Definition of Terminal Velocity: Terminal velocity is the constant maximum velocity attained by an object falling through a fluid when the net force acting on it becomes zero. This occurs when the sum of the upward forces (buoyant force and viscous drag force) equals the downward force (gravitational force). Expression for Terminal Velocity: Consider a spherical object of radius $r$ and density $\rho$ falling through a viscous liquid of density $\sigma$ and coefficient of viscosity $\eta$. Step 1: Identify the forces acting on the sphere. Gravitational force ($F_g$) acting downwards:* $$F_g = mg = (\text{volume of sphere}) \times (\text{density of sphere}) \times g$$ $$F_g = \frac{4}{3}\pi r^3 \rho g$$ Buoyant force ($F_b$) acting upwards (by Archimedes' principle):* $$F_b = (\text{volume of sphere}) \times (\text{density of liquid}) \times g$$ $$F_b = \frac{4}{3}\pi r^3 \sigma g$$ Viscous drag force ($F_v$) acting upwards (by Stokes' Law):* $$F_v = 6\pi \eta r v$$ where $v$ is the instantaneous velocity of the sphere. Step 2: Determine terminal velocity. When the sphere reaches terminal velocity ($v_t$), its acceleration becomes zero ($a = 0$). At this point, the net force is zero, meaning the upward forces balance the downward force. $$F_g = F_b + F_v$$ $$\frac{4}{3}\pi r^3 \rho g = \frac{4}{3}\pi r^3 \sigma g + 6\pi \eta r v_t$$ Step 3: Solve for $v_t$. Rearrange the equation to isolate $v_t$: $$6\pi \eta r v_t = \frac{4}{3}\pi r^3 \rho g - \frac{4}{3}\pi r^3 \sigma g$$ $$6\pi \eta r v_t = \frac{4}{3}\pi r^3 (\rho - \sigma) g$$ Divide both sides by $6\pi \eta r$: $$v_t = \frac{4\pi r^3 (\rho - \sigma) g}{3 \times 6\pi \eta r}$$ $$v_t = \frac{4\pi r^3 (\rho - \sigma) g}{18\pi \eta r}$$ Simplify the expression: $$\boxed{v_t = \frac{2 r^2 (\rho - \sigma) g}{9 \eta}}$$ Graphical Variation of Velocity with Time: When an object is dropped in a viscous fluid, its velocity initially increases from zero. As the velocity increases, the viscous drag force also increases, which reduces the net accelerating force. Eventually, the sum of the upward buoyant force and viscous drag force equals the downward gravitational force. At this point, the net force becomes zero, acceleration becomes zero, and the object attains a constant maximum velocity called the terminal velocity ($v_t$). The graph of velocity ($v$) versus time ($t$) shows an exponential increase in velocity from zero, which then levels off to a constant value (terminal velocity) as time progresses. ` Velocity ^ | / | / | / | / | / | / |/_________________ Terminal Velocity (v_t) | +----------------------> Time 0 `