A uniform rod and a block of equal mass are arranged as shown. Initially, the rod is in horizontal position. When the system is released from this position, the acceleration of the tip of the rod is (3/K)g. Find the value of K. [pulley and string are ideal]
Physics · Mixed Chapter
JEE Mains

Question

A uniform rod and a block of equal mass are arranged as shown. Initially, the rod is in horizontal position. When the system is released from this position, the acceleration of the tip of the rod is (3/K)g. Find the value of K. [pulley and string are ideal]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let the rod length be L, its mass m, angular acceleration α and tension T in the vertical cord. About the hinge: TL−mg(L/2)=Iα=(mL²/3)α. Step 2: Immediately after release ω=0, so the rod-tip vertical acceleration is a=αL. Hence a=3T/m−(3/2)g. Step 3: By string constraint, the equal-mass suspended block accelerates downward with the same magnitude a. Its equation is mg−T=ma, i.e. T/m=g−a. Step 4: Substitution gives a=3(g−a)−1.5g ⇒4a=1.5g ⇒a=3g/8. Thus K=8.