The density of a rod of length L varies linearly with position along its length, such that it increases to thrice its value from one end to the other. This rod is placed on a frictionless horizontal surface and a horizontal force of magnitude F is applied to a point on the rod. At what point should this force be applied such that the rod slides on the surface without rotating?

The density of a rod of length L varies linearly with position along i

1.	The density of a rod of length L varies linearly with position along its length, such that it increases to thrice its value from one end to the other. This rod is placed on a frictionless horizontal surface and a horizontal force of magnitude F is applied to a point on the rod. At what point should this force be applied such that the rod slides on the surface without rotating?
Answer» CONSIDER an element dx at a distance x from one end of the rod of length L.The center of mass of the rod is X cm = ∫ 0L λdx∫ 0L xλdx or X cm = ∫ 0L (A+Bx)dx∫ 0L x(A+Bx)dx = [Ax+Bx 2 /2] 0L [Ax 2 /2+Bx 3 /3] 0L = AL+BL 2 /2AL 2 /2+BL 3 /3 = 63AL 2 +2BL 3 × 2AL+BL 2 2 = 3L(2A+BL)L 2 (3A+2BL) = 3(2A+BL)L(3A+2BL) EXPLANATION:mark as brainlist and follow krna mat bhulna

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