Physics

The position of a particle at time $t$ is given by the relation $x = \dfrac { v _ { 0 } } { \alpha } \left( 1 - e ^ { - \alpha t } \right)$. The dimensional formula for $\alpha ^ { 2 } \ v _ { 0 } ^ { 3 }$ will be :

SOLUTION
$x=\cfrac { { v }_{ 0 } }{ \alpha } \left( 1-{ e }^{ -\alpha t } \right) ....(A)$

From equation (A) since exponential is dimensionless

$\alpha t$ is dimensionless quantity

$\Rightarrow \alpha t=\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right]$

$\quad \therefore \left[ \alpha \right] =\cfrac { \left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right] }{ \left[ T \right] }$

$\Rightarrow \left[ \alpha \right] =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ -1 } \right] ...(1)$

Also,

$\left[ \cfrac { { v }_{ 0 } }{ \alpha } \right] =\left[ x \right]$

$\left[ { v }_{ 0 } \right] =\left[ x \right] \left[ \alpha \right] ...(2)$

from (2) and (1)

$\left[ { v }_{ 0 } \right] =\left[ x \right] \left[ \alpha \right] =\left[ L \right] \left[ { M }^{ 0 }{ L }^{ 0}{ T }^{ -1 } \right]$

$=\left[ { M }^{ 0 }{ L }^{ 1 }{ T }^{ -1 } \right] ....(3)$

Dimensional formula of $\alpha ^2\ {v_0}^3$ can be obtained from from (1) and (3)

= $[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ -1 }]^2$ $\left[ { M }^{ 0 }{ L }^{ 1 }{ T }^{ -1 } \right]^3$

$=\left[ { M }^{ 0 }{ L }^{ 3 }{ T }^{ -5 } \right]$

You're just one step away

Subjective Medium Published on 18th 08, 2020
Questions 244531
Subjects 8
Chapters 125
Enrolled Students 202

Realted Questions

Q1 Single Correct Medium
Given that v is the speed, r is radius and g is acceleration due to gravity. Which of the following is dimensionless?
• A. $\dfrac{{ v }^{ 2 }r}{g}$
• B. $\dfrac{{ v }^{ 2 }}{rg}$
• C. $\dfrac{{ v }^{ 2 }g}{r}$
• D. ${ v }^{ 2 }rg$

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q2 Single Correct Medium
If  $S = \dfrac { d } { t } + \dfrac { e } { ( f - t ) ^ { 2 } } ,$  where  $S$  is the displacement of the body in time  $t .$  Find the dimensions of  $e$  and  $f$
• A. $\mathrm { LT } , \mathrm { T }$
• B. $\mathrm { LT } ^ { 2 } , \mathrm { T }$
• C. $\mathrm { L } ^ { 2 } \mathrm { T } , \mathrm { T } ^ { - 3 }$
• D. $LT, T ^ { 2 }$

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q3 Single Correct Medium
The least count of a stop-watch is $(1/5)\ s$. The time of $20$ oscillations of a pendulum is measured to be $25\ s$. What is the maximum percentage error in this measurement ?
• A. $8\%$
• B. $1\%$
• C. $4\%$
• D. $16\%$

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q4 Subjective Medium
If momentum $p$, area $A$ and time $T$ are taken to be fundamental quantities, then power has dimensional formula.

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020

Q5 Single Correct Medium
Light years is the unit of:
• A. time
• B. length
• C. mass
• D. none of these

Asked in: Physics - Units and Measurement

1 Verified Answer | Published on 18th 08, 2020