Work and the work-energy principle | Physics | Khan Academy

Summary

The script explains the concept of work in physics, detailing how force exerted on an object over a distance results in energy transfer. It introduces the formula W=Fdcosθ, where W is work, F is force, d is displacement, and θ is the angle between them. The script clarifies that only the component of force parallel to displacement does work, while perpendicular components do not. It also discusses the implications of positive and negative work on an object's energy and introduces the work-energy principle, showing how net work equals the change in kinetic energy, affecting the object's speed.

Takeaways

• 🔧 To transfer energy to an object, a force must be exerted on it.
• 📐 The formula for work done by a force is W = F * d * cos(theta), where W is work, F is force, d is displacement, and theta is the angle between the force and displacement.
• 🔄 Cosine theta accounts for the fact that only the component of the force in the direction of displacement does work.
• 🔄 The perpendicular component of the force does no work.
• 📏 The unit of work is joules, which is also the unit for energy.
• ⬆️ Positive work means the force is giving energy to the object.
• ⬇️ Negative work means the force is taking energy away from the object.
• ⊥ Work done by a force perpendicular to displacement or with no displacement is zero.
• 💡 Holding a weight above your head does no work because there's no displacement.
• 🔄 The net work done on an object can be found by summing the individual works of all forces acting on it.
• 🚀 The net work done on an object is equal to the change in its kinetic energy, as described by the work-energy principle.

Q & A

What is the definition of work in physics?

In physics, work is defined as the amount of energy transferred to an object when a force is exerted on it.

What is the formula to calculate the work done by a force?

The formula to calculate the work done by a force is W = F * d * cos(theta), where W is work, F is the magnitude of the force, d is the displacement, and theta is the angle between the force and the displacement.

What does the cosine term represent in the work formula?

The cosine term in the work formula represents the component of the force that is in the direction of the displacement, as only this component contributes to doing work.

What are the units of work?

The units of work are newton-meters, which is also known as a joule, the same unit used to measure energy.

How does the direction of force relative to displacement affect the work done?

If the force is in the same direction as the displacement, the work done is positive. If it is in the opposite direction, the work done is negative. If the force is perpendicular to the displacement, the work done is zero.

What is the significance of positive and negative work?

Positive work means the force is giving energy to the object, causing it to speed up. Negative work means the force is taking energy away, causing it to slow down.

Can work be zero even if a force is applied?

Yes, work can be zero if the force is perpendicular to the displacement or if there is no displacement at all, such as when holding a weight stationary.

How can we calculate the net work done on an object?

The net work done on an object can be calculated by summing the individual amounts of work done by each force acting on the object.

What is the relationship between net work and kinetic energy?

The net work done on an object is equal to the change in its kinetic energy, as described by the work-energy principle.

How does the work-energy principle relate to the final and initial kinetic energies?

The work-energy principle states that the net work done on an object is equal to the difference between its final and initial kinetic energies, expressed as 1/2 * m * (v_final^2 - v_initial^2).

What happens to an object's kinetic energy if the net work done on it is positive?

If the net work done on an object is positive, its kinetic energy increases, meaning the object speeds up.

What does it mean for an object if the net work done on it is zero?

If the net work done on an object is zero, it means the object's kinetic energy remains constant, indicating no change in its speed.

Outlines

00:00

🔧 Work Done by a Force

This paragraph explains the concept of work in physics, which is the energy transferred to an object by exerting a force on it. The formula for work done (W) is W = Fd cos(theta), where F is the force, d is the displacement, and theta is the angle between the force and displacement. Work is measured in joules, which are also units of energy. The paragraph discusses how only the component of the force that aligns with the direction of displacement contributes to work. It also explains that positive work indicates energy transfer to the object, negative work indicates energy taken away, and work is zero when the force is perpendicular to displacement or if there is no displacement. The concept of net work is introduced, which is the total work done by all forces acting on an object. The net work can be calculated by summing individual works or using the simplified formula W = mad, where 'a' is acceleration and 'd' is displacement, assuming the net force is constant and aligned with displacement.

