Length

Examples of Length in the following topics:

• Fitting a line by eye

  • We want to describe the relationship between the head length and total length variables in the possum data set using a line.
  • In this example, we will use the total length as the predictor variable, x, to predict a possum's head length, y.
  • A scatterplot showing head length against total length for 104 brushtail possums.
  • A point representing a possum with head length 94.1mm and total length 89cm is highlighted.
  • The figure on the left shows head length versus total length, and reveals that many of the points could be captured by a straight band.

• Length

  • Length is one of the basic dimensions used to measure an object.
  • In other contexts "length" is the measured dimension of an object.
  • Length is a measure of one dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed).
  • In the physical sciences and engineering, when one speaks of "units of length", the word "length" is synonymous with "distance".
  • There are several units that are used to measure length.

• Area and Arc Length in Polar Coordinates

  • If you were to straighten a curved line out, the measured length would be the arc length.
  • Using polar coordinates allows us to integrate along the length of the arc in order to compute its length.
  • The length of $L$ is given by the following integral:
  • Thus $\Delta \theta$, the length of each subinterval, is equal to $b-a$ (the total length of the interval), divided by $n$, the number of subintervals.
  • Their length can be calculated with calculus.

• Length

  • Length is a physical measurement of distance that is fundamentally measured in the SI unit of a meter.
  • Length can be defined as a measurement of the physical quantity of distance.
  • Many qualitative observations fundamental to physics are commonly described using the measurement of length.
  • Many different units of length are used around the world.
  • The basic unit of length as identified by the International System of Units (SI) is the meter.

• Arc Length and Speed

  • The length of a curve can be difficult to measure.
  • Adding up all these lengths together would be equivalent to stretching the curve out straight and measuring its length.
  • The length of the curve is called the arc length.
  • In order to calculate the arc length, we use integration because it is an efficient way to add up a series of infinitesimal lengths.
  • Adding up each tiny hypotenuse yields the arc length.

• Arc Length and Speed

  • Arc length and speed are, respectively, a function of position and its derivative with respect to time.
  • Since length is a magnitude that involves position, it is easy to deduce that the derivative of a length, or position, will give you the velocity—also known as speed—of a function.
  • Let's start this atom by looking at arc length with calculus.
  • The arc length is the length you would get if you took a curve, straightened it out, and then measured the length of that line .
  • The arc length can be found using geometry, but for the sake of this atom, we are going to use integration.

• Length Contraction

  • Let's look at the results with the aether again.If we have a rod of length $L_0$ in the primed frame what it is length in the unprimed frame.
  • We have define the length to be the extent of an object measured at a particular time.

• Dots, Ties, and Borrowed Divisions

  • A half note is half the length of a whole note; a quarter note is half the length of a half note; an eighth note is half the length of a quarter note, and so on.
  • (See Duration:Note Length. ) The same goes for rests.
  • (See Duration: Rest Length. ) But what if you want a note (or rest) length that isn't half of another note (or rest) length?
  • In other words, the note keeps its original length and adds another half of that original length because of the dot.
  • For example, the first dot after a half note adds a quarter note length; the second dot would add an eighth note length.

• Bond Lengths

  • Various spectroscopic methods also exist for estimating the bond length between two atoms in a molecule.
  • The bond length is the average distance between the nuclei of two bonded atoms in a molecule.
  • Bonds lengths are typically in the range of 1-2 Å, or 100-200 pm.
  • For example, the bond length of $C - C$ is 154 pm; the bond length of $C = C$ is 133 pm; and finally, the bond length of $C \equiv C$ is 120 pm.
  • The minimum energy occurs at the equilibrium distance r0, which is where the bond length is measured.

• Length Contraction

  • Length contraction is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer.
  • Now let us imagine that we want to measure the length of a ruler.
  • Consequently, the length of the ruler will appear to be shorter in your frame of reference (the phenomenon of length contraction occurred).
  • For example, at a speed of 13,400,000 m/s (30 million mph, .0447c), the length is 99.9 percent of the length at rest; at a speed of 42,300,000 m/s (95 million mph, 0.141c), the length is still 99 percent.
  • Observed length of an object at rest and at different speeds