Problem 12 - Basic Engineering Mathematics

Question

The law connecting friction \(F\) and load \(L\) for an experiment is of the form \(F=a L+b\), where \(a\) and \(b\) are constants. When \(F=5.6, L=8.0\) and when \(F=4.4\), \(L=2.0\). Find the values of \(a\) and \(b\) and the value of \(F\) when \(L=6.5\).

Step-by-Step Solution

Answer

\(a = 0.2\), \(b = 4.0\), and \(F = 5.3\) when \(L = 6.5\).

Step 1: Setup Equations with Given Values

Using the first point where \(F = 5.6\) and \(L = 8.0\), we substitute into the equation \(F = aL + b\), yielding: \[ 5.6 = 8a + b \] Similarly, using the second point where \(F = 4.4\) and \(L = 2.0\), we get the equation: \[ 4.4 = 2a + b \]

Step 2: Solve the System of Equations for a and b

We have two equations: 1. \(5.6 = 8a + b\)2. \(4.4 = 2a + b\)Subtract the second equation from the first to eliminate \(b\):\[ (8a + b) - (2a + b) = 5.6 - 4.4 \]\[ 6a = 1.2 \]Solving for \(a\), we get:\[ a = \frac{1.2}{6} = 0.2 \].

Step 3: Substitute a to Find b

We substitute \(a = 0.2\) back into one of the original equations. Using equation 2: \[ 4.4 = 2(0.2) + b \]\[ 4.4 = 0.4 + b \]Solving for \(b\), we get: \[ b = 4.4 - 0.4 = 4.0 \].

Step 4: Use the Values of a and b to Calculate F when L=6.5

With \(a = 0.2\) and \(b = 4.0\), substitute into the equation to find \(F\) when \(L = 6.5\):\[ F = 0.2 \cdot 6.5 + 4.0 \]Calculate:\[ F = 1.3 + 4.0 = 5.3 \].

Key Concepts

Understanding Systems of Equations

A system of equations is a collection of two or more equations with a common set of variables. In the context of the given exercise, we have two linear equations that need to be solved together to find the values of the unknown variables 'a' and 'b'.
These equations describe the relationship between friction \( F \) and load \( L \). Each equation reflects a different set of experimental observations. - Equation 1: \( 5.6 = 8a + b \)- Equation 2: \( 4.4 = 2a + b \)The objective is to find values of \( a \) and \( b \) that satisfy both equations simultaneously.
This process involves solving for one variable in terms of the other, and then substituting back to find the second variable.

What are Constants in Linear Equations?

In linear equations, constants are fixed values that do not change. They are not multiplied by variables and provide translation and scaling in the equation.
In our exercise, the constants 'a' and 'b' determine the slope and intercept of the line described by the equation \( F = aL + b \).
- \( a \) is the coefficient of \( L \), often considered the rate of change or the slope in a linear equation.
- \( b \) is the constant term, which acts as the y-intercept in the equation when plotted, representing where the line crosses the y-axis.
Understanding the role of constants helps predict how changes in one point of the line (like varying \( L \)) will shift \( F \), the friction, in the system.

How the Substitution Method Helps Solve Equations

The substitution method is a useful approach for solving systems of equations. By substituting one equation into another, you can eliminate a variable and simplify the system down to a single equation. This method was used in the exercise to find the values of \( a \) and \( b \).
- We first solved the second equation \( 4.4 = 2a + b \) to isolate one of the variables (in this case, \( b \)).
- By knowing \( a \) from the subtraction of the two initial equations, we could substitute \( a = 0.2 \) into either of the equations to solve for \( b \).
Using this method allows for a straightforward approach to solving equations, as one can handle one variable at a time until all unknown quantities are determined.