Work and the work-energy principle | Physics | Khan Academy
Summary
TLDRThe 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
🔧 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.
🚀 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.
Mindmap
Keywords
💡Work
💡Force
💡Displacement
💡Theta (θ)
💡Cosine Theta
💡Joules
💡Net Work
💡Net Force
💡Acceleration
💡Kinetic Energy
💡Work-Energy Principle
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.
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