Hooke's I aw states that the force F required to compress or...

Name: Problem 17, Chapter 2: Algebra and Trigonometry Real Mathematics, Real People
Uploaded: 2021-07-03T19:10:28-08:00
Duration: 4 min
Channel: Kimberly Waterbury

Hooke's I aw states that the force $F$ required to compress or stretch a spring (within its elastic limits) is proportional to the distance $d$ that the spring is compressed or stretched from its original length. That is, $F=k d,$ where $k$ is the measure of the stiffness of the spring and is called the spring constant. The table shows the elongation $d$ in centimeters of a spring when a force of $F$ kilograms is applied. (a) Sketch a scatter plot of the data. (b) Find the equation of the line that seems to best fit the data. (c) Use the regression feature of a graphing utility to find a linear model for the data. (d) Use the model from part (c) to estimate the elongation of the spring when a force of 55 kilograms is applied.

Hooke's I aw states that the force $F$ required to compress or stretch a spring (within its elastic limits) is proportional to the distance $d$ that the spring is compressed or stretched from its original length. That is, $F=k d,$ where $k$ is the measure of the stiffness of the spring and is called the spring constant. The table shows the elongation $d$ in centimeters of a spring when a force of $F$ kilograms is applied.
(a) Sketch a scatter plot of the data.
(b) Find the equation of the line that seems to best fit the data.
(c) Use the regression feature of a graphing utility to find a linear model for the data.
(d) Use the model from part (c) to estimate the elongation of the spring when a force of 55 kilograms is applied.

Key Concepts

Linear Relationships

A linear relationship is one in which two variables change at a constant rate with respect to each other. In the context of Hooke's Law, the relationship between force and displacement is linear, meaning that as one variable increases, the other increases proportionally, and this relationship can be graphically represented by a straight line.

Prediction

Prediction in the context of a linear model involves using the derived regression equation to estimate the value of one variable based on a given value of another. Once a linear model has been established, for example using regression analysis, it can be used to predict outcomes such as the elongation of a spring for a force value that was not originally measured.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It helps in determining the line that best fits the scatter plot data points, which allows for predicting the dependent variable for given values of the independent variable, embodying the underlying trend such as the force-displacement relationship in Hooke's Law.

Hooke's Law

Hooke's Law is a principle in physics that states the force applied to an elastic object, such as a spring, is directly proportional to the displacement or deformation of that object, within its elastic limits. This relationship is often expressed in the formula F = k * d, where F is the force, d is the displacement, and k is the spring constant indicating the stiffness of the spring.

Scatter Plots

Scatter plots are graphical representations used to display pairs of data points on a coordinate axis, enabling visualization of the relationship between two variables. They are particularly useful in identifying patterns, trends, or correlations in the data, such as the linearity suggested by Hooke's Law in a spring's force and displacement measurements.

Transcript

00:01 We are given a problem that talks about hook's law, and we are given different x and y values, or k and d values in this case.

00:13 And we have to create a scatter plot, find the best fit equation, and the linear equation, which is a model with your graphing calculator, and then we have to use that one graph.

00:22 And everything right there, this is all my data right here.

00:26 So i added a data and statistics page and i plotted those points.

00:32 Those are the points and it definitely looks like a line.

00:35 So we did the scatter plot.

00:37 The next says figure out the equation of the best fit line.

00:41 Now we could pick any two of these points like 80, 5 .3 and pick any other one.

00:53 Let's pick this one, which is 20 and 1 .4.

00:56 So if we're asked to find the equation of the best fit line, that's this right here.

01:05 You can pick any two points you want.

01:08 The slope would be 5 .3 minus 1 .4, which is 3 .9.

01:15 So i'd have y equals 3 .9.

01:20 Well, we would say k, oh no, x, y, it would be kd.