Question

1.Turpentine oil is flowing through a tube of length l and radius r. The pressure difference between the two ends of the tube is P. The viscosity of the 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. Determine the dimensions of η.

2. The position x of a particle depends upon time t according to the equation x = at + bt2. Determine the dimensions and units of a and b. What are the physical quantities denoted by a and b?

3. The velocity v of a particle depends upon time t according to the relation

𝑣=𝑎𝑡+ 𝑏/𝑡+𝑐

What are the dimensions of a, b and c?

Answer 1

1 We have:

$$[\eta]=\frac{P(r_2-x_2)}{4vl}=\frac{\text{Pa}·\text{m}}{\text{m/s}\cdot\text{m}}=\\\space\\ =\frac{\text{Pa}}{\text{m/s}}=\frac{\text{kg}}{\text{m}^2\text{s}}=\text{N}·\text{s}/\text{m}^2.$$

2 The physical quantities denoted by a and b are initial velocity and acceleration: x is in meters, so

\text{m}=[a]·\text{s},\\ [a]=\text{m/s}.\\ \text{m}=[b]·\text{s}^2,\\ [b]=\text{m/s}^2.

3 Same approach as in (2):

\text{m/s}=[a]·\text{s},\\ [a]=\text{m/s}^2.\\ \text{m/s}=[b]/\text{s},\\ [b]=\text{m}.\\ \text{m/s}=[c].