The world's fastest elevatoc, in Taiwan's Taipei 101...

The world's fastest elevatoc, in Taiwan's Taipei 101 skyscraper (Fig. $6.19 \lambda$, ascends at the rate of $1010 \mathrm{~m} / \mathrm{min}$. Counterweights balance the weight of the elevator car, so the motor doesn't have to lift the car's weight. If the motor produces $330 \mathrm{~kW}$ of power, what's the max imum number of 67 -kg people the elevalor can accommodate? (Your answer someshat owerestimales the actual raled lead of 24 peopple.) FIGURE 6.19 Problem 89

Key Concepts

Relationship Between Force, Velocity, and Power

In a lifting scenario, power is also expressed as the product of force and velocity. This relationship is important because it shows that for a fixed power output, increasing the speed of lifting requires handling less mass, or conversely, increasing the mass requires a reduction in speed if the power output remains unchanged.

Counterweight Systems in Mechanical Design

Counterweight systems balance a significant portion of the load in lifting mechanisms like elevators. By offsetting the weight of the car, the motor is required to do work only on the additional weight (such as passengers), illustrating an efficient design strategy in mechanical systems.

Power

Power is the rate at which work is performed or energy is transferred over time. In mechanical systems, it quantifies how quickly energy is used to perform tasks such as lifting a mass against gravity.

Work and Energy in Lifting

When lifting an object, work is done against the force of gravity, which increases the gravitational potential energy of the object. This work is calculated by multiplying the weight of the object by the height it is raised, linking the concept of energy directly to the forces involved.

Transcript

00:01 Hi there, let's analyze and answer this problem.

00:04 The situation is about the world's fastest elevator, which is found in taiwan.

00:10 It's called taipei 101 skyscraper.

00:14 Okay, so the situation and the given values are this.

00:18 This world's fastest elevator is able to move upward at the rate of, so let's say it's a constant rate of 1 ,010 meters per minute.

00:31 It.

00:32 So it's possible because of a very powerful motor that enables it to move at that speed.

00:41 So if the elevator motor has a power output of 330 kilowatts, we are to determine n, wherein n is the number of 67 kilogram person who can ride in the elevator, considering the given power output.

01:04 So how do we answer this problem? we can answer this problem by using the relationship between power output and force applied and the velocity of that particular system.

01:22 In this case, the system is the world's fastest elevator.

01:26 So we say that the power output in terms of applied force and velocity.

01:34 So we actually have here, let's call this as power, i mean the applied force, dot.

01:42 So it's a dot product between two vectors.

01:46 We have the applied force and the velocity vector here.

01:49 Since the two vectors are in the same direction.

01:55 Therefore, for this particular case, the power output is simply the product of the magnitudes of the two vectors.

02:04 So we can simply write it as applied force times the speed.

02:09 We're no longer looking at it as velocity.

02:12 So it's just a magnitude.

02:14 Okay.

02:15 Now, so the force.

02:19 That has to be applied here must be equal to the total weight of n people whose mass is 67 kilograms okay therefore we can say here that this applied force must be equivalent to the mass of a 67 kilogram person times acceleration through to gravity, but this is only for a single person.

02:47 We are to determine the n number of people.

02:50 So this is the total applied force.

02:53 Okay, therefore we can say that the power output is just equivalent to n mass g times the speed loop...

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