An
object placed on a tilted surface will often slide
down the surface. The rate at which the object slides down
the surface is dependent upon how tilted the surface
is; the greater the tilt of the surface, the faster
the rate at which the object will slide down it. In physics,
a tilted surface is called an inclined plane. Objects
are known to accelerate down inclined planes because of an
unbalanced force. To understand this type of motion, it is
important to analyze the forces acting upon an object on an
inclined plane. The diagram at the right depicts the two
forces acting upon a crate which is positioned on an
inclined plane (assumed to be friction-free). As shown in
the diagram, there are always at least two forces
acting upon any object that is positioned on an inclined
plane - the force of gravity and the normal force. The
force of gravity (also
known as weight) acts in a downward direction; yet the
normal force acts in a
direction perpendicular to the surface (in fact,
normal means "perpendicular").

The first peculiarity of inclined plane
problems is that the normal force is not directed in
the direction which we are accustomed to. Up to this point
in the course, we have always seen normal forces acting in
an upward direction, opposite the direction of the force of
gravity. But this is only because the objects were always on
horizontal surfaces and never upon inclined planes. The
truth about normal forces is not that they are always
upwards, but rather that they are always directed
perpendicular to the surface that the object is on.

The task of determining the net force
acting upon an object on an inclined plane is a difficult
manner since the two (or more) forces are not directed in
opposite directions. Thus, one (or more) of the forces will
have to be resolved into perpendicular components so that
they can be easily added to the other forces acting upon the
object. Usually, any force directed at
an angle to the horizontal is resolved into horizontal and
vertical components. However, this is not the process
that we will pursue with inclined planes. Instead, the
process of analyzing the forces acting upon objects on
inclined planes will involve resolving the weight vector
(F_{grav}) into two perpendicular components. This
is the second peculiarity of inclined plane problems. The
force of gravity will be resolved into two components of
force - one directed parallel to the inclined surface and
the other directed perpendicular to the inclined surface.
The diagram below shows how the force of gravity has been
replaced by two components - a parallel and a perpendicular
component of force.

The perpendicular
component of the force of gravity is directed opposite the
normal force and as such balances the normal force. The
parallel component of the force of gravity is not balanced
by any other force. This object will subsequently accelerate
down the inclined plane due to the presence of an unbalanced
force. It is the parallel component of the force of gravity
which causes this acceleration. The parallel component of
the force of gravity is the net force.

The task of determining the magnitude of
the two components of the force of gravity is a mere manner
of using the equations. The equations for the parallel and
perpendicular components are:

In the absence of friction and other forces (tension,
applied, etc.), the acceleration of an object on an incline
is the value of the parallel component (m*g*sine of angle)
divided by the mass (m). This yields the equation

(in the absence of friction and
other forces)

In
the presence of friction or other forces (applied force,
tensional forces, etc.), the situation is slightly more
complicated. Consider the diagram shown at the right. The
perpendicular component of force still balances the normal
force since objects do not accelerate perpendicular to the
incline. Yet the frictional force must also be considered
when determining the net force. As in all net force
problems, the net force
is the vector sum of all the forces. That is, all the
individual forces are added together as vectors. The
perpendicular component and the normal force add to 0 N. The
parallel component and the friction force add together to
yield 5 N. The net force is 5 N, directed along the incline
towards the floor.

The above problem (and all inclined plane
problems) can be simplified through a useful trick known as
"tilting the head." An inclined plane problem is in every
way like any other net force problem with the sole exception
that the surface has been tilted. Thus, to transform
the problem back into the form with which you are more
comfortable, merely tilt your head in the same
direction that the incline was tilted. Or better yet,
merely tilt the page of paper (a sure remedy for TNS
- "tilted neck syndrome" or "taco neck syndrome") so that
the surface no longer appears level. This is illustrated
below.

Once the force of gravity has been resolved into its two
components and the inclined plane has been tilted, the
problem should look very familiar. Merely ignore the force
of gravity (since it has been replaced by its two
components) and solve for the net force and
acceleration.

As
an example consider the situation depicted in the diagram at
the right. The free-body diagram shows the forces acting
upon a 100-kg crate which is sliding down an inclined plane.
The plane is inclined at an angle of 30 degrees. The
coefficient of friction between the crate and the incline is
0.3. Determine the net force and acceleration of the
crate.

