Physics

# If $F$ is given by $F$ = (${\alpha / \beta}$) (${1 - e ^{{- \beta t^2} / m}}$), where $F$ is force and $m$ & $t$ are mass and time, then dimension of $\alpha$ will be :

##### SOLUTION
Since, $F=\cfrac { \alpha }{ \beta } \left( 1-{ e }^{ -\cfrac { \beta { t }^{ 2 } }{ m } } \right)$

Since exponential is an dimensionless quantity

Hence,
$-\cfrac { \beta { t }^{ 2 } }{ m }$ is a dimensionless quantity

$\therefore$  $\left[ \cfrac { \beta { t }^{ 2 } }{ m } \right] =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right]$

$\Rightarrow \beta \left[ \cfrac { { T }^{ 2 } }{ M } \right] =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right]$

Hence, $\quad \beta =\cfrac { \left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right] \left[ M \right] }{ \left[ { T }^{ 2 } \right] }$

and the two quantities cannot subtracted added until they don't have same dimension

$\left[ F \right] =\cfrac { \alpha }{ \beta }$

since, $\left[ F \right] =\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right]$

$\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right] =\cfrac { \left[ \alpha \right] }{ \left[ { M }^{ 1 }{ L }^{ 0 }{ T }^{ -2 } \right] }$

$\therefore \left[ \alpha \right] =\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right] \left[ { M }^{ 1 }{ L }^{ 0 }{ T }^{ -2 } \right]$

$\left[ \alpha \right] =\left[ { M }^{ 2 }{ L }^{ 1 }{ T }^{ -4 } \right]$

You're just one step away

Subjective Medium Published on 18th 08, 2020
Questions 244531
Subjects 8
Chapters 125
Enrolled Students 218

#### Realted Questions

Q1 Single Correct Medium
Match the physical quantities given in Column I with suitable dimensions expressed in Column II.
 Column I Column II a) Angular momentum g) ${ ML }^{ 2 }{ T }^{ -2 }$ b) Latent heat h)  ${ ML }^{ 2 }{ Q }^{ -2 }$ c) Torque i)  ${ ML }^{ 2 }{ T }^{ -1 }$ d) Capacitane j)  ${ ML }^{ 3 }{ T }^{ -1 }{ Q }^{ -2 }$ e) Inductance k)  ${ M }^{ -1 }{ L }^{ -2 }{ T }^{ 2 }{ Q }^{ 2 }$ f) Resistivity l)  ${ L }^{ 2 }{ T }^{ -2 }$
• A. a-i, b-l, c-g, d-k, e-h, f-j
• B. a-l, b-i, c-k, d-g, e-j, f-h
• C. a-i, b-l, c-h, d-j, e-g, f-k
• D. a-h, b-j, c-g, d-k, e-i, f-l

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q2 Single Correct Medium
Write the dimensions of $a/b$ in the relation $P = \dfrac{a - t^2}{bx}$; where $P$ is the pressure, $x$ is the distance, and $t$ is the time.
• A. $M^{-1}L^0T^{-2}$
• B. $ML^0T^{-2}$
• C. $ML^0T^{2}$
• D. $MLT^{-2}$

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q3 Single Correct Medium
A student wishes to find the volume of a small, irregularly-shaped stone.
A ruler and a measuring cylinder containing some water area available. Which apparatus is needed?
• A. neither the ruler nor the measuring cylinder
• B. the measuring cylinder only
• C. the ruler and the measuring cylinder
• D. the ruler only

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q4 Single Correct Medium
The absolute value of which of the following quantity of a substance cannot be determined?
• A. Entropy
• B. Enthalpy
• C. Volume
• D. Heat capacity

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q5 Assertion & Reason Medium
##### ASSERTION

Force cannot be added to pressure.

##### REASON

Because their dimensions are different.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020