Question #1904

The angular position of a particle is given by theta(t) = b + ct + dt2, where b, c, and d are constants. The radial displacement of the particle is r (t) = a theta(t) where a is a constant. Determine (a) the angular velocity, (b) the angular acceleration alpha, (c) the linear velocity, (d) the transverse acceleration, and (e) the centripetal acceleration of the particle.

Expert's answer

a) The angular velocity, ω =dθ (t)/dt = c + 2d* t.

b) The angular acceleration, α = d ω /dt = 2d.

c) The linear velocity, v = dr/dt = a* dθ/dt = a( c + 2d*t.)

d) The transverse acceleration, at = dv/dt = a {2d}

e) the centripetal acceleration of the particle

ac = ω^{2}*r = (c+2d*t)^{2}*a*(b + ct + dt^{2}) = 2a(c^{2}+4cdt+4d^{2}t^{2})*(b + ct + dt^{2}) =

=2a(bc^{2} + tc^{3} + dct^{2} + 4dbct + 4dc^{2}t^{2 }+ 4d^{2}ct^{3} + 4d^{2}bt^{2} + 4d^{2}ct^{3} + 4d^{3}t^{4})

b) The angular acceleration, α = d ω /dt = 2d.

c) The linear velocity, v = dr/dt = a* dθ/dt = a( c + 2d*t.)

d) The transverse acceleration, at = dv/dt = a {2d}

e) the centripetal acceleration of the particle

ac = ω

=2a(bc

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