How Force, Power, Torque and Energy
Work by Karim
Nice
If you've read many of our articles, you've seen a lot of
terminology thrown around -- words such as mass,
force, torque, work, power and
energy. What do these words really mean, and are they
interchangeable?
In this edition of How Stuff
Works, we will help to bring all of this terminology
together, give some examples of when each is used and even try
a few calculations along the way to get the hang of it.
Throughout this article, we will refer to different types
of units. In most of the world, the International System of
Units (SI - from the French Le Système International
d'Unités), also referred to as the metric system, is
accepted as the standard set of units. This system contains
most of the metric units you are used to, like meters and
kilograms, but also includes units for many other physical and
engineering properties. Even the United States has officially
adopted the SI system of units, but English Engineering
Units (like pounds and feet) are still in everyday use.
Before we jump into explaining these terms, we need to start
with some basics. We'll start with mass, and work our way up
to energy.
What is Mass?
Common Units of
Mass
SI: Gram
(g) 1 g = 0.001 kg Kilogram
(kg) 1 kg = 2.2 lbm 1 kg =
0.0685 slug
English: Pound mass
(lbm) 1 lbm = 0.4536 kg Slug
(slug) 1 slug = 14.5939 kg
Generally, mass is
defined as the measure of how much matter an object or body
contains -- the total number of subatomic particles
(electrons, protons and neutrons) in the object. If you
multiply your mass by the pull of Earth's gravity, you get
your weight. So if your body weight is fluctuating, by eating
or exercising, it is actually the number of atoms that is
changing. It is important to understand that mass is
independent of your position in space. Your body's mass on the
moon is the same as its mass on the earth, because the number
of atoms is the same. The earth's gravitational pull, on the
other hand, decreases as you move farther away from the earth.
Therefore, you can lose weight by changing your elevation, but
your mass remains the same. You can also lose weight by living
on the moon, but again your mass is the same.
Mass is important for calculating how fast things
accelerate when we apply a force to them. What determines how
fast a car can accelerate? You probably know that your car
accelerates slower if it has five adults in it. We'll explore
this relation between mass, force and acceleration in a little
more detail after we talk about force.
What is Force?
Common Units of
Force
SI: newton
(N) 1 N = 0.225 lb
English: Pound (lb) 1
lb = 4.448 N
One
type of force that everyone is familiar with is weight.
This is the amount of force that the earth exerts on you.
There are two interesting things about this force:
It pulls you down, or, more exactly, towards the center
of the earth.
It is proportional to your mass. If you have more mass,
the earth exerts a greater force on you.
When you step on the bathroom scale, you exert a force on
the scale. The force you apply to the scale compresses a
spring, which moves the needle. When you throw a baseball, you
apply a force to the ball, which makes it speed up. An
airplane engine creates a force, which pushes the plane
through the air. A car's tires exert a force on the ground,
which pushes the car along.
Force causes acceleration. If you apply a force to a
toy car (for example, by pushing on it with your hand), it
will start to move. This may sound simple, but it is a very
important fact. The movement of the car is governed by
Isaac Newton's Second Law, which forms the foundation
for classical mechanics. Newton's Second Law states that the
acceleration (a) of an object is directly proportional to
the force (F) applied, and inversely proportional to the
object's mass (m). That is, the more force you apply to an
object, the greater the rate of acceleration; and the more
mass the object has, the lower the rate of acceleration.
Newton's second law is usually summarized in equation form:
a = F/m, or F = ma
To honor Newton's achievement, the standard unit of force
in the SI system was named the newton. One newton (N)
of force is enough to accelerate one kilogram (kg) of mass at
a rate of one meter per second, per second (m/s2). In fact, this is really how
force and mass are defined. A kilogram is the amount of
weight at which 1 N of force will accelerate at a rate of 1
m/s2. In English units, a
slug is the amount of mass that 1 pound of force
will accelerate at 1 ft/s2,
and a pound mass is the amount of mass that 1 lb of force will
accelerate at 32 feet/s2.
The Earth exerts enough force to accelerate objects that
are dropped at a rate of 9.8 m/s2, or 32 feet/s2. This gravity force is often
referred to as g in equations. If you drop something
off a cliff, for each second it falls it will speed up by 9.8
m/s. So, if it falls for five seconds, it will reach a speed
of 49 m/s. This is a pretty fast rate of acceleration. If a
car accelerated this fast, it would reach 60 mph in less than
three seconds!
Usually, when we talk about forces, there is more than one
force involved, and these forces are applied in different
directions. Let's look at a diagram of a car. When the car is
sitting still, gravity exerts a downward force on the car
(this force acts everywhere on the car, but for simplicity, we
can draw the force at the car's center of mass). But the
ground exerts an equal and opposite upward force on the tires,
so the car does not move.
