Question

For a certain mechanical element the constitutive relation is Y = AV^3, where Y is a system variable, a is a constant, and V is velocity. Give each answer below in terms of V. (a) If Y is the force, i.e. (Y = F), find an expression for the power in the element? (b) If Y is the linear momentum, i.e. (Y = p), find an expression for the energy stored in the element?

Answer 1

Answer:

a) Power = Av⁴

b) Energy = Av⁴/2 or (A²v⁶)/2m

Explanation:

Given Y = Av³

A = a constant, v = Velocity, Y = system variable.

a) If Y = Force, F, Find Power in the element.

Power is the dot product of Force and velocity and it's a scalar quantity.

i.e. P = F.v = F.v (cos θ) where θ is the angle between the force and velocity vector.

But in this case, average power is simply given by Fv.

P(avg) = Fv = Yv = (Av³) × v = Av⁴

b) If Y = linear momentum, p, Find the energy stored in the element.

Energy is related to linear momentum by the relationship between kinetic energy and linear momentum.

p = mv and E = mv²/2 = (mv)(v)/2, so,

E = pv/2

For this question, p = Y = Av³

E = Yv/2 = (Av³)v/2 = Av⁴/2

Kinetic energy is often related to momentum through this expression too,

p = mv; p² = m²v²

E = mv²/2; E = (mv²/2) × (m/m) = m²v²/2m = p²/2m

Therefore, E = Y²/2m = (Av³)²/2m = (A²v⁶)/2m

Hope this helps!