Question

Tangential acceleration of a particle moving in a circle of radius 1m varies with time t as ( initial velocity of particle is zero ). Time after which total acceleration of particle makes and angle of 30 degree with radial acceleration is

Answer 1

see figure from graph,

slope of graph , $\frac{a_t}{t}$ = tan60° = √3

⇒$a_t$ = √3t ........(1)

⇒dv/dt = √3 t

⇒∫dv = √3 ∫t dt

⇒v = (√3/2)t²

⇒v² = (3/4)t⁴

centripetal acceleration,$a_c$ = v²/r = (3/4)t⁴/1 = (3/4)t⁴

given, angle between tangential acceleration and centripetal acceleration, θ = 30°

$\frac{a_t}{a_c}$ = tan30° = 1/√3

⇒√3t/(3/4)t⁴ = 1/√3

⇒4/(√3)t³ = 1/√3

⇒t³ = 4

⇒t = (4)⅓

hence, t = 2⅔