A force defined by <math> <mrow> <mi>F</mi> <mo>=</mo> <mi>α</mi> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>β</mi> <mi>t</mi> </mrow> </math> acts on a particle at a given time <math> <mrow> <mi>t</mi> </mrow> </math> . The factor which is dimensionless, if <math> <mrow> <mi>α</mi> </mrow> </math> and <math> <mrow> <mi>β</mi> </mrow> </math> are constants, is:

From principle of homogeneity [ F ] = [ α t 2 ] = [ β t ]  [α]=[F][t2] and [β]=[F][t]  ∴ [ α ] [ t ] = [ β ]  ∴αtβ= dimensionless

A force defined by <math> <mrow> <mi>F</mi> <mo>=</mo> <mi>α</mi> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>β</mi> <mi>t</mi> </mrow> </math> acts on a particle at a given time <math> <mrow> <mi>t</mi> </mrow> </math> . The factor which is dimensionless, if <math> <mrow> <mi>α</mi> </mrow> </math> and <math> <mrow> <mi>β</mi> </mrow> </math> &nbsp;are constants, is:

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Physics Class 11th Units and Measurement

A force defined by $F = α t^{2} + β t$ acts on a particle at a given time $t$ . The factor which is dimensionless, if $α$ and $β$ are constants, is:

Option 1 -

$β t / α$

Option 2 -

$α t / β$

Option 3 -

$α β t$

Option 4 -

$α β / t$

2 Views | Posted a month ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 2

Detailed Solution:

From principle of homogeneity

$[F] = [α t^{2}] = [β t]$

$[α] = \frac{[F]}{[t^{2}]} and [β] = \frac{[F]}{[t]}$

$∴ [α] [t] = [β]$

$∴ \frac{α t}{β} = dimensionless$

From principle of homogeneity <math> <mrow> <mrow> <mo>[</mo> <mrow> <mi>F</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>α</mi> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>β</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> </mrow> </math>  <math> <mrow><mrow><mo>[</mo><mrow><mi>α</mi></mrow><mo>]</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>[</mo><mrow><mi>F</mi></mrow><mo>]</mo></mrow></mrow><mrow><mrow><mo>[</mo><mrow><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>]</mo></mrow></mrow></mfrac> and <mrow><mo>[</mo><mrow><mi>β</mi></mrow><mo>]</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>[</mo><mrow><mi>F</mi></mrow><mo>]</mo></mrow></mrow><mrow><mrow><mo>[</mo><mrow><mi>t</mi></mrow><mo>]</mo></mrow></mrow></mfrac></mrow> </math>  <math> <mrow> <mo>∴</mo> <mrow> <mo>[</mo> <mrow> <mi>α</mi> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>β</mi> </mrow> <mo>]</mo> </mrow> </mrow> </math>  <math> <mrow><mo>∴</mo><mfrac><mrow><mi>α</mi><mi>t</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>=</mo> dimensionless </mrow> </math>

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A force defined by $F = α t^{2} + β t$ acts on a particle at a given time $t$ . The factor which is dimensionless, if $α$ and $β$ are constants, is:

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A force defined by F = α t 2 + β t acts on a particle at a given time t . The factor which is dimensionless, if α and β are constants, is:

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A force defined by $F = α t^{2} + β t$ acts on a particle at a given time $t$ . The factor which is dimensionless, if $α$ and $β$ are constants, is: