The rotor of an electric motor has an angular velocity of...

Name: Problem 2, Chapter 17: Vector Mechanics for Engineers: Statics and Dynamics
Uploaded: 2020-06-29T23:33:49-08:00
Duration: 7 min 29 s
Channel: Nathan Silvano
Description: The rotor of an electric motor has an angular velocity of 3600 rpm when the load and power are cut off. The 110-lb rotor, which has a centroidal radius of gyration of 9 in., then coasts to rest. Knowing that the kinetic friction of the rotor produces a couple with a magnitude of 2.5 lb?ft, determine the number of revolutions that the rotor executes before coming to rest.

The rotor of an electric motor has an angular velocity of 3600 rpm when the load and power are cut off. The 110-lb rotor, which has a centroidal radius of gyration of 9 in., then coasts to rest. Knowing that the kinetic friction of the rotor produces a couple with a magnitude of 2.5 lb?ft, determine the number of revolutions that the rotor executes before coming to rest.

Key Concepts

Work-Energy Principle in Rotational Motion

The work-energy principle states that the work done by a force (or torque, in rotational dynamics) on an object is equal to its change in kinetic energy. In the context of rotational motion, the work done by the frictional torque is equal to the loss of rotational kinetic energy, which enables the analysis of how many revolutions occur before the system comes to rest.

Torque and Angular Deceleration

Torque, the rotational equivalent of force, causes an angular acceleration or deceleration in rotating bodies. In the presence of friction, a constant torque opposing the motion results in a steady angular deceleration. By linking the frictional torque to the change in angular velocity over time, one can determine the distance (in terms of revolutions) that the object covers before stopping.

Moment of Inertia

The moment of inertia quantifies an object’s resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. When given a centroidal radius of gyration, the moment of inertia is determined by multiplying the mass (or weight, with proper conversion) by the square of that radius, serving as a key factor in energy and angular deceleration calculations.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a rotating object, expressed as 1/2 times the moment of inertia times the square of the angular velocity. It represents the work required to either set the object into rotation or bring it to rest, thereby connecting the energy stored in rotation to the forces and torques acting on the system when deceleration is involved.

Transcript

00:01 Alright, so in this problem, we just need to calculate the number of revolutions that the rotor executes before stopping.

00:11 So, first of all, let's remember that to calculate this, we should use the principle of work and energy.

00:22 Because we know that since this rotor is moving, we know he has a connecting energy, therefore we have that the initial energy, therefore we have that the initial energy.

00:31 Initial connecting energy plus the potential energy generated by the force, the momentum, is going to be equal the final connecting energy.

00:46 So we just need to discover each one of these quantities to solve what we really want, which is the number or revolutions.

00:59 Okay, so how we're going to manage to calculate this? first of all, we know that in the first position, the initial position, we have an angular speed that is equal 3 ,600 rotations per minute.

01:23 And that is also equal 120 pi rads per second.

01:33 Radians per second.

01:37 Okay, so we know that to calculate the connecting energy of rotation, k1, is going to be equal half of the moment of inertia that multiplies omega -1 square.

01:57 Okay, so what is the moment of inertia of this water? so let's put here the moment of inertia is going to be equal m, which is the mass, multiply by the radius is square.

02:15 But the problem here in this problem is that we do not have the mass of the water.

02:20 So we need to use the weight of the model because we know that mass is just the weight divided by the gravity's acceleration.

02:35 Therefore, we can simplify our kinetic energy in here to half omega -1 square, which we already have this, and this is going to multiply the moment of radiation.

02:50 The moment of inination right now is going to be the weight divided by g that multiplies the radius square.

03:00 Okay, so that's the equation we should solve.

03:04 So first of all, let's substitute all the numerical values in here.

03:10 So k -1 is going to be equal half omega.

03:17 Omega is 120 pi square.

03:22 The multiplies the weight the weight is going to be 1 .10 divided by the gravity which is going to be it's going to be let me see 32 .2 because we are not in the international system of units therefore we cannot use 9 .8 so this multiply the radius which is just 9 divided by 12 square...

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