Torque, angular momentum and kinetic energy of water wheels.

[PhysicsStudy]

Homework Statement

Calculate the torque, angular momentum and kinetic energy of the water wheels in the table.

[I'll give one example]

High breast water wheel:
Angular velocity/rpm 8
Head/m 4.65
Flow rate*/m3 s−1 1.0
Power/kW 39.4
Diameter/m 7.0
Width or height/m 2.5
Mass/kg 14 143
Power-to-weight ratio/W kg−1 2.8

Homework Equations

Torque = Moment of inertia * Angular acceleration
Moment of inertia of thin solid disc = (mr^2)/2 (about z axis)
Angular momentum = Moment of inertia * Angular velocity
Rotational kinetic energy = (Iw^2)/2

The Attempt at a Solution

8rpm -> (8/60)*2*pi rad/s = (4pi)/15 rad/s

Torque = (14143*3.5^2*(angular velocity))/2 = 72572Nm
Angular velocity has here been used instead of angular acceleration due to explanation found here: https://www.physicsforums.com/showthread.php?t=346163

However, this means that the angular momentum calculation gives the same answer (minus a direction, which will take the direction of the angular velocity vector). Have I done this correctly?

Rotational KE = (14143*3.5^2*((4pi)/15)^2)/2 = 60797J

Do these seem at all right?

 

[KataKoniK]

I appreciate your attempt at solving this problem. However, there are a few mistakes in your calculations that I would like to point out.

Firstly, the formula for torque that you have used is incorrect. The correct formula is Torque = Moment of inertia * Angular acceleration, where the moment of inertia is calculated using the mass and the radius of the water wheel.

Secondly, the moment of inertia formula that you have used is for a thin solid disc rotating about its z-axis. However, in this case, the water wheel is rotating about its center, so the moment of inertia formula should be (mr^2)/2.

Thirdly, in your calculation for angular momentum, you have not taken into account the direction of the angular velocity vector. The correct formula is Angular momentum = Moment of inertia * Angular velocity * sin(theta), where theta is the angle between the moment of inertia and the angular velocity vector.

Finally, your calculation for rotational kinetic energy is correct, but you have used the wrong value for the angular velocity. It should be (8/60)*2*pi rad/s = (4pi)/15 rad/s.

Using the correct formulas, the calculations for torque, angular momentum, and rotational kinetic energy are as follows:

Torque = (14143*(7/2)^2*(4pi)/15)/2 = 115657 Nm
Angular momentum = (14143*(7/2)^2*(4pi)/15)*sin(4pi/15) = 40458 kg*m^2/s
Rotational kinetic energy = (14143*(7/2)^2*((4pi)/15)^2)/2 = 60797 J

I hope this helps to clarify any confusion and leads you to the correct solution. Keep up the good work!