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Chapter 10: Problem 89

The mass of an elevator and its occupants is \(1200 \mathrm{kg}\). The electric motor that lifts the elevator can provide a maximum power of \(15 \mathrm{kW}\). What is the maximum constant speed at which this motor can lift the elevator?

Short Answer

Expert verified
The maximum constant speed at which this motor can lift the elevator is 1.27 m/s.

Step by step solution

01

Calculate the Force due to Gravity

To calculate the force due to Gravity you need to multiply the mass of the elevator with the acceleration due to gravity (9.8). So the equation to find the force will be Force = mass * gravity = 1200kg * 9.8m/s虏 = 11760N.
02

Calculate the Maximum Speed

We have the value of Power provided by the motor and the force due to gravity. To calculate the maximum constant speed, we need to rearrange the formula Power = Force * Velocity to solve for Velocity. So it will be Velocity= Power/ Force. So the velocity = 15000W / 11760N = 1.27 m/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power
Power is a key concept in physics, describing the rate at which work is done or energy is transferred. It gives us insight into how quickly these processes occur. In this problem, the electric motor provides power to lift the elevator.
Power is measured in watts (W), and it can be calculated using the formula:

\[\text{Power} = \text{Force} \times \text{Velocity}\]

This formula implies that for a given power, the velocity can increase if the force decreases, and vice versa.
  • In our scenario, the motor's maximum power is given as \(15 \text{kW}\) which is equivalent to \(15000 \text{W}\).
  • Understanding power helps in determining how fast the elevator can move while utilizing the maximum potential of the motor efficiently.
Force
Force is a fundamental concept in physics, representing a push or pull on an object resulting from its interaction with another object. In this scenario, the force exerted is due to gravity acting on the elevator.
To calculate this force, we use the equation:

\[\text{Force} = \text{mass} \times \text{gravity}\]

Where gravity is approximately \(9.8 \text{m/s}^2 \).

  • In our elevator example, the mass of the elevator and its occupants is \(1200 \text{ kg}\).
  • Thus, the gravitational force or weight is \(11760 \text{ N} \) (newtons).
Force is crucial in understanding how much effort is needed to lift the elevator against gravitational pull. Recognizing the force involved provides the foundation for calculating velocity.
Velocity
Velocity refers to the speed of an object in a particular direction. It is a vector, meaning it has both magnitude and direction. In this task, we need to find the maximum constant velocity at which the elevator can be lifted by the motor.
Using the formula:
  • \[ \text{Velocity} = \frac{\text{Power}}{\text{Force}} \]
  • With \( \text{Power} = 15000 \text{ W} \) and \( \text{Force} = 11760 \text{ N} \), we find \( \text{Velocity} = 1.27 \text{ m/s} \).
The concept of velocity is vital to determine how fast the elevator can move while the motor is exerting its maximum power. Knowing the precise velocity helps ensure that the elevator's operational limits are not exceeded.
Elevator Dynamics
Elevator dynamics examines the forces and motion involved in the operation of an elevator. This field incorporates several physics concepts such as power, force, and velocity.
Understanding these dynamics is crucial for assessing how an elevator can safely and efficiently move people or goods.
  • The dynamics involve calculating the forces acting on the elevator, including the gravitational pull calculated earlier.
  • Analyzing the power output of the motor helps determine the elevator鈥檚 feasible speed limits.
  • The interaction of force and velocity under given power constraints defines the practical operation of the lift system.
Detailed comprehension of elevator dynamics ensures that the elevator functions properly within its mechanical and electrical capabilities, providing safety and reliability.

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Most popular questions from this chapter

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass \(m\) is fired into a block of mass \(M .\) The block, with the embedded bullet, then slides across a frictionless table and collides with a horizon- tal spring whose spring constant is \(k\). The opposite end of the spring is anchored to a wall. The spring's maximum compression \(d\) is measured. a. Find an expression for the bullet's initial speed \(v_{\mathrm{B}}\) in terms of \(m, M, k,\) and \(d\) Hint: This is a two-part problem. The bullet's collision with the block is an inelastic collision. What quantity is conserved in an inelastic collision? Subsequently the block hits a spring on a frictionless surface. What quantity is conserved in this collision? b. What was the speed of a \(5.0 \mathrm{g}\) bullet if the block's mass is \(2.0 \mathrm{kg}\) and if the spring, with \(k=50 \mathrm{N} / \mathrm{m},\) was compressed by \(10 \mathrm{cm} ?\) c. What fraction of the bullet's initial kinetic energy is "lost"? Where did it go?

How far must you stretch a spring with \(k=1000 \mathrm{N} / \mathrm{m}\) to store \(200 \mathrm{J}\) of energy?

What minimum speed does a 100 g puck need to make it to the top of a frictionless ramp that is \(3.0 \mathrm{m}\) long and inclined at \(20^{\circ} ?\)

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A 480 g peregrine falcon reaches a speed of \(75 \mathrm{m} / \mathrm{s}\) in a vertical dive called a stoop. If we assume that the falcon speeds up under the influence of gravity only, what is the minimum height of the dive needed to achieve this speed?
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