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| 1. |
Prove that f=mv2/r |
| Answer» \xa0Let us consider a particle travel with constant speed from A to B in a circular path of radius r in a short time Δtmaking an angular displacement Δθ as shown in figure.\xa0distance S travelled is the arc length AB and it is given by, S = v×Δt ..................(1)\xa0Arc distance S also written geometrically as, S = r×Δθ ...........................(2)\xa0from (1) and (2), we write , Δθ/Δt = v/r ...............(3)\xa0Eqn.(3) is rate of angular displacement or angular velocity.\xa0For small angular displacement made in short time Δt, we can write = dθ/dt = v/r ......................(4)\xa0In right side figure, velocity vectors at A and B are drawn by merging their initial points.The angle between these vectors also Δθ.\xa0from the right side figure, we write, sin (Δθ/2) = (Δv/2) / v or Δv/2 = v sin(Δθ/2) .................(5)\xa0For small angle Δθ, eqn.(5) is written as, Δv/2 = v Δθ/2 or Δv/Δθ = v or dv/dθ = v .....................(6)\xa0using eqn.(4) and eqn(6), acceleration directed towards centre, dv/dt = ( dv/dθ ) (dθ/dt) = v×(v/r) = ( v2\xa0/ r )\xa0If m is mass, then force F = mass×acceleration = m×(v2\xa0/ r )\xa0 | |