Download of Force - Physics

Physics 12 Gizmo – Inclines
Simple Machine
Imagine you had to load 50 barrels of root beer onto a truck, each weighing 250 pounds (117 kg).
It would be an enormously difficult task to lift them up directly. But what if you could roll the barrels
up a long ramp? If the slope were gentle enough, the job would be much easier.
Redirection of Force
An inclined plane is a simple machine; a basic tool that helps us by redirecting or amplifying force.
In this activity you will learn how an inclined plane redirects force.
1.
On the SIMULATION pane of the Gizmo™, a brick lies halfway up an inclined plane.
Check that the Coeff. of friction ( μ) is set to 0.00, and click Play (
a.
).
How long did it take the brick to slide down the ramp? What was the
Velocity of the brick at the bottom?
b.
Click Reset (
). Experiment with the Gizmo to find the relative
effects of Weight and Angle on the motion of the brick. Which factor
has no effect at all?
2.
Select the FREE-BODY DIAGRAM tab, and check that the Magnitude mode is
chosen. This diagram shows the forces on the brick. The Weight of the brick is a
force that points straight downward, but the inclined plane breaks this force into two
components: one parallel to the plane (W | | ) and one perpendicular (W ).
a.
Which force (W | | or W ) will cause the brick to slide down the
plane?
b.
What is the ratio of W | | to the Weight of the brick? What is the
ratio of W
c.
to the Weight?
Use a calculator to find the sine, cosine, and tangent of the Angle of
the inclined plane. Do any of these resemble the ratios you just
calculated?
d.
Use this relationship to derive a formula for W | | and W
in terms of
weight (W) and Angle (θ). Test your formula with a variety of weights
and angles.
3.
Since the brick pushes downward on the plane, the plane must push upward on the
brick. This is called the normal force and is represented by a red arrow on the
diagram.
a.
b.
How does the magnitude of the normal force relate to W ?
Given the directions and magnitudes of W , W | | , and the normal
force, what force represents the total net force on the brick?
c.
Select the SIMULATION tab and note the Mass of the brick. Use
the formula F = ma to predict what the acceleration (a) of the brick
should be.
d.
Click Play. Select the TABLE tab and look at the acceleration
column. Was your predicted value correct? (Note: Because the brick is
accelerating downwards, acceleration is negative.)
4.
Click Reset and select the CONTROLS tab. Set the Coeff. of friction ( μ) to 0.25.
(To quickly set a slider to a certain value, type the value in the field to the right of
the slider and press Enter.) Set the Angle to 30°, and select the FREE-BODY
DIAGRAM tab.
a.
The light blue arrow on the diagram represents the force of friction.
How is the direction of this force related to the direction of movement?
b.
What is the ratio between the force of friction and the value of W ?
How does this relate to the coefficient of friction, μ?
c.
Write a formula for the force of friction based on μ and W . Use this
formula to predict the force of friction when the Weight is 100 N, the
Angle is 30°, and the Coeff. of friction ( μ) = 0.50. Use the Gizmo to
check your answer. (Note: The force of friction can never exceed W||. At
most they will be equal.)
d.
Based on the current force of friction, the mass of the brick, and
W | | , predict the acceleration of the brick. Use the Gizmo to check your
answer.
e.
Challenge: Write a formula for the acceleration of the brick based on
its mass (m), the angle (θ), and the coefficient of friction (μ). (Recall
that the weight of the brick is equal to the product of its mass and
gravitational acceleration, 9.82 m/s2.)
Mechanical Advantage and Efficiency
In the previous activity, you found that ramps split the weight of an object into two forces, one of
which is cancelled by the normal force. In this activity you will find how this property makes them
useful for lifting objects.
1.
Click Reset. Set the Angle to 48°, the Coeff. of friction ( μ) to 0.00, and the
Weight to 300 N. Change the External force to On, and a toy car appears to push
the brick up the ramp. Check that the Applied force is 100 N and click Play.
a.
Does the car apply enough force to push the brick uphill?
b.
Click Reset. Increase the Applied force until you find a force
sufficient to push the brick up the plane. What is this force?
c.
Select the FREE-BODY DIAGRAM tab. How does this Applied
force relate to the value of W | | ?
d.
How much force would have been required if a forklift were used to
lift the object directly, without the use of a ramp? Why does it take less
force to lift the object using an inclined plane?
2.
An inclined plane can make it easier to lift an object by changing the amount of force
that must be overcome to move it. The mechanical advantage (MA) of an inclined
plane is defined as
a.
Use the results from the last exercise to create a formula for
mechanical advantage in terms of weight (W) and W | | .
b.
In the first activity, you found the formula for W | | in terms of W and
the angle, θ. Substitute this formula into your equation to create an
equation for mechanical advantage in terms of just the angle θ.
c.
Click Reset, set the Coeff. of friction ( μ) to 0.20, and find the
minimum Applied force required to push the brick up the ramp in this
case. How does this force compare to the force of friction (displayed on
the FREE-BODY DIAGRAM tab) plus W | | ? How does this affect the
mechanical advantage of the inclined plane?
3.
The car that pushes the brick up the ramp does work on the brick. If the car pushes
the brick up the ramp by exerting a force F along a distance of d meters, the amount
of work done is W = F d. Set the Angle to 42°, the Coeff. of friction ( μ) to 0.00,
the Weight to 100 N, and the Applied force to 200 N. On the SIMULATION pane,
drag the brick to a height of 0.6 m. Click Play.
a.
