CBSE Class 9 Answered
F=mv^2/r. How.
Asked by suman04naha | 13 Aug, 2019, 07:46: PM
Expert Answer
Let us consider a particle travel with constant speed from A to B in a circular path of radius r in a short time Δt
making an angular displacement Δθ as shown in figure.
distance S travelled is the arc length AB and it is given by, S = v×Δt ..................(1)
Arc distance S also written geometrically as, S = r×Δθ ...........................(2)
from (1) and (2), we write , Δθ/Δt = v/r ...............(3)
Eqn.(3) is rate of angular displacement or angular velocity.
For small angular displacement made in short time Δt, we can write = dθ/dt = v/r ......................(4)
In right side figure, velocity vectors at A and B are drawn by merging their initial points.
The angle between these vectors also Δθ.
from the right side figure, we write, sin (Δθ/2) = (Δv/2) / v or Δv/2 = v sin(Δθ/2) .................(5)
For small angle Δθ, eqn.(5) is written as, Δv/2 = v Δθ/2 or Δv/Δθ = v or dv/dθ = v .....................(6)
using eqn.(4) and eqn(6), acceleration directed towards centre, dv/dt = ( dv/dθ ) (dθ/dt) = v×(v/r) = ( v2 / r )
If m is mass, then force F = mass×acceleration = m×(v2 / r )
Answered by Thiyagarajan K | 13 Aug, 2019, 10:50: PM
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