Answer
Given,
θ(t) = At² + Bt⁴
where A = 1.10
B= 1.60
a) calculating the unit of A and B
Unit of A = unit of θ / unit of t²
= rad/s²
Unit of B = unit of θ / unit of t⁴
= rad/s⁴
(b)
calculating angular speed at time t= 3 s
mass m = 14 kg
diameter d = 49 cm
radius r= d / 2 = 24.5 cm = 0.245 m
Angular momentum of the sphere
L = I w
I = moment of inertia
I = ( 2/ 3) m r²
I =0.667 x 14 x 0.245²
I = 0.5605 kg m²
angular speed
ω = dθ / dt
= 2At + 4Bt^3
= ( 2 x 1.10 x 3) + ( 4 x 1.6 x 3³ )
= 179.4 rad / s
So, angular momentum
L =0.5605 x 179.4
L = 100.55 kg m²/s
(c)now, calculating net torque
T = I α
α = angular acceleration = dω / dt
= 2 A + 12 B t²
= 2 x 1.1 + 12 x 1.6 x 3²
= 175 rad / s²
T = 175 x 0.5605
T = 98.08 N m
