In an experiment, a spherical ball of radius ' r ' and density ' ' falls freely through the glycerin of density ' ', as shown in figure. [figure in the original paper] a) Write the componets of…

a) Components of forces acting on the body

The ball experiences three forces when falling through glycerin:

• Weight (mg): Acts downward due to gravity. $mg=\rho Vg=\rho \left(\frac{4}{3}\pi r^3\right)g$
• Buoyant force (F_b): Acts upward due to displaced glycerin. $F_b=\sigma Vg=\sigma \left(\frac{4}{3}\pi r^3\right)g$
• Viscous drag (F_v): Acts upward, opposing motion (Stokes’ law for laminar flow). $F_v=6\pi \eta rv$

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b) Velocity vs. time graph

• Initially, velocity increases due to net downward force.
• As velocity increases, viscous drag $F_v$ grows until it balances weight and buoyancy.
• Terminal velocity $v_t$ is reached when $F_v=mg-F_b$.

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c) Expression for terminal velocity

At terminal velocity, net force = 0: $$mg-F_b-F_v=0$$ Substitute expressions: $$\rho \left(\frac{4}{3}\pi r^3\right)g-\sigma \left(\frac{4}{3}\pi r^3\right)g-6\pi \eta r{v}_t=0$$ Solve for $v_t$: $$v_t=\frac{2}{9}\frac{(\rho -\sigma )g{r}^2}{\eta }$$ Final expression: $$\boxed{v_t=\frac{2}{9}\frac{(\rho -\sigma )g{r}^2}{\eta }}$$