Chapter 14: Work

Table of Contents

14.1 Introduction to Work

Work describes the transfer of energy to or from an object when a force is applied over a distance. It is a scalar quantity (not a vector) and is measured in units of joules (J). Understanding the concept of work and its relationship to energy is crucial for analyzing various physical systems, from simple machines to complex energy transfer processes. In this chapter, we will explore the key concepts, important equations, and real-life examples related to work.

Work in physics is different than what you think of in every day life, but there are some similarities. Here, mechanical work is being done on the hay bale in moving it some distance.

14.2 Key Concepts

14.2.1 Definition of Work

Work is done on an object when a force is applied to the object, and it moves a distance parallel to some component of that force. That may sound complicated, but, mathematically, work (W) is defined simply as:

$W=F d \cos{(\theta)}$

where $F$ is the force applied, $d$ is the displacement of the object, and $\theta$ is the angle between the force and the displacement vectors. In other words, we could write:

$W=F_{||}d$

where $F_{||}$ is “f parallel,” the component of $F$ that is in the same direction as $d$ .

14.2.2 Positive and Negative Work

Work can be positive, negative, or zero, depending on the direction of the force and the displacement:

Positive work: When the force and displacement are in the same direction $(0^o \le \theta < 90^o)$ , the work is positive. This means that energy is transferred to the object.
Negative work: When the force and displacement are in opposite directions $(90^0 < \theta \le 180^o)$ , the work is negative. This means that energy is transferred out of the object.
Zero work: When the force is perpendicular to the displacement $(\theta = 90^o)$ , the work is zero. This means that no energy is transferred to or from the object.

14.3 Conservative and Non-Conservative Forces

When a force acts on an object, it can either be classified as a conservative force or a non-conservative force. Conservative forces are forces that conserve mechanical energy, meaning that the work done by the force is independent of the path taken by the object. Examples of conservative forces include gravitational and elastic forces.

Non-conservative forces, on the other hand, are forces that do not conserve mechanical energy, meaning that the work done by the force is dependent on the path taken by the object. Examples of non-conservative forces include friction and air resistance.

14.3.1 Examples of Conservative and Non-Conservative Forces

To better understand the difference between these two types of forces, let’s consider an example of a ball rolling down a hill. As the ball rolls down the hill, it experiences both a gravitational force and a frictional force. The gravitational force is a conservative force, which means that the work done by the force is independent of the path taken by the ball. This means that the work done by gravity on the ball as it rolls down the hill is the same whether the ball takes a straight path or a curved path.

In contrast, the frictional force is a non-conservative force, which means that the work done by the force is dependent on the path taken by the ball. This means that the work done by friction on the ball as it rolls down the hill is greater if the ball takes a curved path compared to a straight path. This is because the force of friction acts in opposition to the direction of motion of the ball which causes the ball to lose more mechanical energy along the longer, curved path.

The distinction between conservative and non-conservative forces is important because it allows us to better understand the relationship between work and energy. When a conservative force does work on an object, the energy is transferred from one form to another without any loss of energy. In contrast, when a non-conservative force does work on an object, the energy is transformed from one form to another, but some energy is lost due to factors such as friction or air resistance.

Deadlifting a barbell is a lot of work. Or is it? Each rep leaves the barbell with a displacement of 0 meters. So, the mechanical work done is zero Joules.

14.4 Real-life Examples

Pushing a shopping cart: When you push a shopping cart, you apply a force in the direction of its displacement. The work done on the cart is positive.
Climbing stairs: When you climb stairs, you perform negative work against the gravitational force, which results in an increase in your gravitational potential energy.
Friction: Frictional forces do negative work on moving objects, causing them to lose kinetic energy and slow down.

14.5 Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:

$W_{NET} = \Delta KE = \dfrac{1}{2}mv_f^2-\dfrac{1}{2}mv_i^2$

where $W_{NET}$ is the net work, $\Delta KE$ is the change in kinetic energy, $m$ is the mass of the object, $v_f$ is the final velocity, and $v_i$ is the initial velocity.

This theorem provides a valuable tool for solving problems involving work, forces, and motion.

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This course uses Canvas for homework assignments, quizzes, and exams. These assignments are open to everyone. Anyone is allowed to enroll in the Canvas course. In fact, this is highly encouraged as it will help you track your progress as you go through the course. Graded feedback will help you get an idea for what your grade would actually be in a Physics 1 college course. Use this link to enroll in the Canvas course .

Continue to Chapter 15: Introduction to Energy

Back to Chapter 13: Friction

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14.1 Introduction to Work