Question

In a horizontal circular path of radius R ,total acceleration vector always makes an angle of 60 with the velocity vector.if at t=0,velocity of the particle is v* then find time taken to complete one revolution

Answer 1

Answer:

Time period to complete one revolution is given as

$t = sqrt3 R (\frac{1}{v_o} - \frac{1}{v_o e^{\frac{2\pi}{\sqrt3}}})$

Explanation:

As we know that total acceleration makes 60 degree with velocity vector

so we have

$tan 60 = \frac{a_c}{a_t}$

so we have

$\sqrt3 \frac{dv}{dt} = \frac{v^2}{R}$

$\sqrt3 \frac{dv}{v^2} = \frac{1}{R} dt$

$\sqrt3 (\frac{1}{v_o} - \frac{1}{v}) = \frac{t}{R}$

Now we also know that

$\sqrt3 v\frac{dv}{ds} = \frac{v^2}{R}$

$\sqrt3 \frac{dv}{v} = \frac{1}{R} ds$

for one complete round

$\sqrt3 ln\frac{v}{v_o} = 2\pi$

so we have

$v = v_o e^{\frac{2\pi}{\sqrt3}}$

now we have time period given as

$t = sqrt3 R (\frac{1}{v_o} - \frac{1}{v_o e^{\frac{2\pi}{\sqrt3}}})$

#Learn

Topic : Circular motion

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