Begin the above problem by finding the
force of gravity acting upon the crate and the components of
this force parallel and perpendicular to the incline. The
force of gravity is 980 N and the components of this force
are F_{parallel} = 490 N (980 N • sin 30
degrees) and F_{perpendicular} = 849 N (980 N •
cos30 degrees). Now the normal force can be determined to be
849 N (it must balance the perpendicular
component of the weight vector). The force of friction
can be determined from the value of the normal force and the
coefficient of friction; F_{frict} is 255 N
(F_{frict} = "mu"*F_{norm}= 0.3 • 849
N). The net force is the vector sum of all the forces. The
forces directed perpendicular to the incline balance; the
forces directed parallel to the incline do not balance. The
net force is 235 N (490 N - 255 N). The acceleration is 2.35
m/s/s (F_{net}/m = 235 N/100 kg).

Practice

The two diagrams below depict the free-body diagram for a
1000-kg roller coaster on the first drop of two different
roller coaster rides. Use the above principles of vector
resolution to determine the net force and acceleration of
the roller coaster cars. Assume a negligible affect of
friction and air resistance. When done, click the button to
view the answers.

The affects of the incline angle on the
acceleration of a roller coaster (or any object on an
incline) can be observed in the two practice problems above.
As the angle is increased, the acceleration of the object is
increased. The explanation of this relates to the components
which we have been drawing. As the angle increases, the
component of force parallel to the incline increases and the
component of force perpendicular to the incline decreases.
It is the parallel component of the weight vector which
causes the acceleration. Thus, accelerations are greater at
greater angles of incline. The diagram below depicts this
relationship for three different angles of increasing
magnitude.

Roller
coasters produce two thrills associated with the initial
drop down a steep incline. The thrill of acceleration is
produced by using large angles of incline on the first drop;
such large angles increase the value of the parallel
component of the weight vector (the component which causes
acceleration). The thrill of weightlessness is
produced by reducing the magnitude of the normal force to
values less than their usual values. It is important to
recognize that the thrill of weightlessness is a feeling
associated with a lower than usual normal force. Typically,
a person weighing 700 N will experience a 700 N normal force
when sitting in a chair. However, if the chair is
accelerating down a 60-degrees incline, then the person will
experience a 350 Newton normal force. This value is less
than normal and contributes to the feeling of weighing less
than one's normal weight - i.e.,
weightlessness.

Check
Your Understanding

The following questions are intended to test your
understanding of the mathematics and concepts of inclined
planes. Once you have answered the question, click the
button to see the answers.

1. Two boys are playing ice hockey on a neighborhood
street. A stray puck travels across the friction-free
ice and then up the friction-free incline of a driveway.
Which one of the following ticker tapes (A, B, or C)
accurately portrays the motion of the puck as it travels
across the level street and then up the driveway?

Explain your answer.

2. Little Johnny stands at the bottom of the driveway and
kicks a soccer ball. The ball rolls northward up the
driveway and then rolls back to Johnny. Which one of the
following velocity-time graphs (A, B, C, or D) most
accurately portrays the motion of the ball as it rolls up
the driveway and back down?

Explain your answer.

3. A golf ball is rolling across a horizontal section of
the green on the 18th hole. It then encounters a steep
downward incline (see diagram). Friction is involved. Which
of the following ticker tape patterns (A, B, or C) might be
an appropriate representation of the ball's motion?

Explain why the inappropriate patterns are
inappropriate.

4.
Missy dePenn's eighth frame in the Wednesday night bowling
league was a disaster. The ball rolled off the lane, passed
through the freight door in the building's rear, and then
down the driveway. Millie Meater (Missy's teammate), who was
spending every free moment studying for her physics test,
began visualizing the velocity-time graph for the ball's
motion. Which one of the velocity-time graphs (A, B, C, or
D) would be an appropriate representation of the ball's
motion as it rolls across the horizontal surface and then
down the incline? Consider frictional forces.

5. Three lab partners - Olive N. Glenveau, Glen Brook,
and Warren Peace - are discussing an incline problem (see
diagram). They are debating the value of the normal force.
Olive claims that the normal force is 250 N; Glen claims
that the normal force is 433 N; and Warren claims that the
normal force is 500 N. While all three answers seem
reasonable, only one is correct. Indicate which two answers
are wrong and explain why they are wrong.

6.
Lon Scaper is doing some lawn work when a 2-kg tire escapes
from his wheelbarrow and begins rolling down a steep hill (a
30° incline) in San Francisco. Sketch the parallel and
perpendicular components of this weight vector. Determine
the magnitude of the components using trigonometric
functions. Then determine the acceleration of the tire.
Ignore resistance force.

Finally, determine which one of the velocity-time graph
would represent the motion of the tire as it rolls down the
incline.

Explain your answer.

7. In each of the following diagrams, a 100-kg box is
sliding down a frictional surface at a constant speed of 0.2
m/s. The incline angle is different in each situation.
Analyze each diagram and fill in the blanks.