05:02

🚀 Work-Energy Principle

The second paragraph delves into the work-energy principle, which links the net work done on an object to its change in kinetic energy. The kinetic energy of an object is given by the formula 1/2mv^2, where 'm' is mass and 'v' is velocity. The net work done is equated to the change in kinetic energy, expressed as the difference between the final and initial kinetic energies: W_net = 1/2mv_final^2 - 1/2mv_initial^2. This principle is crucial in understanding how forces affect the motion of objects. If the net work is positive, the object speeds up; if negative, it slows down; and if zero, the object maintains a constant speed. The paragraph emphasizes the importance of this principle in analyzing the motion of objects under the influence of various forces.

Keywords

💡Work

Work, in the context of physics, refers to the transfer of energy to an object by applying a force. It is a measure of how much energy is given to or taken away from an object. The video defines work mathematically as W = F * d * cos(theta), where W is the work done, F is the force applied, d is the displacement of the object, and theta is the angle between the force and displacement. Work is central to understanding how forces affect the energy of an object.

💡Force

Force is defined as any interaction that, when unopposed, will change the motion of an object. In the video, force is described as the agent responsible for doing work on an object, and its magnitude is crucial in calculating the amount of work done. The script mentions that only the component of the force that aligns with the direction of displacement contributes to doing work.

💡Displacement

Displacement is the change in position of an object. In the video, displacement (d) is one of the variables in the work formula, representing how far an object moves while a force is being applied to it. The script clarifies that if there is no displacement, no work is done, as seen in the example of holding a weight above your head.

💡Theta (θ)

Theta, often denoted as θ, represents the angle between the force applied and the direction of displacement. The script explains that the component of force parallel to the displacement does work, and this is where cosine theta comes into play in the work formula, as it accounts for the effective component of force that contributes to work.

💡Cosine Theta

Cosine theta is a trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the context of the video, it is used in the work formula to determine the effective component of the force that is parallel to the displacement and thus does work on the object.

💡Joules

Joules are the units of work and energy in the International System of Units (SI). The video mentions that work is measured in joules, which are the same units used for energy. This connection is logical as work is a form of energy transfer, as seen when a force does positive or negative work, indicating energy given to or taken away from an object.

💡Net Work

Net work refers to the total work done on an object by all forces acting upon it. The video discusses calculating net work by summing the individual works of each force or, more simply, by considering the net force and displacement. It's a crucial concept for understanding the overall effect of multiple forces on an object's energy.

💡Net Force

Net force is the vector sum of all the forces acting on an object. The video simplifies the concept by assuming all forces align with the displacement direction, thus eliminating the need for the cosine theta term in the net work calculation. Net force is key to understanding how an object's motion changes under the influence of multiple forces.

💡Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In the video, acceleration is related to net force through Newton's second law (F_net = m * a), where m is mass, and a is acceleration. The script uses this relationship to derive a formula for net work in terms of velocities, connecting work to changes in kinetic energy.

💡Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is calculated as 1/2 * m * v^2, where m is mass, and v is velocity. The video concludes by stating that the net work done on an object is equal to the change in its kinetic energy, a principle known as the work-energy principle. This principle is fundamental to understanding how work affects an object's state of motion.

💡Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates the work done on an object to the change in its kinetic energy. The video explains that if the net work is positive, the object speeds up (increase in kinetic energy), and if it's negative, the object slows down (decrease in kinetic energy). If the net work is zero, the kinetic energy remains constant.

Highlights

Work done by a force is defined as the energy transferred to an object.

The formula for work done is W = F * d * cos(theta), where W is work, F is force, d is displacement, and theta is the angle between force and displacement.

Work is measured in joules, which is the same unit as energy.

Cosine theta accounts for the component of force that aligns with the direction of displacement.

A positive work value indicates that the force is giving energy to the object.

A negative work value suggests that the force is taking energy away from the object.

Work done is zero if the force is perpendicular to the displacement or if there's no displacement.

Holding a weight above your head does no work because there's no displacement.

Net work done on an object can be found by summing individual work done by each force.

Assuming forces align with displacement simplifies the calculation by eliminating the cosine theta term.

Net force is equal to mass times acceleration, which can replace F in the work equation.

The work-energy principle relates net work to the change in kinetic energy of an object.

Kinetic energy is given by 1/2 * m * v^2, where m is mass and v is velocity.

The net work done on an object is equal to the change in its kinetic energy.

