Help with a problem please?

Question

ii) under uniform circular motion, radial forces do no work and therefore produce no power because the force is perpendicular to the particles velocity. A radial force with position vector r originating from (0,0) for a particle of mass m in uniform circular motion is expressed by :

F = m dv/dt r = rcos(theta) i + rsin(theta) j and v = dr /dt

w (omega) = dtheta /dt

Verify no power is produced by a radial force . . . by performing the operation P = F . v = 0

in the above equation i think the period is the bigger dot meaning F dotted into v. . . but any help wold be appreciated

Answer 1

F = m dv/dt = ma

P = power = F dot v = m[a dot v] = m [a v cos 90] =0

here acceleration and velocity are at cross (90 deg) so dot product vanishes by cos 90

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for circle >> x^2 +y^2 =r^2

x = r cos θ  > y = r sin θ

vector r = x i + yj = r [cos(θ) i + sin (θ) j]

r = constant so no derivative with time

dx/dt = - r sin θ, dy/dt = r cos θ

velocity vector = v = vx i + vy j = r [- sin θ i + cos θ j]

a = acceleration = - r [cos θ i + sin θ j]

F = - m r [cos θ i + sin θ j]

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P= F dot v = - mr^2 {[cos θ i + sin θ j] DOT [- sin θ i + cos θ j]

P= - mr^2 {- sin θ cos θ + sin θ cos θ} = 0 proved

i dot i = 1 = j dot j

i dot j = j dot i =0

complete help