Question

When studying whiplash resulting from rear end collisions, the rotation of the head is of primary interest. an impact test was performed, and it was found that the angular acceleration of the head is defined by the relation α = 700cosθ + 70sinθ, where α is expressed in rad/s2 and θ in radians. knowing that the head is initially at rest, determine the angular velocity of the head when θ = 30°?

Answer 1

Like how velocity is the integral of acceleration, angular velocity is the integral of angular acceleration. We call angular velocity ω and we need to integrate that function to get it. Remember ∫cos(x) = sin(x) and ∫sin(x) = -cos(x) and integrate α to get the equation for ω, which is ω = 700sin(θ) - 70cos(θ), then we need to convert 30° into radians, multiply it by (π/180°) to do the conversion, which gives θ = π/6. Plugging numbers in gives ω = 700sin(π/6) - 70cos(π/6) = 289.37 rad/s.