The velocity of upper layer of water in a river is 36 kmh⁻¹. Shearing stress between horizontal layers of water is 10⁻³ Nm⁻². Depth of the river is __________ m. (Co-efficient of viscosity of water is 10⁻² Pa.s)

Solution

Physics sir · 4 steps

The riverbed is still and the top moves at 10 m/s, so the velocity gradient is just v over the depth.

1. Set up the gradient. Water touching the riverbed does not move, and the top layer moves at v. Take the velocity to rise uniformly with height, so the velocity gradient is v/d, where d is the depth.

   Shearing stress = η v/d ⇒ d = (η v)/stress

2. Convert the speed. 36 km h⁻¹ = 36 × 1000/3600 m s⁻¹ = 10 m s⁻¹.
3. Put in numbers. d = (10⁻² × 10)/10⁻³ = 10⁻¹/10⁻³ = 100 m.
4. So. The depth of the river is 100 m.

The common slip: 360. Using 36 directly instead of converting it to 10 m s⁻¹ gives 360, not 100.

The idea to keep. Newton's law of viscosity is stress = η × velocity gradient. A layer resting on a fixed surface has zero speed there, so the gradient is the top speed divided by the thickness. Always work in SI units.

Answer from the NTA final answer key. Solution by PrepWiser.