Physics Study Guide
Nonlinear Relationships
Physics Study Guide
Nonlinear Relationships
Nonlinear Relationships
Figure 1-17 shows the distance a brass ball falls versus time. Note that the graph is not a straight line, meaning the relationship is not linear. There are many types of nonlinear relationships in science. Two of the most common are the quadratic and inverse relationships. The graph in Figure 1-17 is a quadratic relationship, represented by the following equation.
Quadratic Relationship Between Two Variables
$y = ax^2 + bx + c$
A quadratic relationship exists when one variable depends on the square of another.
A computer program or graphing calculator easily can find the values of the constants a, b, and c in this equation. In this case, the equation is d = 5t².
See the Math Handbook in the back of the book for more on making and using line graphs.
An object is suspended from spring 1, and the spring’s elongation (the distance it stretches) is $X_1$. Then the same object is removed from the first spring and suspended from a second spring. The elongation of spring 2 is $X_2$. $X_2$ is greater than $X_1$.
- On the same axes, sketch the graphs of the mass versus elongation for both springs.
- Is the origin included in the graph? Why or why not?
- Which slope is steeper?
- At a given mass, $X_2 = 1.6 \, X_1$. If $X_2 = 5.3 \, \text{cm}$, what is $X_1$?
The graph in Figure 1-18 shows how the current in an electric circuit varies as the resistance is increased. This is an example of an inverse relationship, represented by the following equation.
Inverse Relationship $y = \dfrac{a}{x}$
A hyperbola results when one variable depends on the inverse of the other.
The three relationships you have learned about are a sample of the simple relations you will most likely try to derive in this course. Many other mathematical models are used. Important examples include sinusoids, used to model cyclical phenomena, and exponential growth and decay, used to study radioactivity. Combinations of different mathematical models represent even more complex phenomena.
24. The mass values of specified volumes of pure gold nuggets are given in Table 1–4.
- Plot mass versus volume from the values given in the table and draw the curve that best fits all points.
- Describe the resulting curve.
- According to the graph, what type of relationship exists between the mass of the pure gold nuggets and their volume?
- What is the value of the slope of this graph? Include the proper units.
- Write the equation showing mass as a function of volume for gold.
- Write a word interpretation for the slope of the line.