A uniform plank of mass m, free to move in the horizontal direction only, is placed at the top of a solid cylinder of mass 2m and radius R. The plank is attached to a fixed wall by means of a light spring of spring constant k. There is no slipping between the cylinder and the plank, and between the cylinder and the ground. Find the time period of small oscillations of the system if T = π√(xm/k), where x = _____.
Physics · Mixed Chapter
JEE Mains

Question

A uniform plank of mass m, free to move in the horizontal direction only, is placed at the top of a solid cylinder of mass 2m and radius R. The plank is attached to a fixed wall by means of a light spring of spring constant k. There is no slipping between the cylinder and the plank, and between the cylinder and the ground. Find the time period of small oscillations of the system if T = π√(xm/k), where x = _____.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: For rolling without slipping on the ground and under the plank, velocity of the cylinder centre is v/2 if the plank moves at v, and cylinder angular speed is ω=v/(2R). Step 2: Kinetic energy of plank is ½mv². Cylinder translation is ½(2m)(v/2)²=mv²/4. Cylinder rotation with I=(1/2)(2m)R²=mR² is ½mR²(v/2R)²=mv²/8. Step 3: Total K=(7/8)mv²=½(7m/4)v²; effective mass for the spring coordinate is 7m/4. Step 4: T=2π√[(7m/4)/k]=π√(7m/k). Hence x=7.