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Magazine Chapter 3: Problem 12
Use the following table, which gives the valve lift \(L\) (in mm) of a certain cam as a function of the angle \(\theta\) (in degrees ) through which the cam is turned. Plot the values. Find the indicated values by reading the graph. $$\begin{array}{l|l|l|l|l|l|l|l|l} \theta\left(^{\circ}\right) & 0 & 20 & 40 & 60 & 80 & 100 & 120 & 140 \\ \hline L(\mathrm{mm}) & 0 & 1.2 & 2.3 & 3.3 & 3.8 & 3.0 & 1.6 & 0 \end{array}$$ For \(L=2.0 \mathrm{mm},\) find \(\theta.\)
Short Answer
Expert verified
\(\theta\) is approximately \(30^\circ\) for \(L=2.0\, \text{mm}\).
Step by step solution
01
Understanding the Table
The table provides a set of values for the valve lift \(L\) corresponding to different angles \(\theta\). The goal is to use these points to create a graph that will help in finding the angle \(\theta\) when the valve lift \(L\) is 2.0 mm.
02
Plotting the Given Values
Using graph paper or graphing software, plot the given points where \(\theta\) is along the x-axis and \(L\) is along the y-axis. The points to plot are: (0, 0), (20, 1.2), (40, 2.3), (60, 3.3), (80, 3.8), (100, 3.0), (120, 1.6), (140, 0). Connect these points smoothly to create a curve that represents the relationship between \(\theta\) and \(L\).
03
Drawing the Horizontal Line for L=2.0mm
On the graph, draw a horizontal line at \(L = 2.0\, \text{mm}\) across the y-axis. This line will intersect the curve at certain points. We need to find these intersection points to determine the corresponding \(\theta\) values.
04
Finding \(\theta\) Values from the Graph
Check where the horizontal line at \(L = 2.0\, \text{mm}\) intersects the plotted curve. Read off the \(\theta\) values directly from the x-axis where these intersections occur. If the line intersects the curve between two plotted points, estimate the value of \(\theta\) using interpolation.
05
Interpolating to Estimate \(\theta\)
Since the graph shows that \(L=2.0 \mathrm{mm}\) occurs approximately between \(\theta = 20^\circ\) and \(\theta = 40^\circ\), do a linear interpolation to get an accurate estimate. For example, if \(L=2.3\, \text{mm}\) at \(40^\circ\) and \(L=1.2\, \text{mm}\) at \(20^\circ\), the estimated \(\theta\) for \(L=2.0\, \text{mm}\) would be closer to \(20^\circ\) but before reaching \(40^\circ\). An approximate \(\theta\) would be around \(30^\circ\).
06
Confirm the Estimated \(\theta\) Value
Double-check the graph and the calculated interpolation to ensure the estimation is accurate. Make any necessary adjustments based on graphical inaccuracies. The value of \(\theta\) for \(L=2.0\, \text{mm}\) should be around \(30^\circ\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Techniques
Graphing is a valuable skill in trigonometry, especially when you're visualizing relationships between two variables, such as angle (\(\theta\)) and valve lift (\(L\)). The key is to accurately plot the given data points on a coordinate system. On graph paper, place (\(\theta\)) on the x-axis and (\(L\)) on the y-axis. Make sure to take equal intervals on both axes to ensure the accuracy of the graph.
Use these tips to improve your graphing skills:
Use these tips to improve your graphing skills:
- Label the Axes: Clearly label both the x-axis (angle (\(\theta\)) ) and y-axis (valve lift (\(L\)) ).
- Scaled Intervals: Choose uniform intervals that suit the range of your data.
- Plot Points Carefully: Double-check each point鈥檚 coordinates before plotting.
- Smooth the Curve: After plotting, connect the dots smoothly to reflect the continuous nature of the data.
Interpolation
Interpolation is a method used in mathematics to estimate unknown values within two known values. In your graph, after drawing a smooth curve through the plotted points, you may notice that the horizontal line at (\(L = 2.0\) mm) does not exactly intersect a plotted data point.
To handle this, use linear interpolation. Here's how:
Apply these steps consistently to improve your ability to estimate intermediate or missing values accurately on the graph.
To handle this, use linear interpolation. Here's how:
- Select the Interval: Identify the interval where the unknown value lies. For (\(L = 2.0\)) mm, it would be between (\(\theta = 20^{\circ}\)) and (\(\theta = 40^{\circ}\)), according to your table.
- Find the Differences: Calculate the difference in (\(L\)) between these two points ((\(L = 2.3\)) mm and (\(L = 1.2\)) mm).
- Apply the Formula: Use the linear interpolation formula to find (\(\theta\)). The step-by-step calculation will give you a more precise approximation of where (\(L = 2.0\)) mm occurs.
Apply these steps consistently to improve your ability to estimate intermediate or missing values accurately on the graph.
Angle Measurement
Understanding angle measurement is central to solving problems involving rotations or periodic movements, such as those found in trigonometry. Angles can be measured in degrees or radians, but for this exercise, we focus on degrees. One complete revolution in a circle equals (\(360^{\circ}\)).
Here's why angle measurement is crucial:
Familiarity with converting between degrees and radians can enhance your proficiency in trigonometry, providing a more comprehensive understanding of how angular measurements relate to cyclic functions in various contexts.
Here's why angle measurement is crucial:
- Defining Position: It helps in defining the position on a circular path, which directly correlates with many real-world scenarios involving rotation, like cams in machinery.
- Facilitates Computation: Degrees are common for computations involving trigonometric functions and their inverses.
- Simplifies Analysis: Provides an intuitive understanding when visualizing data on a graph, like interpreting points on a circular arc.
Familiarity with converting between degrees and radians can enhance your proficiency in trigonometry, providing a more comprehensive understanding of how angular measurements relate to cyclic functions in various contexts.
Function Analysis
Function analysis is the step where the relationship between the independent variable (\(\theta\)) and the dependent variable (\(L\)) is explored to make predictions or understand behavior. In this scenario, (\(L\)) as a function of (\(\theta\)) was plotted to visualize how valve lift changes with the cam angle.
Here's how to approach function analysis:
Here's how to approach function analysis:
- Identify Trends: Look at the plotted graph and identify any patterns, like increasing or decreasing trends. Notice if the curve represents a periodic or continuous function.
- Determine Characteristics: Check if the function is linear or non-linear. Analyze intervals of increase or decrease, maxima or minima, and symmetrical properties if any.
- Application of Analysis: Use the insights derived from the graph to make informed predictions or solve problems. Like determining angle intersections for specific lift values using interpolation.
