Water drives a waterwheel (or turbine) of radius R=3.0 m as shown in Fig. 47 . The water enters

Water drives a waterwheel (or turbine) of radius R=3.0 m as...

Water drives a waterwheel (or turbine) of radius $R=3.0 \mathrm{m}$ as shown in Fig. $47 .$ The water enters at a speed $v_{1}=7.0 \mathrm{m} / \mathrm{s}$ and exits from the waterwheel at a speed $v_{2}=3.8 \mathrm{m} / \mathrm{s} .$ (a) If 85 $\mathrm{kg}$ of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 $\mathrm{s}$ , how much power is delivered to the wheel?

Water drives a waterwheel (or turbine) of radius $R=3.0 \mathrm{m}$ as shown in Fig. $47 .$ The water enters at a speed $v_{1}=7.0 \mathrm{m} / \mathrm{s}$ and exits from the waterwheel at a speed
$v_{2}=3.8 \mathrm{m} / \mathrm{s} .$ (a) If 85 $\mathrm{kg}$ of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 $\mathrm{s}$ , how much power is delivered to the wheel?

Key Concepts

Mass Flow Rate

Mass flow rate is a measure of the quantity of mass that passes through a given surface or control volume per unit time. It plays a crucial role in fluid dynamics and enables the calculation of momentum and energy fluxes in systems involving continuous mass movement.

Angular Momentum

Angular momentum is the rotational counterpart of linear momentum and quantifies the amount of rotation an object has, taking into account its mass, shape, and rotational speed. It is defined as the cross product of an object's position vector and its linear momentum, and is conserved in isolated systems, making it central to the analysis of rotational dynamics.

Torque

Torque is the measure of the force that can cause an object to rotate about an axis. It is the rotational analogue of force and is mathematically defined as the rate of change of angular momentum. Torque is fundamental in determining how forces lead to rotational acceleration in mechanical systems.

Angular Velocity

Angular velocity describes how fast an object rotates or revolves relative to another point, typically expressed in radians per second. It provides a link between the rotational motion of an object and its linear speed at points along its radius, which is essential in analyses involving rotational kinematics and dynamics.

Power in Rotational Systems

Power in rotational systems is the rate at which energy is transferred or work is done by a rotating object. It is closely related to both torque and angular velocity, with power calculated as their product, providing insight into the efficiency and performance of mechanical systems involving rotation.

Transcript

00:01 In this problem, we're going to talk about angular momentum and torque.

00:04 So first, remember that the angular momentum l associated with rotating particle is equal to the mass of the particle times the radius of rotation times the linear accelerate, i'm sorry, the linear speed v.

00:25 Also, remember that the torque is equal to the change in angular moment.

00:34 Divided by the change in time.

00:37 So it's delta l divided by delta t.

00:40 And finally what we're going to need to our exercise is that the work exerted can be calculated as the torque exerted over a particle times the variation in the angular position of that particle.

01:02 So now i can go into our problem.

01:04 We have a water wheel that has a radius of 3 meters.

01:10 And we have water that enters the water wheel with an initial speed of 7 meters per second and exits the water wheel with a final speed of 3 .8 meters per second.

01:27 And in question a, we have to consider that the flux of water through the water wheel is 85 kilograms per second.

01:36 And we have to calculate what rate, i'm sorry, what is the rate with which the water delivers angular momentum to the water wheel.

01:53 So the water loses angular momentum and this angular momentum ends up with the water wheel.

02:00 Okay, so let's calculate what is the loss in angular momentum of the water? notice that the initial angular momentum is the mass of the water.

02:11 This actually is the mass divided by time.

02:14 So this is the mass of the water times the radius of the water wheel times the initial velocity, while the final one is the mass times the radius times the final velocity.

02:26 So the change in angular momentum is mr times vf minus vi.

02:34 Okay, so this is 85 kilograms.

02:38 In one second, we're going to calculate the change in angular momentum per unit time.

02:45 So it's 85 kilograms per second times three meters times 7 minus 3 .8 that is 3 .2 meters per second...

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