1. Explain Newton's law of viscosity. 2. Derive the formula of pressure at any point in the liquid. 3. State Archimedes' principle. Explain why the weight of a body is less in the... 1. Explain Newton's law of viscosity. 2. Derive the formula of pressure at any point in the liquid. 3. State Archimedes' principle. Explain why the weight of a body is less in the water than in the liquid. 4. Find the coefficient of viscosity with the given data. 5. Calculate the pressure on the tank at its bottom.
Understand the Problem
The question asks for an explanation of Newton's law of viscosity, derivation of a formula related to pressure, and various calculations concerning a fluid's properties, including pressure and viscosity in a tank filled with water.
Answer
- Newton's law of viscosity: $\tau = \mu \cdot \frac{du}{dy}$ - Pressure formula: $P = P_0 + \rho gh$ - Archimedes' principle: buoyant force = weight of displaced fluid - Coefficient of viscosity: $\mu = \frac{F}{6 \pi r v}$ - Pressure at tank bottom: $P = 68600 \, \text{Pa}$
Answer for screen readers
- Newton's law of viscosity states that $\tau = \mu \cdot \frac{du}{dy}$.
- Pressure at a depth $h$ is $P = P_0 + \rho gh$.
- Archimedes' principle states that the buoyant force equals the weight of the fluid displaced.
- Coefficient of viscosity $\mu = \frac{F}{6 \pi r v}$.
- The pressure at the bottom of the tank is: $$ P = 1000 \times 9.8 \times 7 = 68600 , \text{Pa} $$
Steps to Solve
- Explain Newton's Law of Viscosity
Newton's law of viscosity states that the shear stress ($\tau$) between two layers of fluid is directly proportional to the velocity gradient (shear rate, $\frac{du}{dy}$) between the layers. Mathematically, this is represented as: $$ \tau = \mu \cdot \frac{du}{dy} $$ where $\mu$ is the dynamic viscosity of the fluid.
- Derive the Formula of Pressure at Any Point in the Liquid
The pressure at a depth $h$ in a liquid is given by: $$ P = P_0 + \rho gh $$ where $P_0$ is the pressure at the surface of the liquid, $\rho$ is the density of the liquid, and $g$ is the acceleration due to gravity.
- State Archimedes' Principle
Archimedes' principle states that any object submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object. The weight of a body appears less in water due to the upward buoyant force acting against the weight of the body.
- Find the Coefficient of Viscosity with the Given Data
Using the formula stemming from Stokes' law: $$ F = 6 \pi \mu r v $$ where $F$ is the force acting on the sphere, $\mu$ is the coefficient of viscosity, $r$ is the radius, and $v$ is the velocity of the sphere. Rearranging gives: $$ \mu = \frac{F}{6 \pi r v} $$
- Calculate the Pressure on the Tank at Its Bottom
Using the formula derived in step 2 at the bottom of a tank of height $h$: $$ P = \rho g h $$ Substituting the values ($\rho = 1000 , \text{kg/m}^3$, $g = 9.8 , \text{m/s}^2$, and $h = 7 , \text{m}$), we calculate: $$ P = 1000 \times 9.8 \times 7 $$
- Newton's law of viscosity states that $\tau = \mu \cdot \frac{du}{dy}$.
- Pressure at a depth $h$ is $P = P_0 + \rho gh$.
- Archimedes' principle states that the buoyant force equals the weight of the fluid displaced.
- Coefficient of viscosity $\mu = \frac{F}{6 \pi r v}$.
- The pressure at the bottom of the tank is: $$ P = 1000 \times 9.8 \times 7 = 68600 , \text{Pa} $$
More Information
Newton's law of viscosity helps us understand how fluids behave under stress. The derived pressure formula is essential for hydrodynamics and engineering applications. Archimedes' principle is crucial in buoyancy problems, showing how the weight of submerged objects changes.
Tips
- Misunderstanding the relationship between shear stress and viscosity.
- Incorrectly applying the pressure formula, such as neglecting atmospheric pressure at the surface.
- Forgetting to convert units when substituting in formulas.
AI-generated content may contain errors. Please verify critical information
