A small steel ball is dropped into a long cyl | vedclass

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(B) When a steel ball is dropped in a viscous liquid like glycerine,it experiences three forces: gravitational force $(mg)$ downwards,buoyant force $(

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(B) When a steel ball is dropped in a viscous liquid like glycerine,it experiences three forces: gravitational force $(mg)$ downwards,buoyant force $(F_B)$ upwards,and viscous drag force $(F_v = 6 \pi \eta r v)$ upwards.The equation of motion is: $mg - F_B - F_v = ma$Substituting the expressions for forces: $(ho \frac{4}{3} \pi r^3) g - (ho_L \frac{4}{3} \pi r^3) g - 6 \pi \eta r v = m \frac{dv}{dt}$Let $K_1 = \frac{4}{3} \pi r^3 g (ho - ho_L) / m$ and $K_2 = \frac{6 \pi \eta r}{m}$. Then the equation becomes: $\frac{dv}{dt} = K_1 - K_2 v$Integrating this differential equation with initial conditions ($v=0$ at $t=0$): $\int_0^v \frac{dv}{K_1 - K_2 v} = \int_0^t dt$This yields: $v = \frac{K_1}{K_2} (1 - e^{-K_2 t})$As $t 	o \infty$,the velocity approaches a constant terminal velocity $v_T = \frac{K_1}{K_2}$. The graph of $v$ versus $t$ is an exponential curve that starts from the origin and asymptotically approaches the terminal velocity,which corresponds to Graph $B$.

$A$ small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity-time graph for the transit of the ball?

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