In rising from the bottom of a lake, to the top, the temperature of an air bubble remains unchanged, скачать в хорошем качестве

In rising from the bottom of a lake, to the top, the temperature of an air bubble remains unchanged,

In rising from the bottom of a lake, to the top, the temperature of an air bubble remains unchanged, but its diameter gets doubled. If the pressure on the surface is equal to h meter of mercury column and relative density of mercury is ρ, then the depth of lake in meter, is [NEET] (a) 8ρh m (b) 7ρh m (c) ρh m (d) 12ρh m 🌊 *Physics Behind Air Bubble Expansion Problem* 🌊 This problem focuses on the application of gas laws and pressure concepts to understand how an air bubble behaves as it rises through a liquid. The key aspects revolve around **Boyle's Law**, **fluid pressure variation**, and the relationship between volume and radius. Let’s explore the physics at play. --- 🧪 *Key Physical Concepts* #### *1. Boyle’s Law* Boyle's Law states that for a gas at constant temperature (isothermal process), the product of pressure (\(P\)) and volume (\(V\)) remains constant: \[ P \cdot V = \text{constant}. \] This implies that as pressure decreases, the volume of a gas increases. In the problem, the temperature of the air bubble remains constant, allowing us to apply Boyle’s Law directly. #### *2. Fluid Pressure Variation* The pressure in a fluid increases with depth due to the weight of the liquid column above it: \[ P = P_{\text{surface}} + \rho_{\text{liquid}} g h, \] where: \( P_{\text{surface}} \): Pressure at the liquid's surface, equivalent to the atmospheric pressure. \( \rho_{\text{liquid}} \): Density of the liquid. \( g \): Acceleration due to gravity. \( h \): Depth below the surface. #### *3. Volume and Radius Relation* The volume of a spherical air bubble is proportional to the cube of its radius (\(r\)): \[ V \propto r^3. \] If the diameter of the bubble doubles as it rises, its radius also doubles, and its volume increases eightfold: \[ V_{\text{final}} = 2^3 \cdot V_{\text{initial}} = 8 \cdot V_{\text{initial}}. \] #### *4. Relationship Between Liquid and Mercury Pressures* The pressure at the surface is given in terms of a mercury column, with its height \(h\) and relative density (\(\rho_{\text{Hg}}\)): \[ P_{\text{surface}} = \rho_{\text{Hg}} g h. \] --- 📚 *Analyzing the Problem* 1. **Initial and Final Conditions**: At the bottom of the lake, the bubble experiences high pressure due to the weight of the water column. As it rises, the pressure decreases, causing the bubble to expand. At the surface, the pressure equals \( P_{\text{surface}} \). 2. **Using Boyle’s Law**: At constant temperature: \[ P_{\text{bottom}} V_{\text{initial}} = P_{\text{surface}} V_{\text{final}}. \] Substituting \( V_{\text{final}} = 8 \cdot V_{\text{initial}} \): \[ P_{\text{bottom}} = 8 \cdot P_{\text{surface}}. \] 3. **Calculating Depth**: The pressure at the bottom of the lake is: \[ P_{\text{bottom}} = P_{\text{surface}} + \rho_{\text{water}} g H, \] where \( H \) is the depth of the lake. Equating pressures: \[ 8 \cdot P_{\text{surface}} = P_{\text{surface}} + \rho_{\text{water}} g H. \] Rearrange to find \( H \): \[ \rho_{\text{water}} g H = 7 \cdot P_{\text{surface}}. \] Substituting \( P_{\text{surface}} = \rho_{\text{Hg}} g h \): \[ H = 7ρh. \] --- 🔬 *Theoretical Insights* This problem beautifully demonstrates the interplay between fluid mechanics and gas laws. The expansion of the bubble reflects the effect of decreasing pressure as the bubble rises, while the proportional relationship between pressures highlights how different fluids (water and mercury) are connected in terms of their densities. The solution provides a practical example of *Boyle's Law* in real-life scenarios, such as diving and underwater explorations, where pressure changes significantly impact gas volumes. --- 📌 **Answer**: *(b) 7ρh* --- 📲 *Stay Inspired with Physics Daily!* Follow us for more interesting physics concepts and problem-solving techniques: 👉 [Physics Behind Everything](https://www.instagram.com/physics_beh...) --- 🔖 *Relevant Hashtags for This Topic:* #NEET2025 #FluidMechanics #BoylesLaw #PhysicsExplained #ProblemSolving #UnderwaterPhysics do just one support to us that you don't forget to like and share our video..( if you like our content )... also don't forget to subscribe our channel... it will give a huge support to us from your end... It will be helpful for iit, neet and as well as for cbse, icse,and other state boards examination. for any further query contact us at : 7843900879 #physics #onlinemyschool #physicsbehindeverything facebook :   / pbevns   twitter : https://twitter.com/behind_physics?s=09 Instagram : https://www.instagram.com/physics_beh... @pbe @physicsbehind everything