The plot of a linear relation given an equation

In summary, for small values of q, the function sin(x) can be approximated by x. Applying this to the given equation, we get ω(q) = qa√(f/m). Simplifying further, we get ω(q) = 2√(f/m)qa/2.
  • #1
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Homework Statement


[tex]\omega (q)= \sqrt{( \frac{4f}{m})} sin\frac{qa}{2}[/tex]

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Homework Equations


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The Attempt at a Solution


I don't understand the linear line given on the graph. For low q (or as q tends to zero) it says the relationship is linear. But as q tends to zero for the given equation I don't see how the equation goes to:

##\omega (q)= qa \sqrt{\frac{f}{m}}##

please could someone explain this, it's not a homework question, just going through the notes we have been given.

thanks in advance for any help
 
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  • #2
When the argument of the function sin(x) is small, x<<1, sin(x)≈x. So ##ω(q)=\sqrt{\frac{4f}{m}} \sin(\frac{qa}{2}) = \sqrt{\frac{4f}{m}} \frac{qa}{2}##.
Pull out 4 from the square root, ##ω(q)=2\sqrt{\frac{f}{m}} \frac{qa}{2}##, simplify by 2.
 
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  • #3
ehild said:
When the argument of the function sin(x) is small, x<<1, sin(x)≈x. So ##ω(q)=\sqrt{\frac{4f}{m}} \sin(\frac{qa}{2}) = \sqrt{\frac{4f}{m}} \frac{qa}{2}##.
Pull out 4 from the square root, ##ω(q)=2\sqrt{\frac{f}{m}} \frac{qa}{2}##, simplify by 2.

cant believe i couldn't see that! thanks very much for your help!
 

1. What exactly is a linear relation?

A linear relation is a mathematical relationship between two variables that can be represented by a straight line on a graph. It follows a specific form, y = mx + b, where m is the slope and b is the y-intercept.

2. Can you explain the components of a linear relation equation?

The y-intercept, b, is the point where the line crosses the y-axis. It represents the initial value of y when x is equal to 0. The slope, m, represents the rate of change between the two variables. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

3. How do you plot a linear relation given an equation?

To plot a linear relation, you need to find at least two points on the line. You can do this by substituting different values for x in the equation and solving for y. Once you have two points, plot them on a graph and connect them with a straight line. You can also find the y-intercept and use the slope to plot additional points.

4. What is the significance of the slope in a linear relation?

The slope in a linear relation represents the rate of change between the two variables. It tells us how much the value of y changes for every unit change in x. A positive slope indicates a positive relationship where y increases as x increases. A negative slope indicates an inverse relationship where y decreases as x increases.

5. Can a linear relation have a slope of 0?

Yes, a linear relation can have a slope of 0. This means that the line is horizontal and has no change in the y-values for any change in x. In this case, the equation of the line would be y = b, where b is the y-intercept. This type of relation is called a constant function.

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