∙A 22,500 N elevator is to be accelerated upward by...

\bullet A $22,500$ N elevator is to be accelerated upward by connecting it to a counterweight using a light (but strong!) cable passing over a solid uniform disk-shaped pulley. There is no appreciable friction at the axle of the pulley, but its mass is 875 $\mathrm{kg}$ and it is 1.50 $\mathrm{m}$ in diameter. (a) How heavy should the counterweight be so that it will accelerate the elevator upward through 6.75 $\mathrm{m}$ in the first 3.00 $\mathrm{s}$ , starting from rest? (b) Under these conditions, what is the tension in the cable on each side of the pulley?

Key Concepts

Tension in Cable-Pulley Systems

In systems where a cable passes over a pulley, tensions in the cable segments can differ if the pulley has a non-negligible moment of inertia. The difference in tension creates a net torque on the pulley that accelerates it rotationally. Understanding this concept is key to analyzing how forces are transmitted between connected masses in such systems.

Newton's Second Law for Translation

This concept states that the net force acting on an object is equal to its mass multiplied by its acceleration (F=ma). In problems involving moving masses like an elevator and a counterweight, it is used to relate the forces due to weight and tension to the acceleration of the masses, allowing one to compute the net force required for a given acceleration.

Rotational Dynamics of a Rigid Body

Rotational dynamics deals with the motion of objects that rotate about an axis, and it is governed by the equation ? = I?, where ? is the net torque, I is the moment of inertia, and ? is the angular acceleration. In systems that include a pulley, the rotational inertia of the pulley affects the dynamics, causing a difference between the tensions on either side of the pulley.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a uniform disk, the moment of inertia is calculated using the formula I = (1/2) mR^2. This is crucial for determining how the mass and geometry of a rotating pulley influence its angular acceleration when a net torque is applied.

Kinematic Equations for Constant Acceleration

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, acceleration, and time. In problems where an object must travel a specified distance in a given time from rest, these equations are used to determine the required acceleration, which then feeds into the force and torque analysis.

Transcript

0:00 Hello.

00:03 In problem number 13, we're looking at an elevator that will be accelerated upward by connecting it to a counterweight over a uniform cylindrical pulley.

00:18 So let's take a look at the numbers we have right now.

00:23 We know that this is the elevator and it is got a 22 ,500 newton weight.

00:37 And we don't know the size of our counterweight.

00:45 The pulley has a diameter of 1 .5, which gives us then a radius here of 0 .75.

00:55 We know the mass of that pulley, 875 kilograms.

01:03 So first question i ask this is what is the size of this counterweight.

01:16 And we also need to know what's the tension in both sides on the rope on both sides.

01:26 Now, we get us a little help here in that they tell us that the elevator will be accelerating upward, a distance of 6 .75.

01:37 So the distance that this elevator will be moving is 6 .75 meters.

01:44 And it does that in three seconds.

01:50 We know also that it's the velocity initial is zero.

01:59 So that helps us a little bit there.

02:03 And gives us a few things to help us.

02:05 First, we can use this information to find, our acceleration of our elevator.

02:13 I'm going to do this on the other page here.

02:16 So we know using our kinematics formulas that we can find the acceleration.

02:25 We were given this displacement of this elevator is 6 .75.

02:30 We know it started from rest, and that occurred in three seconds.

02:39 Solving 4a, we get an acceleration of 1.

02:43 1 .5 meters per second squared.

02:50 So that helps us out.

02:56 And remember it's going up, which means this counterweight would be coming down.

03:00 Now, we need to look at kind of the three separate things going on here.

03:05 First, i'm going to look at the elevator itself from a three -body standpoint.

03:11 We know it has a downward weight of 22 ,500 newtons.

03:18 And we have enough for tension.

03:21 And i'm going to call this tension the left tension.

03:26 Now, looking at the counterweight, similar, free body.

03:31 We just don't know what that weight is, but we know this, and i'm going to call this the right tension.

03:41 Now, we do have some torque on our pulley system here that we will want to look at.

03:50 We know that there is a torque that's going to rotate this pulley clockwise, and that is due to the tension in the right cable.

04:03 And then we also have the tension in the left cable that's going to do a counterclockwise rotation.

04:12 Calculate here, because we will need to know our moment of inertia.

04:18 Moment of inertia for a cylinder.

04:26 So our moment of inertia of this cylinder is one -half mr -squared.

04:52 In this case, it's one -half.

04:57 Our mass is 875 times diameter squared.

05:07 This actually is 246 kilogrammeater square.

05:13 Okay...

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