A force defined by $ F = \alpha t^2 + \beta t $ acts on a particle at a given time $ t $. The factor which is dimensionless, if $ \alpha $ and $ \beta $ are constants, is:
A force defined by $ F = \alpha t^2 + \beta t $ acts on a particle at a given time $ t $. The factor which is dimensionless, if $ \alpha $ and $ \beta $ are constants, is:
- \(\frac{\beta t}{\alpha}\)
- \(\frac{\alpha t}{\beta}\)
- αβt
- \(\frac{\alpha\beta}{t}\)
The Correct Option is B
Solution and Explanation
Concepts:
Dimensional analysis, Physics
Explanation:
The objective is to identify the dimensionless factor by analyzing the dimensions of each term in the force equation F = αt² + βt.
The dimension of force (F) is [MLT⁻²].
For the term αt² to be dimensionally consistent with force, α must have the dimension [MLT⁻⁴], given that t² has the dimension [T²].
Similarly, for the term βt to be dimensionally consistent with force, β must have the dimension [MLT⁻³], given that t has the dimension [T].
The dimensions of each option are now evaluated:
a) αβt: Dimension: [MLT⁻⁴][MLT⁻³][T] = [M²L²T⁻⁶T] = [M²L²T⁻⁵] (Not dimensionless)
b) αβ/t: Dimension: [MLT⁻⁴][MLT⁻³]/[T] = [M²L²T⁻⁷] (Not dimensionless)
c) βt/α: Dimension: [MLT⁻³][T]/[MLT⁻⁴] = [MLT⁻²]/[MLT⁻⁴] = [T²] (Not dimensionless)
d) αt/β: Dimension: [MLT⁻⁴][T]/[MLT⁻³] = [MLT⁻³]/[MLT⁻³] = [1] (Dimensionless)
Consequently, the dimensionless factor is αt/β.
Step-by-Step Solution:
Step 1: Establish the dimension of force F: [MLT⁻²].
Step 2: Deduce the dimension of α from the term αt²: [MLT⁻⁴].
Step 3: Deduce the dimension of β from the term βt: [MLT⁻³].
Step 4: Examine the dimensions of each provided option to determine the dimensionless factor.
Step 5: Conclude that αt/β is dimensionless.
Final Answer: αt/β
