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Dimensions in Physics, Lecture notes of Physics

Physics

https://www.youtube.com/watch?v=3rjMrL8Lnvs&t=55s 1. Dimensions of Physical Quantities 2. Dimensions of Constants of Proportionality 3. Dimensions of Equations

Typology: Lecture notes

2019/2020

Uploaded on 04/19/2020

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Chapter 1
Dimension
11 Physics
1
Presented by
Shehzaib Yousuf Khan
Mechanical Engineer
(NEDUET)
ESY Learning| 2019-20 | Board of Intermediate Education (BIEK) Karachi
https://www.youtube.com/watch?v=3rjMrL8Lnvs&t=55s
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Chapter 1

Dimension

11 Physics

Presented by

Shehzaib Yousuf Khan

Mechanical Engineer

(NEDUET)

ESY Learning| 2019-20 | Board of Intermediate Education (BIEK) Karachi

https://www.youtube.com/watch?v=3rjMrL8Lnvs&t=55s

Outline

1. Dimension of Physical Quantities

2. Dimension of Constants

3. Dimension of Equations

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

Dimension

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

Quantity Unit Dimension

Length ๐‘š ๐ฟ

Mass ๐‘˜๐‘” ๐‘€

Time ๐‘  ๐‘‡

Temperature ๐พ ๐œƒ

Electric Current (^) ๐ด ๐ผ

Luminous Intensity ๐‘๐‘‘ ๐ฝ

Amount of Substance ๐‘š๐‘œ๐‘™ ๐‘

โ€œThe representation of the nature a physical quantity.โ€

Most commonly used

Length could be

1. Distance (๐‘†) or Displacement (

2. Radius (r) or Diameter (D)

3. Breadth (b) or Width (w) or Height (h)

4. Force Arm (r) or Couple Arm (r)

5. Circumference (C)

6. Amplitude (x)

All these are represented dimensionally as L

Symbol: Bracket [] around physical quantity e.g. [v]

Dimension of Physical Quantities

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

1. Velocity:

โ€œDisplacement per unit timeโ€

By taking dimensions on both sides

[๐‘ฃ] =
[๐‘†]
[๐‘ก]

Dimensions: ๐‘† = ๐ฟ ๐‘ก = ๐‘‡

โˆ’ 1

2. Acceleration:

โ€œThe rate of change of velocity with respect to timeโ€

By taking dimensions on both sides

[๐‘Ž] =
[ฮ”๐‘ฃ]
[๐‘ก]

Dimensions: ฮ”๐‘ฃ = ๐ฟ๐‘‡

โˆ’ 1

๐‘ก = ๐‘‡

โˆ’ 1

โˆ’ 1 โˆ’ 1

โˆ’ 2

Dimension of Physical Quantities

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

5. Linear Momentum:

โ€œThe product of mass and velocityโ€

By taking dimensions on both sides

๐‘ = ๐‘š [๐‘ฃ]
[๐‘ก]

Dimensions: ๐‘š = ๐‘€ ๐‘† = ๐ฟ ๐‘ก = ๐‘‡

โˆ’ 1

Dimension of Physical Quantities

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

6. Force:

According to 2

nd

Law of motion

By taking dimensions on both sides

๐น = ๐‘š [๐‘Ž]
[๐‘ก]
๐‘† /[๐‘ก]
[๐‘ก]
[๐‘ก][๐‘ก]

Applying dimensions on both sides

Dimensions:

2

โˆ’ 2

Dimension of Physical Quantities

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

8. Torque:

By taking dimensions on both sides

๐œ = ๐น [๐‘‘]
[๐‘ก]
[๐‘‘]
[๐‘ก][๐‘ก]
[๐‘‘]

Dimensions:

๐‘† = [๐‘‘] = ๐ฟ

Applying dimensions on both sides

2

2

2 ๐‘‡

โˆ’ 2

Dimension of Physical Quantities

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

9. Work:

By taking dimensions on both sides

๐‘Š = ๐น [๐‘†]
[๐‘ก]
[๐‘†]
[๐‘ก][๐‘ก]
[๐‘†]

Dimensions:

Applying dimensions on both sides

2

2

2 ๐‘‡

โˆ’ 2

Dimension of Constant of Proportionality

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

1. Find the dimension of C in the equation:

3

By taking dimensions on both sides

3

[๐ถ] =

3

Dimensions:

[๐‘†] = ๐ฟ

Applying dimensions on both sides

3

โˆ’ 3

Dimension of Constant of Proportionality

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

2. Stokes Law โ€“ Coefficient of Viscosity:

By taking dimensions on both sides

๐น = 6๐œ‹ ๐œ‚ ๐‘… [๐‘ฃ]

Numbers are dimensionless i.e. 6๐œ‹ = 1 ,

therefore

๐‘š [๐‘Ž] = ๐œ‚ ๐‘…

Dimensions:

[๐‘…] = ๐ฟ

Applying dimensions on both sides

โˆ’ 1 ๐‘‡

โˆ’ 1

๐‘† /[๐‘ก]
[๐‘…] ๐‘ก

Dimension of Equations

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

st

Equation of Motion:

๐‘“

๐‘–

By taking dimensions on both sides

๐‘“

๐‘–

Here, ๐‘ฃ

๐‘“

๐‘–

[๐‘ก]
[๐‘ก]
[๐‘ก]

โ€œAll equations in physics must have the same dimensions on both sides.โ€

[๐‘ก]
[๐‘ก]
[๐‘ก]

Applying dimensions on both sides

Numbers are dimensionless, therefore neglect 2

Dimensions:

Hence, 1

st

equation of motion is dimensionally correct

Dimension of Equations

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

nd

Equation of Motion:

๐‘–

2

By taking dimensions on both sides

๐‘–

2

Numbers are dimensionless, therefore neglect ยฝ

also ๐‘ฃ

๐‘–

= ๐‘ฃ = [๐‘†]/[๐‘ก]

2

[๐‘ก]
[๐‘ก]

2

[๐‘ก]
[๐‘ก]
๐‘† = ๐‘† + S

Applying dimensions on both sides

Numbers are dimensionless, therefore neglect 2

Dimensions:

Hence, 2

nd

equation of motion is dimensionally correct

Dimension of Equations

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

Time Period of Spring System:

By taking dimensions on both sides

๐‘‡ = [2๐œ‹]
[๐‘š]
[๐‘˜]

According to Hookeโ€™s Law: ๐น = ๐‘˜๐‘ฅ

Also, Numbers are dimensionless, therefore

[๐‘š]
[๐‘˜]
[๐‘š]
๐น /[๐‘ฅ]
[๐‘š][๐‘ฅ]
๐‘š [๐‘Ž]
[๐‘ฅ]
[๐‘Ž]
[๐‘ฅ]
[ฮ”๐‘ฃ]/[๐‘ก]
[๐‘ฅ][๐‘ก]
[ฮ”๐‘ฃ]
[๐‘ฅ][๐‘ก]
[๐‘†]/[๐‘ก]

Dimensions:

๐‘† = [๐‘ฅ] = ๐ฟ

2

2

Hence, equation of time period is dimensionally correct

2

[๐‘†]

Applying dimensions on both sides

Dimension of Equations

ESY Learning| Shehzaib Yousuf Khan | Board of Intermediate Education (BIEK) Karachi

Wave Velocity:

By taking dimensions on both sides

๐‘ฃ = ๐‘“ [๐œ†]

Here, ๐‘“ = 1 /๐‘‡

[๐‘‡]
[๐‘ก]
[๐‘‡]

Dimensions:

๐‘† = [๐œ†] = ๐ฟ
๐‘ก = [๐‘‡] = ๐‘‡

Applying dimensions on both sides

Hence, equation of wave velocity is dimensionally correct