Figure 1. Animation of
forces on a car
When the car begins to accelerate, some new forces come in
to play. The rear wheels exert a force against the ground in a
horizontal direction; this makes the car start to accelerate.
When the car is moving slowly, almost all of the force goes
into accelerating the car. The car resists this acceleration,
with a force equal to its mass multiplied by its
acceleration. You can see in Figure 1 how the
force arrow starts out large because the car accelerates
rapidly at first. As it starts to move, the air exerts a force
against the car, which grows larger as the car gains speed.
This aerodynamic drag force acts in the opposite direction of
the force of the tires, which is propelling the car, so it
subtracts from that force, leaving less force available for
acceleration.
Eventually, the car will reach its top speed, the point at
which it cannot accelerate any more. At this point, the
driving force is equal to the aerodynamic drag, and no force
is left over to accelerate the car.
What is
Torque? Torque is a force that tends to
rotate or turn things. You generate a torque any time you
apply a force using a wrench. Tightening the lug nuts on your
wheels is a good example. When you use a wrench, you apply a
force to the handle. This force creates a torque on the lug
nut, which tends to turn the lug nut.
English units of torque are pound-inches or pound-feet; the
SI unit is the Newton-meter. Notice that the torque units
contain a distance and a force. To calculate the torque, you
just multiply the force by the distance from the center. In
the case of the lug nuts, if the wrench is a foot long, and
you put 200 pounds of force on it, you are generating 200
pound-feet of torque. If you use a two-foot wrench, you only
need to put 100 pounds of force on it to generate the same
torque.
A car engine creates torque, and uses it to spin the
crankshaft. This torque is created exactly the same way; a
force is applied at a distance. Let's take a close look at
some of the engine parts:
Figure 2. How torque is
generated in one cylinder of a four-stroke engine.
The combustion of gas in the cylinder creates pressure
against the piston. That pressure creates a force on the
piston that pushes it down. The force is transmitted from the
piston to the connecting rod, and from the connecting rod into
the crankshaft. In Figure 2, notice that the point
where the connecting rod attaches to the crank shaft is some
distance from the center of the shaft. The horizontal distance
changes as the crankshaft spins, so the torque also changes,
since torque equals force multiplied by
distance.
You might be wondering
why only the horizontal distance is important in determining
the torque in this engine. You can see in Figure 2 that when
the piston is at the top of its stroke, the connecting rod
points straight down at the center of the crankshaft. No
torque is generated in this position, because only the force
that acts on the lever in a direction perpendicular to the
lever generates a torque.
If you have ever tried to loosen really tight lug nuts on
your car, you know a good way to make a lot of torque is to
position the wrench so that it is horizontal, and then stand
on the end of the wrench -- this way you are applying all of
your weight at a distance equal to the length of the wrench.
If you were to position the wrench with the handle pointing
straight up, and then stand on the top of the handle (assuming
you could keep your balance), you would have no chance of
loosening the lug nut. You might as well stand directly on the
lug nut.
Figure 3. A simulated
dynamometer test of two different engines. Click here
for the large version.
Figure 3 shows the the maximum torque and power
generated by two different engines. One engine is a
turbo-charged Caterpillar C-12 diesel truck engine. This
engine weighs about 2,000 pounds, and has a displacement of
732 cubic inches (12 liters). The other engine is a highly
modified Ford Mustang Cobra engine, with a displacement of 280
cubic inches (4.6 liters); it has an added supercharger and
weighs about 400 pounds. They both produce a maximum of about
430 horsepower (hp), but only one of these engines is suitable
for pulling a heavy truck. The reason lies partly in the
power/torque curve shown above.
When the animation pauses, you can see that the Caterpillar
engine produces 1,650 lb-ft of torque at 1200 RPM, which is
377 hp. At 5,600 RPM, the Mustang engine also makes 377 hp,
but it only makes 354 lb-ft of torque. If you have read the
article on gears, you
might be thinking of a way to help the Mustang engine produce
the same 1650 lb-ft of torque. If you put a gear reduction of
4.66:1 on the Mustang engine, the output speed would be
5600/4.66 RPM, or 1200 RPM, and the torque would be 4.66 * 354
lb-ft or 1,650 lb-ft -- exactly the same as the big
Caterpillar engine.
Now you might be wondering, why don't big trucks use small
gas engines instead of big diesel engines? In the scenario
above, the big Caterpillar engine is loafing along at 1,200
RPM, nice and slow, producing 377 horsepower. Meanwhile, the
small gas engine is screaming along at 5,600 RPM. The small
gas engine is not going to last very long at that speed and
power output. The big truck engine is designed to last years,
and to drive hundreds of thousands of miles each year it
lasts.
