Suppose the dimensional formula for acceleration, velocity length are `alpha beta^(-2), alphabeta^(-1) and alpha gamma`. Then the dimensional formula for the coefficient of friction is

Options

A. `alphabeta//gamma `
B. `alphagamma//beta`
C. `alpha^(0)beta^(0)gamma`
D. `alpha^(0)beta^(0)gamma^(-1)`

Explanation

To find the dimensional formula for the coefficient of friction given the dimensional formulas for acceleration, velocity, and length, we can follow these steps:

### Step 1: Understand the formula for the coefficient of friction
The coefficient of friction (μ) is defined as the ratio of the force of friction to the normal force. In terms of dimensions, it can also be expressed as:
\[ \mu = \frac{v^2}{a \cdot l} \]
where \(v\) is velocity, \(a\) is acceleration, and \(l\) is length.

### Step 2: Substitute the given dimensional formulas
We are given:
- Velocity \(v\) has the dimensional formula \(\alpha \beta^{-1}\)
- Acceleration \(a\) has the dimensional formula \(\alpha \beta^{-2}\)
- Length \(l\) has the dimensional formula \(\alpha \gamma\)

Now we can substitute these into the formula for the coefficient of friction:
\[ \mu = \frac{(\alpha \beta^{-1})^2}{(\alpha \beta^{-2})(\alpha \gamma)} \]

### Step 3: Simplify the expression
First, calculate \(v^2\):
\[ v^2 = (\alpha \beta^{-1})^2 = \alpha^2 \beta^{-2} \]

Now substitute this into the equation for \(\mu\):
\[ \mu = \frac{\alpha^2 \beta^{-2}}{(\alpha \beta^{-2})(\alpha \gamma)} \]

Next, simplify the denominator:
\[ \text{Denominator} = \alpha \beta^{-2} \cdot \alpha \gamma = \alpha^2 \beta^{-2} \gamma \]

Now we can rewrite \(\mu\):
\[ \mu = \frac{\alpha^2 \beta^{-2}}{\alpha^2 \beta^{-2} \gamma} \]

### Step 4: Cancel out common terms
Cancel \(\alpha^2\) and \(\beta^{-2}\) from the numerator and denominator:
\[ \mu = \frac{1}{\gamma} \]

### Step 5: Write the final dimensional formula
Thus, the dimensional formula for the coefficient of friction is:
\[ \mu = \gamma^{-1} \]

### Final Answer
The dimensional formula for the coefficient of friction is \(\gamma^{-1}\).

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