooke's Law.
• What is Hooke's Law?

  • States that force is proportional to extension
  • Applies within the elastic limit of materials
  • Can be expressed as $F =$$kx$ (where $F$ is force, $k$ is the spring constant, and $x$ is extension)
• If a material is loaded beyond the proportional limit, what type of behavior will it exhibit?

  Plastic behavior
• What are the two types of deformation that occur beyond the limit of proportionality?

  • Elastic deformation
  • Returns to original shape when force is removed
  • Plastic deformation
  • Results in permanent changes to shape
• What quantity does the x-axis of the graph represent?

  Extension in meters
• What law describes the relationship in the linear region?

  Hooke's Law
• How does the relationship between force and extension change after the limit of proportionality?

  The relationship is no longer straight-line
• How does elastic deformation differ from plastic deformation?

  Elastic deformation is reversible; plastic is not