Question

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

The kinetic energy doesn't increase because there's no change in velocity.

Step-by-step solution

Understand the Scenario

To solve this problem, we must first understand the scenario: a package is resting in an elevator moving upwards at a constant speed. The elevator exerts an upward force on the package.

Analyze Constant Speed

Since the elevator is moving at a constant speed, this implies there is no acceleration in the motion of the package. Therefore, the net force acting on the package is zero.

Equilibrium of Forces

The upward force exerted by the elevator on the package is balanced by the downward gravitational force (weight) acting on the package. This equilibrium means there is no change in velocity.

Work and Kinetic Energy Principle

When work is done, it can result in a change in kinetic energy. However, since the elevator speed is constant, the kinetic energy of the package does not increase as there's no change in velocity.

Key concepts

Kinetic Energy

Kinetic energy is the energy of motion. Any object that moves has kinetic energy proportional to its mass and the square of its velocity. This relationship is given by the equation: \[ KE = \frac{1}{2}mv^2 \] where \( KE \) is kinetic energy, \( m \) is mass, and \( v \) is velocity.
When thinking about kinetic energy in the context of our problem with the package in the elevator, notice that a change in velocity is necessary to alter the kinetic energy of an object. Because the package is moving at a constant speed, its velocity does not change.
This means that while work is done by the elevator, the kinetic energy remains constant because velocity remains unchanged. It is important to understand such scenarios to grasp why energy does not always convert in expected ways.

Constant Speed

Constant speed means that an object is moving continuously at the same rate. There is no acceleration or deceleration. For an object to maintain a constant speed, the forces acting on it must be balanced.
In the case of the elevator, the upward force from the elevator is exactly balanced by the downward gravitational force on the package. This situation creates constant speed, where the velocity is steady over time.
Since there's no acceleration, the net force acting on the object is zero.

• This concept ensures that the motion is uninterrupted and unchanging in the upward direction.
• It also clarifies why there is no change in the kinetic energy of the package.

Understanding constant speed is crucial to notice when immediate changes in motion or energy are not occurring, despite ongoing forces.

Force Equilibrium

Force equilibrium is when the sum of all forces acting on an object is zero, resulting in no change in the object’s motion. This equilibrium does not mean there are no forces acting; rather, any applied forces are balanced by opposing forces.
For the package in the elevator, force equilibrium is evident because:

• The upward force exerted by the elevator counteracts the downward gravitational force perfectly.
• These balanced forces lead to a constant velocity as there is no net force to change the motion.

When forces are in equilibrium, the system is stable in terms of motion. This is why, even when the elevator does work, the kinetic energy does not increase, as both movement and energy change occur predominantly through unbalanced forces.

Gravitational Force

Gravitational force is the pull exerted by the Earth on objects due to its mass. It acts towards the center of the Earth, contributing to the weight of the object.Gravitational force is calculated as: \[ F_g = mg \] where \( F_g \) is the gravitational force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
In our problem, the package experiences a downward gravitational force. As the elevator provides an upward force to counter this, the system is in balance.

• This equilibrium allows the elevator to maintain a constant speed.
• It illustrates how gravitational force interacts with other forces to determine the motion path and energy changes.

Recognizing gravitational force’s effect helps in understanding force dynamics and why certain energy states, such as kinetic energy, remain unchanged despite ongoing work.