If net work is positive, the object speeds up; if negative, it slows down.

If net work is zero, the object maintains a constant speed.

The derivation of the work-energy principle simplifies calculations but does not rely on assumptions.

Transcripts

In order to transfer energy to an object,

you've got to exert a force on that object.

The amount of energy transferred by a force

is called the work done by that force.

The formula to find the work done

by a particular force on an object

is W equals F d cosine theta.

W refers to the work done by the force F. In other words,

W is telling you the amount of energy

that the force F is giving to the object.

F refers to the size of the particular force doing

the work.

d is the displacement of the object, how far it moved

while the force was exerted on it.

And the theta and cosine theta refers

to the angle between the force doing

the work and the displacement of the object.

You might be wondering what this cosine theta is doing in here.

This cosine theta is in this formula

because the only part of the force that does work

is the component that lies along the direction

of the displacement.

The component of the force that lies perpendicular

to the direction of motion doesn't actually do any work.

We notice a few things about this formula.

The units for work are Newton's times meters,

which we called joules.

Joules are the same unit that we measure energy in,

which makes sense because work is telling you

the amount of joules given to or taken away

from an object or a system.

If the value of the work done comes out

to be positive for a particular force,

it means that that force is trying

to give the object energy.

The work done by a force will be positive

if that force or a component of that force

points in the same direction as the displacement.

And if the value of the work done comes out to be negative,

it means that that force is trying to take away energy

from the object.

The work done by a force will be negative

if that force or a component of that force

points in the opposite direction as the displacement.

If a force points in a direction that's

perpendicular to the displacement,

the work done by that force is 0,

which means it's neither giving nor taking away energy

from that object.

Another way that the work done by a force could be 0

is if the object doesn't move, since the displacement

would be 0.

So the force you exert by holding a very heavy weight

above your head does not do any work on the weight

since the weight is not moving.

So this formula represents the definition

of the work done by a particular force.

But what if we wanted to know the net work or total

work done on an object?

We could just find the individual amounts

of work done by each particular force and add them up.

But there's actually a trick to figuring out

the net work done on an object.

To keep things simple, let's assume that all the forces

already lie along the direction of the displacement.

That way we can get rid of the cosine theta term.

Since we're talking about the net work done on an object,

I'm going to replace F with the net force on that object.

Now, we know that the net force is always

equal to the mass times the acceleration.

So we replace F net with m times a.

So we find that the net work is equal to the mass

times the acceleration times the displacement.

I want to write this equation in terms of the velocities

and not the acceleration times the displacement.

So I'm going to ask you recall a 1-D kinematics

equation that looked like this.

The final velocity squared equals the initial velocity

squared plus 2 times the acceleration

times the displacement.

In order to use this kinematic formula,

we've got to assume that the acceleration is constant,

which means we're assuming that the net force on this object

is constant.

Even though it seems like we're making a lot of assumptions

here, getting rid of the cosine theta

and assuming the forces are constant,

none of those assumptions are actually

required to derive the result we're going to attain.

They just make this derivation a lot simpler.

So looking at this kinematic formula,

we see that it also has acceleration times

displacement.

So I'm just going to isolate the acceleration

times the displacement on one side of the equation

and I get that a times d equals v final squared

minus v initial squared divided by 2.

Since this is what a times d equals,

I can replace the a times d in my net work formula.

And I find that the net work is equal to the mass

times the quantity v final squared minus v

initial squared divided by 2.

If I multiply the terms in this expression,

I get that the net work is equal to 1/2 mass times

the final velocity squared minus 1/2

mass times the initial velocity squared.

In other words, the net work or total work

is equal to the difference between

the final and initial values of 1/2 mv squared.

This quantity 1/2 m times v squared

is what we call the kinetic energy of the object.

So you'll often hear that the net work done on an object

is equal to the change in the kinetic energy of that object.

And this expression is often called the work energy

principle, since it relates the net work done on an object

to the kinetic energy gained or lost by that object.

If the net work done is positive,

the kinetic energy is going to increase

and the object's going to speed up.

If the net work done on an object is negative,

the kinetic energy of that object

is going to decrease, which means it's going to slow down.

And if the net work done on an object is 0,

it means the kinetic energy of that object

is going to stay the same, which means the object maintains

a constant speed.