Select your language

Suggested languages for you:
Log In Start studying!
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

All-in-one learning app

  • Flashcards
  • NotesNotes
  • ExplanationsExplanations
  • Study Planner
  • Textbook solutions
Start studying

Work and Energy

Save Save
Print Print
Edit Edit
Sign up to use all features for free. Sign up now
Work and Energy

In physics, the terms work and energy have very specific meanings that are different from what we refer to in our daily lives.

Work is the measurement of the energy transfer that takes place when an object is moved along with at least a part of the applied force.

In other words, work done is the action on an object, which, in turn, moves an object in a certain direction. Pushing a box across a floor is an example of work done.

Work and Energy, Visual representation of work, StudySmarterFigure 1. A force applied that moves an object in the direction of the force applied creates work.

How can we calculate the work done and what are its units?

We can calculate work done on an object using the following formula:

W = F·d

Here:

  • W = work done.
  • F = force applied.
  • d = distance covered in the direction of the force applied.

Work has the SI unit of Joules (Nm). According to the formula, work can only be done if the object on which the force is applied moves in the same direction as the force. If you apply force on an object and there is no displacement, you have done no work regardless of how tired you feel.

What if the force applied does not move an object in the direction of the force?

While calculating the work done on an object, we must also bear in mind that only the component of the force that is parallel to the direction of the movement is taken into consideration. If you apply a force on a box, and your arms are parallel to the floor, all your efforts will contribute to the work done on the box.

Work and Energy, Force resulting in work, StudySmarterFigure 2. Only the component of the force that is parallel to the movement of the object is the force that results in work. Source: Oğulcan Tezcan, StudySmarter.

If, on the other hand, your arms are not parallel to the floor and you are pushing the box at an angle, only the component of that force that is parallel to the movement will be taken into account while calculating the work done. The component of the applied force that is perpendicular to the direction of the motion is discarded.

In situations where the force applied is at an angle to the displacement and we have to calculate the horizontal distance covered, we use the following formula:

W = F·dcos(θ)

Here, θ is the angle from the horizontal plane.

In figure 2, if the angle theta is zero, the applied force is completely parallel to the direction of motion, and the equation becomes W = F⋅d. If theta is equal to 90° and the direction of the force is downwards, the equation results in zero and the box will not move, as the applied force will be pushing the box into the ground.

If the vertical force happens to be in the upward direction, the box will move in the upward direction (if the force is great enough), and work will be done.

Can there be both positive and negative work?

You might be thinking that work has a force and makes an object move in a certain direction, but this is not the case. Work is a dot product of two vector quantities: force and displacement. And as the dot product of two vector quantities is a scalar quantity, work also becomes a scalar quantity. It has a magnitude but no specific direction.

Just like temperature is a scalar quantity but can have negative values, work can also be positive or negative.

Work is positive if the force applied to an object is in the same direction as the displacement. Work is negative if the applied force is in the opposite direction to the movement of the object.

Energy and its units

Previously, we defined the work done as the energy transferred to an object, which indicates that the energy element plays an important role in determining the total work done.

Energy is the capacity of a body to perform work.

Energy, which determines how much work a body can do, is measured in Joules (Nm). Energy is merely a property of a system that can be transferred to another system.

Concept of Work and Energy

This theorem states that whenever work is done on an object, there will be a change in the kinetic energy of that object. Kinetic energy happens to be the energy of motion. But how can we prove this theorem?

First of all, we are going to make a few assumptions, even though these assumptions are not required to derive the formula to prove the work-energy theorem. They just make the calculations much simpler.

Suppose you have forces acting on an object whose direction is the same as the displacement. Theta will be zero in this case, which reduces the equation to:

W = F·d

Work and Energy, Net force acting on an object, StudySmarterFigure 3. Net force in the direction of the movement.

As we are dealing with the Wnet total work on an object, we can rewrite the above formula as:

Wnet = Fnet·d

Fnet is the total net force applied to an object. We also know that the net force is equal to the product of mass and acceleration. Hence:

Fnet = F= ma

Wnet = m·a·d

The above equation has acceleration, which is why we need to rewrite it in the velocity v form. Recall from the kinematics equations that:

vf2 = vi2 + 2·a·d

a·d = Vf2 - Vi22

Vf in the above equation is the final velocity, Vi is the initial velocity, a is the acceleration, and d is the displacement. To use the above kinematic equation, we need to assume the acceleration to be constant. In that case, the Fnet applied is constant as well.

Putting the values into the Wnet equation yields:

Wnet=m·a·d

a·d=vf2-vi22

Wnet = m(vf2 - vi22) = 12mvf2 - 12mvi2

This quantity of 1/2⋅m⋅v2 is what we call kinetic energy. Finally, we can write the work-energy theorem as:

Wnet = KE

Work and Energy - Key takeaways

  • Work done can be calculated as the dot product of force and displacement vectors.
  • Work done on an object depends only on the component of the force that is parallel to the direction of the movement.
  • Work can be positive or negative.
  • Energy is the capacity of an object to perform work.
  • The work-energy theorem states that whenever work is done on an object, there is a change in the kinetic energy of that object.

Frequently Asked Questions about Work and Energy

They are scalar quantities.

Energy is the ability of a body to do work, while work is the amount of energy transferred to a body in the same direction of the movement.

The units of work and energy are Joules.

Whenever work is done on an object, there is a change in the kinetic energy of that object. In relation, power is the rate of doing work or work done per unit time. 

Final Work and Energy Quiz

Question

Pick the correct definition of energy from the following:

Show answer

Answer

The capacity of a body to do work.

Show question

Question

Pick the correct definition of work from the following:

Show answer

Answer

The product of force and distance moved in the direction of the force.

Show question

Question

Define kinetic energy.

Show answer

Answer

The energy due to the motion of the body.

Show question

Question

Is work done if you push against a wall?

Show answer

Answer

No.

Show question

Question

What are the units of work and energy?

Show answer

Answer

Joules.

Show question

Question

If you push against a box and it displaces in the direction of the force applied, is work done?

Show answer

Answer

Yes.

Show question

Question

Are work and energy scalar or vector quantities?

Show answer

Answer

Scalar.

Show question

Question

Suppose you apply a force on a body that is in opposite direction to the displacement, will the work done be positive or negative?

Show answer

Answer

Negative.

Show question

Question

If you increase the mass of an object, its kinetic energy will:

Show answer

Answer

Increase.

Show question

Question

If the velocity of an object increases two times, its kinetic energy:

Show answer

Answer

Increases four times.

Show question

Question

If the force applied is at an angle, which component of the force is taken into consideration while calculating the work done?

Show answer

Answer

The component which is parallel to the displacement.

Show question

Question

In the formula to calculate the work done by an object, when would most work be done?

Show answer

Answer

If the theta angle is 0°.

Show question

Question

In the formula to calculate the work done by an object, when would the least work be done?

Show answer

Answer

If the theta angle is 90°.

Show question

60%

of the users don't pass the Work and Energy quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Just Signed up?

Yes
No, I'll do it now

Sign up to highlight and take notes. It’s 100% free.