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Which is the most accurate instrument?

The Vernier caliper with the lowest value of least count is the one with the highest accuracy.

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Hint: Finding the least count (L.C.) of a Vernier caliper, tells us about the accuracy of the instrument. In the given question, find the least count of all 4 options in the same unit and compare them. The Vernier caliper with the lowest value of least count is the one with the highest accuracy.

Formula used:

The L.C. is given by:

${\text{L}}{\text{.C}}{\text{. = 1 m}}{\text{.s}}{\text{.d}}{\text{.}} - {\text{1 v}}{\text{.s}}{\text{.d}}{\text{.}}$

Where the value of v.s.d. must be written in terms of m.s.d

Complete step by step solution:

In order to find the accuracy of a Vernier caliper, we should know its least count. So we analyse all the four given options- A) Here, the least count is already given to us, which is ${\text{0}}{\text{.1cm}}$.

B) Here, the value of m.s.d. is given as-

$1m.s.d. = 0.1cm$ , as it is a millimetre scale. $\{ 1cm = 10mm\} $

We also know that,

$10v.s.d. = 9m.s.d.$

So, $1v.s.d. = \dfrac{9}{{10}}m.s.d.$

$L.C. = 1m.s.d. - 1v.s.d.$

On substituting the value of $1v.s.d$we get-

$L.C. = 1m.s.d. - \dfrac{9}{{10}}m.s.d.$

Therefore,$L.C. = \dfrac{1}{{10}}m.s.d.$

Now, we put the value of $1m.s.d.$ here-

$L.C. = \dfrac{1}{{10}} \times 0.1{\text{ }}cm$

$L.C. = 0.01cm$

C) Here, the value of m.s.d. is given as-

$1m.s.d. = 0.1cm$ (Millimetre scale)

Also,

$50v.s.d. = 49m.s.d.$

$1v.s.d. = \dfrac{{49}}{{50}}m.s.d.$

Therefore L.C. is given by-

$L.C. = 1m.s.d. - 1v.s.d.$

\[L.C. = 1m.s.d. - \dfrac{{49}}{{50}}m.s.d.\]

$L.C. = \dfrac{1}{{50}}m.s.d.$

$L.C. = \dfrac{1}{{50}} \times 0.1{\text{ }}cm$

$L.C. = 0.002cm$

D) Here, it is said that there are 20 divisions in $1{\text{ cm}}$, which means,

$1m.s.d. = \dfrac{1}{{20}}cm = 0.05cm$

And $50v.s.d. = 49m.s.d.$

$1v.s.d. = \dfrac{{49}}{{50}}m.s.d.$

Therefore, L.C. is-

$L.C. = 1m.s.d. - \dfrac{{49}}{{50}}m.s.d.$

$L.C. = \dfrac{1}{{50}} \times 0.05 = 0.001{\text{ }}cm$

It is clear that $0.001cm$ is the lowest least count, thus (D) is the correct option. Note: All of the values of the least count should be compared only when they are in the same unit, which is centimeters here. Other units like millimeters can also be used but are less common in the case of Vernier calipers.

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Ratios

Mathematical ratios describe the size and relationship between two or more things, and they come in handy in understanding and performing music. For example, if a string instrument is plucked so that the entire length of the string (called an open string) vibrates, a specific pitch, or tone, is sounded. If you touch the string at the halfway point, and then pluck the string so that half the length of the string vibrates, the pitch is an octave higher than it was with the open string. Mathematically speaking, the ratio of the length of the open string to the length of the octave is 2:1 (or, you could say the length of the open string is two times the length of the octave). Knowing this ratio, you can pick up any stringed instrument and know how to play an octave! And another thing you might have figured out, as the ancient Greek philosopher Pythagoras did, is that the shorter the string on a musical instrument (such as a violin), the higher the pitch of the sound it produces.

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