1. **Problem statement:** Calculate the coefficient of static friction $\mu_s$ for a system where a 25 kg block rests on a rough horizontal plane connected by a string over a smooth pulley to a 15 kg hanging block. The system is on the verge of moving. 2. **Relevant formula and concepts:** The system is in equilibrium at the threshold of motion, so the frictional force $f$ balances the tension caused by the hanging mass. Static friction force: $$f = \mu_s N$$ Normal force $N$ on the horizontal block equals its weight: $$N = m_1 g$$ Tension $T$ equals the weight of the hanging block: $$T = m_2 g$$ At impending motion, friction force equals tension: $$\mu_s m_1 g = m_2 g$$ 3. **Substitute values:** $$\mu_s \times 25 \times g = 15 \times g$$ Cancel $g$ on both sides: $$\mu_s \times 25 = 15$$ 4. **Solve for $\mu_s$:** $$\mu_s = \frac{15}{25} = \frac{3}{5}$$ 5. **Interpretation:** The coefficient of static friction required to prevent motion is $\frac{3}{5}$. **Final answer:** $\boxed{\frac{3}{5}}$ corresponds to option (b).