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Units and Measurement

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Physics Part I

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Physics

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Class 10 Class 12
A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:
left parenthesis straight a right parenthesis space straight y space equals space straight a space sin space 2 straight pi space straight t divided by straight T space

left parenthesis straight b right parenthesis space straight y space equals space straight a space sin space vt

space left parenthesis straight c right parenthesis space straight y space equals space left parenthesis straight a divided by straight T right parenthesis space sin space straight t divided by straight a

straight d right parenthesis space straight y space equals space open parentheses straight a square root of 2 close parentheses space left parenthesis sin space 2 space πt divided by straight T space plus space cos space 2 πt space divided by straight T space right parenthesis 
(a = maximum displacement of the particle, v = speed of the particle, T = time-period
of motion).
Rule out the wrong formulas on dimensional grounds.




The displacement y has the dimension of length.

Therefore, the formula for it should also have the dimension of length.

Trigonometric functions are dimensionless and their arguments are also dimensionless.

Based on these considerations now checking each formula dimensionally, we have

straight a right parenthesis thin space Dimensionally space correct

straight y space equals space straight a space sin space fraction numerator 2 πt over denominator straight T end fraction
Dimension space of space straight y space equals space straight M to the power of straight o straight L to the power of 1 straight T to the power of straight o space space

Dimension space of space straight a space equals space straight M to the power of straight o straight L to the power of 1 straight T to the power of straight o space

Dimsension space of space sin space fraction numerator 2 πt over denominator straight T end fraction space equals space straight M to the power of straight o straight L to the power of straight o straight T to the power of straight o space
So comma space dimension space of space straight L. straight H. straight S space equals space Dimension space of space straight R. straight H. straight S

rightwards double arrow space given space formula space is space dimensionally space correct.

straight b right parenthesis thin space Incorrect space dimensionally.

straight y space equals space straight a space sin space vt space

Dimension space of space straight y space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space space

Dimension space of space straight a space equals space straight M to the power of 0 space end exponent straight L to the power of 1 space straight T to the power of 0 space space

Dimension space of space vt space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of – 1 end exponent space cross times space straight M to the power of 0 space end exponent straight L to the power of 0 space end exponent straight T to the power of 1 space end exponent equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space

The space argument space of space the space trigonometric space function space
must space be space dimensionless comma space which space is space not space so space in space this space case. space

Hence comma space given space formula space is space dimensionally space incorrect.

straight c right parenthesis space Incorrect

Dimension space of space straight y space equals space straight M to the power of 0 space space straight L to the power of 1 space space straight T to the power of 0

Dimension space of space begin inline style straight a over straight T end style equals space straight M to the power of 0 space end exponent space straight L to the power of 1 space space straight T to the power of – 1 end exponent space

Dimension space of space straight t over straight a equals space equals space straight M 0 space straight L to the power of – 1 space end exponent straight T to the power of 1 space
Argument space of space the space trignometric space function space is
space dimensionless. space Hence comma space the space formula space is space dimensionally space
incorrect.

straight d right parenthesis thin space Correct

Dimension space of space straight y space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0

Dimension space of space straight a space equals straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space

Dimension space of space straight t over straight T space equals space straight M to the power of 0 space end exponent straight L to the power of 0 space end exponent straight T to the power of 0 space
Since space the space srgument space of space the space trignometric space
function space is space dimensionless comma space the space dimensions
space of space straight y space and space straight a space are space the space same. space Hence comma space the
given space formula space is space dimensionally space correct. space


103 Views

Give four examples of physical quantities.

Examples of Physical quantitie are force, mass, density and power.
821 Views

What is measurement?

Measurement is the process of attaching a numeric value to an aspect of a natural phenomenon. Measurement is making the quantitative knowledge of a physical quantity. 
1289 Views

What is the need of measurement?

To get the complete knowledge of any physical quantity, measurement is needed. Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. Knowledge without measurement is incomplete and unsatisfactory. 
825 Views

Define unit.

The quantity used as a standard of measurement is called unit. 

Example: Unit of time is second. 

1172 Views

Define physical quantity.

 


A measurable quantity in terms of which laws of Physics can be expressed is called physical quantity.

The result of a measurement of a physical quantity is expressed by a number accompanied by a unit.


878 Views