Turpentine oil is flowing through a tube of length I and radius r. The pressure difference between the two ends of the tube is P. The viscosity of oil is given by η = P(r²-x²)/(4vl), where v is the velocity of oil at a distance x from the axis of the tube. The dimensions of η are

The correct option is [ML⁻¹T⁻¹].

To find the dimensions of the coefficient of viscosity η, we can use the given formula:
$\eta =\frac{P(r^2-x^2)}{4vl}$

Let us determine the dimensions of each term in the formula:
1. Pressure difference (P): Pressure is defined as force per unit area. Therefore, its dimensional formula is:
$\left[P\right]=\frac{\left[\text{Force}\right]}{\left[\text{Area}\right]}=\frac{\left[{\text{MLT}}^{-2}\right]}{\left[{\text{L}}^2\right]}$

2. Term (r² - x²): Since both r (radius) and x (distance from the axis) represent lengths, their dimensions are [L]. The subtraction of two quantities with the same dimensions results in a quantity with the same dimensions:
$[r^2-x^2]=\left[{\text{L}}^2\right]$

3. Velocity (v): Velocity is displacement per unit time:
$\left[v\right]=\left[{\text{LT}}^{-1}\right]$

4. Length (l): The length of the tube has the dimension of length:
$\left[l\right]=\left[\text{L}\right]$

5. Constant (4): The number 4 is a dimensionless constant and has no dimensions.

Now, substituting these dimensions back into the formula for η:
$\left[\eta \right]=\frac{\left[P\right]·[r^2-x^2]}{\left[v\right]·\left[l\right]}$

Substituting the dimensional formulas:
$\left[\eta \right]=\frac{\left[{\text{ML}}^{-1}{\text{T}}^{-2}\right]·\left[{\text{L}}^2\right]}{\left[{\text{LT}}^{-1}\right]·\left[\text{L}\right]}$

Simplifying the numerator:
$\left[{\text{ML}}^{-1}{\text{T}}^{-2}\right]·\left[{\text{L}}^2\right]=\left[{\text{MLT}}^{-2}\right]$

Simplifying the denominator:
$\left[{\text{LT}}^{-1}\right]·\left[\text{L}\right]=\left[{\text{L}}^2{\text{T}}^{-1}\right]$

Combining the numerator and denominator:
$\left[\eta \right]=\frac{\left[{\text{MLT}}^{-2}\right]}{\left[{\text{L}}^2{\text{T}}^{-1}\right]}=\left[{\text{ML}}^{1-2}{\text{T}}^{-2-(-1)}\right]=\left[{\text{ML}}^{-1}{\text{T}}^{-1}\right]$

Thus, the dimensions of the coefficient of viscosity η are [ML⁻¹T⁻¹].