Question

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. \(\frac{\beta t}{\alpha}\)
• B. \(\frac{\alpha t}{\beta}\)
• C. αβt
• D. \(\frac{\alpha\beta}{t}\)

Answer 1

The Correct Option is B

Concepts:

Dimensional analysis, Physics

Explanation:

To determine the dimensionless factor, we need to analyze the dimensions of each term in the given force equation F = αt² + βt.

The dimension of force F is [MLT⁻²].

For the term αt² to have the same dimension as force, α must have the dimension [MLT⁻⁴] because t² has the dimension [T²].

Similarly, for the term βt to have the same dimension as force, β must have the dimension [MLT⁻³] because t has the dimension [T].

Now, we check the dimensions of each option:

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)

Therefore, the dimensionless factor is αt/β.

Step-by-Step Solution:

Step 1: Determine the dimensions of force F: [MLT⁻²].

Step 2: Determine the dimensions of α from the term αt²: [MLT⁻⁴].

Step 3: Determine the dimensions of β from the term βt: [MLT⁻³].

Step 4: Analyze the dimensions of each option to find the dimensionless factor.

Step 5: Identify that αt/β is dimensionless.

Final Answer:  αt/β