\(\mathbf{125V}\)
Step 1: Understanding the volume conservation principle When two soap bubbles coalesce under isothermal conditions, the process follows the principle of volume conservation: \[ V_{\text{final}} = V_1 + V_2 \] where: - \( V_1 = 27V \) (volume of first bubble) - \( V_2 = 64V \) (volume of second bubble)
Step 2: Calculating the new total volume \[ V_{\text{final}} = 27V + 64V \] \[ V_{\text{final}} = 91V \]
Step 3: Applying the radius-volume relation for bubbles Since the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] The radius of the final bigger bubble is given by: \[ r_{\text{final}}^3 = r_1^3 + r_2^3 \] where: \[ r_1^3 = 27V, \quad r_2^3 = 64V \] \[ r_{\text{final}}^3 = 27V + 64V = 91V \] The volume of the bigger bubble formed is given by: \[ V_{\text{new}} = (r_{\text{final}})^3 = (3V + 4V)^3 = (5V)^3 = 125V \]
Step 4: Verifying the correct option Comparing with given options, the correct answer is: \[ \mathbf{125V} \]
Two soap bubbles of radius 2 cm and 4 cm, respectively, are in contact with each other. The radius of curvature of the common surface, in cm, is _______________.
Consider an isolated system of two concentric spherical black bodies. The inner sphere of radius \( R \) is at temperature \( T \), and the outer sphere of radius \( 4R \) is at temperature \( 2T \). The rate of absorption of radiant energy by the outer sphere is: