1. αβ/t 2. β t/α 3. α t/β 4. αβ t

Answer: 3. α t/β Key idea: Every term in a physical formula has the same dimensions as F , so α t² and β t have the dimensions of force. Equating α t² with β t dimensionally gives [α t]=[β] . So α t/β is dimensionless. Option 3. The other combinations have leftover dimensions.

Topic: Dimensions and dimensional analysis· Physics 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:

1. \(\frac{\alpha\beta}{t}\)

2. \(\frac{\beta t}{\alpha}\)

3. \(\frac{\alpha t}{\beta}\)

4. \(\alpha\beta t\)

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Answer: 3. \(\frac{\alpha t}{\beta}\)

Key idea: Every term in a physical formula has the same dimensions as \(F\), so \(\alpha t^2\) and \(\beta t\) have the dimensions of force.

Equating \(\alpha t^2\) with \(\beta t\) dimensionally gives \([\alpha t]=[\beta]\). So \(\frac{\alpha t}{\beta}\) is dimensionless.

Option 3. The other combinations have leftover dimensions.

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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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