# Torque, angular momentum and kinetic energy of water wheels.

• PhysicsStudy
In summary, the correct calculations for torque, angular momentum, and rotational kinetic energy of the water wheel are 115657 Nm, 40458 kg*m^2/s, and 60797 J, respectively. The formulas for torque and moment of inertia used should be (mr^2)/2 and Angular momentum = Moment of inertia * Angular velocity * sin(theta), respectively. The angular velocity used in the calculations should be (4pi)/15 rad/s.
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## 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
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

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?

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!

## 1. What is torque in relation to water wheels?

Torque is the measure of the turning force of a water wheel. It is the product of the force applied to the wheel and the distance from the center of rotation to the point where the force is applied. In other words, it is the force that causes a water wheel to rotate.

## 2. How does angular momentum affect water wheel performance?

Angular momentum is a measure of the rotational motion of a water wheel. The greater the angular momentum, the faster the wheel will rotate. This is important because the speed of rotation determines the amount of work that can be done by the water wheel.

## 3. How is kinetic energy related to water wheels?

Kinetic energy is the energy an object possesses due to its motion. In the case of water wheels, the kinetic energy of the water is transferred to the wheel as it flows over the paddles, causing them to turn. This kinetic energy is then converted into mechanical energy, which can be used to do work.

## 4. What factors affect the torque, angular momentum, and kinetic energy of water wheels?

The torque, angular momentum, and kinetic energy of water wheels can be affected by various factors such as the speed and volume of water flow, the size and shape of the paddles, and the weight and diameter of the wheel. Additionally, the efficiency of the water wheel's design and any external forces, such as friction, can also impact these factors.

## 5. How can the torque, angular momentum, and kinetic energy of water wheels be optimized?

To optimize the torque, angular momentum, and kinetic energy of water wheels, careful consideration must be given to the design and construction of the wheel. Factors such as paddle size and shape, wheel weight and diameter, and water flow rate should be carefully calculated and adjusted to maximize the performance of the water wheel. Additionally, minimizing any external forces, such as friction, can also help to optimize the efficiency of the wheel.

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