Question

Angular velocity of smooth parabolic wire y = 0.8 x^2 about axis of parabola in horizontal
plane. If bead of mass m does not slip at (a, b) find value of angular velocity in rad/sec.​

Answer 1

Given info : Angular velocity of smooth parabolic wire, y = 0.8x² about axis of parabola in horizontal plane. Beads doesn't slip at (a, b)

To find : the angular velocity in rad/s

Solution : bead of mass m doesn't slip at (a, b)

means, component of weight along tangent = tangential force

⇒mgsinθ = mω²r = mω²acosθ

⇒ω² = gtanθ/a ........(1)

here y = 0.8x²

let's differentiate with respect to x

i.e., dy/dx = 1.6x

slope of tangent of curve y = f(x) is dy/dx

so, and slope of curve make an angle θ with positive x - axis is tanθ

so, dy/dx = tanθ = 1.6x

at (a, b) , dy/dx = tanθ = 1.6a ......(2)

From equations (1) and (2) we get,

so, ω² = g(1.6a)/a = g × 1.6 = 10 × 1.6 = 16

⇒ω = 4 rad/s

Therefore the angular velocity is 4 rad/s