In the experiment for measurement of viscosity \( \eta \) of a given liquid with a ball having radius \( R \), consider following statements:
A. Graph between terminal velocity \( V \) and \( R \) will be a parabola.
B. The terminal velocities of different diameter balls are constant for a given liquid.
C. Measurement of terminal velocity is dependent on the temperature.
D. This experiment can be utilized to assess the density of a given liquid.
E. If balls are dropped with some initial speed, the value of \( \eta \) will change.
Analyze each statement.
A: Incorrect, as the graph is not a parabola but rather a more complex function of radius and viscosity.
B: Incorrect, as terminal velocity varies with ball size and density.
C: Correct, as viscosity and terminal velocity are temperature-dependent.
D: Correct, as variations in terminal velocity can reflect differences in liquid density.
E: Correct, as the initial speed affects the drag force and settling time, influencing the measured viscosity.
Let \[ I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}} (x+15)^{\frac{15}{13}}} \] If \[ I(37) - I(24) = \frac{1}{4} \left( b^{\frac{1}{13}} - c^{\frac{1}{13}} \right) \] where \( b, c \in \mathbb{N} \), then \[ 3(b + c) \] is equal to:
For the thermal decomposition of \( N_2O_5(g) \) at constant volume, the following table can be formed, for the reaction mentioned below: \[ 2 N_2O_5(g) \rightarrow 2 N_2O_4(g) + O_2(g) \] Given: Rate constant for the reaction is \( 4.606 \times 10^{-2} \text{ s}^{-1} \).