What is Work? The
work we are talking about here is work in the physics
sense. Not home work, or chores, or your job or any other type
of work. It is good old mechanical work.
Work is simply the application of a force over a
distance, with one catch -- the distance only counts if it
is in the direction of the force you apply. Lifting a weight
from the ground and putting it on a shelf is a good example of
work. The force is equal to the weight of the object, and the
distance is equal to the height of the shelf. If the weight
were in another room, and you had to pick it up and walk
across the room before you put it on the shelf, you didn't do
any more work than if the weight were sitting on the ground
directly beneath the shelf. It may have felt like you did more
work, but while you were walking with the weight you moved
horizontally, while the force from the weight was vertical.
Your car also does work. When it is moving it has to apply
a force to counter the forces of friction and aerodynamic
drag. If it drives up a hill, it does the same kind of work
that you do when lifting a weight. When it drives back down
the hill, however, it gets back the work it did. The hill
helps the car move down.
Work is energy that has been used. When you do work, you
use energy. But sometimes the energy you use can be recovered.
When the car drives up the hill, the work it does to get to
the top helps it get back down. Work and energy are closely
related. The units of work are the same as the units of
energy, which we will discuss later.
What is Power?
Common Units of
Power
SI: Watts
(W) 1000 W = 1 kW Kilowatt
(kW) 1 kW = 1.341 hp
English Horsepower
(hp) 1 hp = 0.746 kW
Power is a measure of how
fast work can be done. Using a lever, you may be able
to generate 200 ft-lb of torque. But could you spin that lever
around 3,000 times per minute? That is exactly what your car
engine does.
The SI unit for power is the watt. A watt breaks
down into other units that we have already talked about. One
watt is equal to one Newton-meter per second (Nm/s). You can
multiply the amount of torque in Newton-meters by the
rotational speed in order to find the power in watts. Another
way to look at power is as a unit of speed (m/s) combined with
a unit of force (N). If you were pushing on something with a
force of 1 N, and it moved at a speed of 1 m/s, your power
output would be 1 watt.
An interesting way to figure out how much power you can
output is to see how fast you can run up a flight of stairs.
Measure the height of a set of stairs that takes you up
about three stories.
Time yourself while you run up the stairs as quickly as
possible.
Divide the height of the stairs by the time it took you
to ascend them. This will give you your speed.
For
instance, if it took you 15 seconds to run up 10 meters, then
your speed was 0.66 m/s (only your speed in the vertical
direction is important). Now you need to figure out how much
force you exerted over those 10 meters, and since the only
thing you hauled up the stairs was yourself, this force is
equal to your weight. To get the amount of power you output,
multiply your weight by your speed.
power (W) = (height of stairs (m) /
Time to climb (s) ) * weight (N) power (hp) = [(height of
stairs (ft) / Time to climb (s) ) * weight (lb)] / 550
English: Foot - pound (ft
lb) 1 ft lb = 1.356 Nm British
Thermal Unit (BTU) 1 BTU = 1055
J 1 BTU = 0.0002931 kWh
Energy is the
final chapter in our terminology saga. We'll need everything
we've learned up to this point to explain energy.
If power is like the strength of a weightlifter, energy is
like his endurance. Energy is a measure of how long we can
sustain the output of power, or how much work we can do;
power is the rate at which we do the work. One common unit of
energy is the kilowatt-hour (kW-hr). You learned in the last
section that a kW is a unit of power. If we are using one kW
of power, a kW-hr of energy will last one hour. If we use 10
kW of power, we will use up the kW-hr in just six minutes.
There are two kinds of energy: potential and kinetic.
Potential
Energy Potential energy is waiting to be
converted into power. Gasoline in a fuel tank, food in
your stomach, a compressed spring and a weight hanging from a
tree are all examples of potential energy.
The human body is a type of energy conversion device. It
converts food into power, which can be used to do work. A car
engine converts gasoline into power, which can also be used to
do work. A pendulum clock is a device that uses the energy
stored in hanging weights to do work.
When you lift an object higher, it gains potential energy.
The higher you lift it, and the heavier it is, the more energy
it gains. For example, if you lift a bowling ball one inch,
and drop it on the roof of your car, it won't do much damage
(please, don't try this). But if you lift the ball 100 feet
and drop it on your car, it will put a huge dent in the roof.
The same ball dropped from a higher height has much more
energy. So, by increasing the height of an object, you
increase its potential energy.
