1. **Problem 10:** A metal cube X of length $L$ is heated gaining thermal energy $Q$, causing a temperature rise of $\Delta T$. A second cube Y of length $2L$, made of the same material, gains thermal energy $2Q$. Find the temperature rise of Y. 2. **Formula and concepts:** The thermal energy gained by a body is related to its mass, specific heat capacity, and temperature change by: $$Q = mc\Delta T$$ where $m$ is mass, $c$ is specific heat capacity (same for both cubes), and $\Delta T$ is temperature change. 3. Since both cubes are made of the same material, $c$ is constant. 4. The mass $m$ of a cube is proportional to its volume: $$m \propto V = L^3$$ 5. For cube X: $$m_X = \rho L^3$$ For cube Y: $$m_Y = \rho (2L)^3 = \rho 8L^3 = 8 m_X$$ 6. Using the formula for cube X: $$Q = m_X c \Delta T$$ 7. For cube Y: $$2Q = m_Y c \Delta T_Y = 8 m_X c \Delta T_Y$$ 8. Substitute $Q = m_X c \Delta T$ from step 6: $$2 m_X c \Delta T = 8 m_X c \Delta T_Y$$ 9. Cancel $m_X c$ from both sides: $$2 \cancel{m_X c} \Delta T = 8 \cancel{m_X c} \Delta T_Y$$ 10. Simplify: $$2 \Delta T = 8 \Delta T_Y$$ 11. Divide both sides by 8: $$\frac{2 \Delta T}{8} = \Delta T_Y$$ 12. Simplify fraction: $$\Delta T_Y = \frac{\cancel{2} \Delta T}{\cancel{8}} = \frac{\Delta T}{4}$$ **Answer for problem 10:** $\boxed{\frac{\Delta T}{4}}$ (Option B) --- 13. **Problem 11:** Which graph represents the variation with displacement of the potential energy $P$ and total energy $T$ of a system undergoing simple harmonic motion (SHM)? 14. **Concepts:** In SHM, total mechanical energy $T$ is constant and equals the sum of kinetic and potential energies. 15. Potential energy $P$ varies with displacement $x$ as: $$P = \frac{1}{2} k x^2$$ which is a U-shaped parabola with minimum at $x=0$. 16. Total energy $T$ is constant and represented by a horizontal line above the minimum of $P$. 17. Both graphs A and B show total energy as a horizontal line and potential energy as a U-shaped curve with minimum at zero displacement. 18. The difference is the position of the graph in the coordinate plane (bottom-left vs bottom-right), but this does not affect the physical interpretation. 19. Therefore, both graphs correctly represent the variation of $P$ and $T$ with displacement in SHM. **Answer for problem 11:** Both graphs A and B correctly represent the variation. --- **Summary:** - Problem 10 answer: $\Delta T_Y = \frac{\Delta T}{4}$ (Option B) - Problem 11 answer: Both graphs A and B correctly represent the variation of potential and total energy in SHM.