A 0.4m diameter, 50KG, solid pulley wheel is used to derive a conveyor belt system on a production line. The belt is very light and it's mass can be taken as zero. The pulley wheel is accelerated from rest at 2 rads/s^2 for 3 secs, then rotated at a constant velocity for a further 10 secs, before being decelerated uniformly back to rest in 2 secs.
1) How far does the belt move during the whole 15 secs?
2) What power is required to accelerate the pulley wheel during the acceleration phase (i.e) during initial 3 secs.
(a) w(angular velocity) = Wo (Intial angular velocity) + & (angular acceleration) x t (time)
(b) @(Pheta) = Wo x t + 1/2&t^2
The Attempt at a Solution
I split the calculations into 3 parts Part A - initial acceleration phase (3 secs). Part B - Constant velocity phase (10 secs). Part C - deacceleration phase.
For part A I used the formula (a) to work out the angular velocity at 6 rads/s. I then used formula (b) to find the angular displacement of 9 rads.
For part B I used the formula (b) to work out the angular displacement as 60 rads. (no acceleration just an intial velocity)
For part C I rearranged formula (a) to find the angular deceleration which was -3 rad/s^2. i then put this value into formula (b) to get 6 rads.
I then added up all the angular displacements to give me 9+60+6 = 75 rads. I then converted this to revolutions, so 75/2Pi to give 11.94. The distance travelled is then the circumference of the wheel (Pi x d) multiplied by the revolutions which I worked out at 15 meters. Thats my answer for question 1.
For question 2 I am now a little confused. I have done it this way.
using the following formulas
Radius of Gyration (k) = 0.707 x radius = 0.707 x 0.2 = 0.1414
Moment if Inertia (J) = mk^2 = 50KG x (0.1414)^2 = 0.999698
I have Torque = J x angular acceleration = 0.999698 x 2 rads/s^2 = 2 N/m
Power = Torque x angular velocity = 2N/m x 6 rads/s = 12 watts.
Now I also have a companion who has attempted this question and he tried it this way:
KE rot = 1/2Jw^2= 17.99 Joules.
He says Torque = KErot/angular displacement = 18/9 = 2 N/m
The power = Torque x speed = 2 x (6 x 2Pi) = 75.4 watts
I am confused as to what the actual answer should be for part 2. Any pointers would be much appreciated.