Let's go back to our experiment in which we ran up the
stairs and found out how much power we used. There is another
way to look at how we calculated our power: We calculated how
much potential energy our body gained when we raised it up to
a certain height. This amount of energy was the work we did by
running up the stairs (force * distance, or our weight * the
height of the stairs). We then calculated how long it took to
do this work, and that's how we found out the power. Remember
that power is the rate at which we do work.
The formula to calculate the potential energy (PE) you gain
when you increase your height is:
PE = Force * Distance
In this case, the force is equal to your weight, which is
your mass (m) times the acceleration of gravity (g), and the
distance is equal to your height (h) change. So the formula
can be written:
PE = mgh
Kinetic Energy Kinetic
energy is energy of motion. Objects that are moving,
such as a rollercoaster, have kinetic energy (KE). If a car
crashes into a wall at 5 mph, it shouldn't do much damage to
the car. But if it hits the wall at 40 mph, the car will most
likely be totaled.
Kinetic energy is similar to potential energy. The more the
object weighs, and the faster it is moving, the more kinetic
energy it has. The formula for KE is:
KE = 1/2 m v2, where m is the mass
and v is the velocity.
One of the interesting things about kinetic energy is that
it increases with the velocity squared. This means that if a
car is going twice as fast, it has four times the energy. You
may have noticed that your car accelerates much faster from 0
mph to 20 mph than it does from 40 mph to 60 mph. Let's
compare how much kinetic energy is required at each of these
speeds. At first glance, you might say that each car is
increasing its speed by 20 mph, and so the energy required for
each increase must be the same. But this is not the case.
We can calculate the kinetic energy required to go from 0
mph to 20 mph by calculating the KE at 20 mph and then
subtracting the KE at 0 mph from that number. In this case, it
would be 1/2 m 202 - 1/2 m
02. Because the second part of
the equation is 0, the KE = 1/2 m 202, or 200 m. For the car going from
40 mph to 60 mph, the KE = 1/2 m 602 - 1/2 m 402; so KE = 1,800 m - 800 m, or 1000
m. Comparing the two results, we can see that it takes a KE of
1,000m to go from 40 mph to 60 mph, whereas it only takes 200
m to go from 0 mph to 20 mph.
There are a lot of other factors involved in determining a
car's acceleration, such as aerodynamic drag, which also
increases with the velocity squared. Gear ratios determine how
much of the engine's power is available at a particular speed,
and traction is sometimes a limiting factor. So it's a lot
more complicated than just doing a kinetic energy calculation,
but that calculation does help to explain the difference in
acceleration times.
Bringing it Together Now
that we know about potential energy and kinetic energy, we can
do some interesting calculations. Let's figure out how high a
pole-vaulter could jump if he had perfect technique. First
we'll figure out his KE, and then we'll calculate how high he
could vault if he used all of that KE to increase his height
(and therefore his PE), without wasting any of it. If he
converted all of his KE to PE, then we can solve the equation
by setting them equal to each other:
1/2 m v2 = m g
h
Since mass is on both sides of the equation, we can
eliminate this term. This makes sense because both KE and PE
increase with increasing mass, so if the runner is heavier,
his PE and KE both increase. So we'll eliminate the mass term
and rearrange things a little to solve for h:
1/2 v2 / g = h
Let's say our pole-vaulter can run as fast as anyone in the
world. Right now, the world record for running 100 m is just
under 10 seconds. That gives a velocity of 10 m/s. We also
know that the acceleration due to gravity is 9.8 m/s2. So now we can solve for the
height:
1/2 10 / 9.8 = 5.1 meters
So 5.1 meters is the height that a pole-vaulter could raise
his center of mass if he converted all of his KE into PE. But
his center of mass is not on the ground; it is in the middle
of his body, about 1 meter off the ground. So the best height
a pole-vaulter could achieve is in fact about 6.1 meters, or
20 feet. He may be able to gain a little more height by using
special techniques, like pushing off from the top of the pole,
or getting a really good jump before takeoff.
Figure 4. Animation of pole vault
In Figure 4 you can see how the pole-vaulter's
energy changes as he makes the vault. When he starts out, both
his potential and kinetic energy are zero. As he starts to
run, he increases his kinetic energy. Then, as he plants the
pole and starts his vault, he trades his kinetic energy for
potential energy. As the pole bends, it absorbs a lot of his
kinetic energy, just like compressing a spring. He then uses
the potential energy stored in the pole to raise his body over
the bar. At the top of his vault, he has converted most of his
kinetic energy into potential energy.
Our calculation compares pretty well with the current world
record of 6.15 meters, set by Sergey Bubka in 1993.
