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Chapter 5: Problem 77

A package rests on the floor of an elevator that is rising with constant speed. The elevator exerts an upward force on the package and thus does positive work on it. Why doesn't the kinetic energy of the package increase?

Short Answer

Expert verified
The kinetic energy doesn't increase because the package's speed is constant, indicating no net force or acceleration.

Step by step solution

01

Understanding the Scenario

We are given that a package is resting on the floor of an elevator, which is moving upward with a constant speed. The elevator exerts an upward force on the package.
02

Identifying the Forces

Since the package is resting on the elevator floor, two forces act primarily on the package: the gravitational force (weight) acting downwards and the normal force from the elevator floor acting upwards.
03

Analyzing Constant Speed

The elevator is moving at a constant speed, meaning there is no acceleration. According to Newton's Second Law, this implies that the net force acting on the package is zero.
04

Understanding Work and Energy

Work is defined as the force applied over a distance. While upward force from the elevator does positive work to counteract gravity, the net work done on the package is zero because the forces are balanced and there is no change in speed or direction.
05

Explaining Kinetic Energy Constancy

Kinetic energy depends on the speed of an object. Since the speed of the package remains constant, there is no change in kinetic energy despite the forces and work involved.

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Key Concepts

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

Constant Speed
Understanding constant speed is essential when analyzing the motion of objects. When an object, such as a package in an elevator, moves at a constant speed, it implies consistency in its velocity. Constant speed suggests that the object moves equal distances in equal times, without speeding up or slowing down.

Consequently, there is no acceleration involved.
  • Velocity remains unchanged.
  • Motion is uniform.
In the context of the elevator moving upwards with a constant speed, it means the forces impacting the package are balanced and there is no net force to alter its velocity.
Net Force
Net force is the overall force acting on an object, considering all individual forces. According to Newton's Second Law, the net force determines whether an object will accelerate or maintain its current motion. If the net force is zero, the object will maintain its existing speed or stay at rest.

In the situation presented by the elevator, the package experiences two main forces:
  • The gravitational force pulling it downward.
  • The upward normal force exerted by the elevator floor.
These forces are equal in magnitude and opposite in direction. Therefore, they cancel out, leading to a net force of zero on the package. This balance accounts for the constant speed of the elevator and the package.
Kinetic Energy
Kinetic energy is the energy that an object possesses due to its motion. It is computed using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object, and \( v \) is its speed.

For the package, since its speed (and hence its velocity) remains constant as the elevator advances, its kinetic energy does not change. Even if work is being done by the elevator to offset the gravitational pull, that work counters the forces without accelerating the package.
  • No change in speed means no change in kinetic energy.
  • Energy transitions are only apparent if there's a velocity change.
Newton's Second Law
Newton's Second Law is a fundamental principle that links the net force acting on an object to its mass and acceleration. It states that \( F_{\text{net}} = ma \), where \( F_{\text{net}} \) is the net force, \( m \) is the mass, and \( a \) is the acceleration.

When applied to the package within the elevator, since the package travels at a constant speed, its acceleration is zero. Thus, using Newton's Second Law:
  • Net force \( F_{\text{net}} = 0 \).
  • This implies the forces are balanced (gravity vs. normal).
This law straightforwardly explains why there is no acceleration in the package鈥攊t simply maintains a steady velocity as dictated by a zero net force.

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Most popular questions from this chapter

Apply How does the kinetic energy of an object change if its speed doubles? Triples?

Gexplain If the rate at which work is done on an object is increased, does the power supplied to that object increase, decrease, or stay the same?

Calculate A \(0.21-\mathrm{kg}\) apple falls from a tree to the ground, \(4.0 \mathrm{~m}\) below. Ignoring air resistance, determine the apple's kinetic energy, \(K E\), the gravitational potential energy of the system, \(P E_{\text {gravity, }}\), and the total mechanical energy of the system, \(E\), when the apple's height above the ground is \(3.0 \mathrm{~m}\).

A particle moves without friction. At point A the particle has a kinetic energy of \(12 \mathrm{~J}\); at point \(\mathrm{B}\) the particle is momentarily at rest, and the potential energy of the system is \(25 \mathrm{~J}\); at point \(\mathrm{C}\) the potential energy of the system is \(5 \mathrm{~J}\). (a) What is the potential energy of the system when the particle is at point \(\mathrm{A}\) ? (b) What is the kinetic energy of the particle at point \(C\) ?

Rank Four forces do the following amounts of work and produce the indicated powers: \begin{tabular}{|c|c|c|} \hline Force & Work & Power \\ \hline A & \(40 \mathrm{~J}\) & \(80 \mathrm{~W}\) \\ \hline B & \(35 \mathrm{~J}\) & \(5 \mathrm{~W}\) \\ \hline C & \(75 \mathrm{~J}\) & \(25 \mathrm{~W}\) \\ \hline D & \(60 \mathrm{~J}\) & \(30 \mathrm{~W}\) \\ \hline \end{tabular} Rank these forces in order of increasing time required to do the work. Indicate ties where appropriate.
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