Select the WORK tab. What was the distance the car pushed the
brick?
b.
Based on the Applied force and the distance, how much work was
done on the brick?
4.
As it was pushed up the ramp, the energy of the brick increased. This energy is
expressed as potential energy and kinetic energy.
a.
Subtract the initial height of the brick from the final height of the
brick. How much height did the brick gain?
b.
Potential energy in this case is equal to weight times height. How
much potential energy did the brick gain as it was lifted?
c.
What was the final velocity of the brick?
d.
Kinetic energy is equal to 1/2 mv 2 . What was the final kinetic
energy of the brick? (Remember to use the mass of the brick in this
calculation.)
e.
How much total energy did the brick gain? How does this compare to
the amount of work done on the brick?
5.
When there is no friction, an inclined plane is an ideal machine. When friction is
present, some energy is wasted. The efficiency of a machine is the percentage of
input work that "gets through" to the object. One way to measure this is to divide
the amount of energy the object gained by the work done.
a.
Based on the last simulation, what is the efficiency of an inclined
plane when there is no friction? (Note: Due to rounding errors, the
efficiency may not be exactly 100%.)
b.
Set the Coeff. of friction (μ) to 0.20 and redo the experiment.
Calculate the efficiency by dividing the work done on the object by the
total energy (potential and kinetic) that the object gained.
c.
How do you think angle and weight affect efficiency? Use the Gizmo
to find out.
6.
Advanced challenge: Show mathematically that the efficiency of an inclined plane is
equal to 1 − (F f / F a ), where Ff is the force of friction and Fa is the applied force.
Sliding Objects
After his famous experiment of dropping cannon balls from the Tower of Pisa, Galileo was
determined to fully describe the physical laws of falling bodies. However, freefall happened too
quickly to be accurately measured using available methods, which could be as crude as counting
heartbeats.
Galileo realized that he could slow down the motion of falling bodies by sliding or rolling them down
an inclined plane. The discoveries that resulted from these experiments form the underpinnings of
our understanding of force, acceleration, energy and gravitation.
Frictionless Sliding
Potential energy (P E ) is the energy an object has due to its position or the configuration of its
parts. Kinetic energy (KE ) is the energy an object has due to its motion. In this activity, you will
investigate how these energies change as an object slides down an inclined plane.
1.
First you will look at a single block sliding down a frictionless ramp. For Ramp 1 be
sure that a Steel block will slide on a Frictionless ramp with an Angle of 20°.
(To quickly set a slider to a particular value, type the value in the field to the right of
the slider and press Enter.) For Ramp 2, No block should be selected.
a.
Select the ENERGY tab, and notice the black bar representing the
potential energy of the block. Click Play (
). Describe how the
potential and kinetic energy change as the block slides down the ramp.
b.
Select the GRAPH tab, which should show a graph of Energy vs.
Time. This graph shows the total energy of the block as it slides down
the plane. Is any energy lost as potential energy is converted to kinetic
energy? Explain how you can tell.
2.
Click Reset (
). On the CONTROLS tab, change the object on Ramp 2 to a
Steel block on a Frictionless ramp with an Angle of 60°.
a.
On the GRAPH tab, select a graph of Potential energy vs. Time.
Press Play and observe the potential energy curve for each ramp. How
are the two curves similar? How are they different?
b.
Based on the graphs, approximately how much potential energy did
each block have at the beginning of the experiment? How much at the
end?
c.
Did the ramp angle affect the potential energy the blocks had at the
beginning or end of the experiment?
d.
In the situation shown in this Gizmo, the formula for potential energy
(in Joules) is P E = mgh, where m is the mass of the block (1 kg), g is
the acceleration due to gravity (9.8 m/s 2 ), and h is the height of the
block (each ramp is 1 m high). How does this explain your above
answers?
3.
Change the graph to Kinetic energy vs. Time.
a.
How much kinetic energy did each block have at the beginning of the
experiment?
b.
Approximately how much kinetic energy did each block have at the
end?
c.
How do these answers compare to the potential energies of the
blocks at the beginning and end?
d.
The formula for kinetic energy is
where m is the mass of the block and v is the velocity of the block.
Solve this formula for the velocity (v).
e.
Calculate the velocity of each block at the bottom of the ramp.
(Recall that the mass of each block is 1 kg.)
f.
Change the graph to Velocity vs. Time to check your answers. (You
can see the exact final velocity of the block on ramp 1 by selecting the
TABLE tab and scrolling to the bottom.)
g.
Did the final velocity depend on the ramp angle? Explain why or why
not.
h.
Run several more experiments on the frictionless ramp with different
ramp angles. Did the final velocity ever change? Based on what you
have discovered about potential and kinetic energy, explain why the
ramp angle does not affect the final velocity of the block.
4.
Click Reset. On the SIMULATION pane, drag the block on Ramp 1 so that it is
halfway down the ramp.
a.
Calculate how much potential energy this block had at the beginning
of the experiment, when it was halfway up the ramp.
b.
How much kinetic energy does the block have at the bottom of the
ramp?
c.
Calculate the velocity the block will have at the bottom of the ramp.
d.
Click Play, and check all of your calculations on the GRAPH tab.
Sliding with Friction
Friction is a general term for forces that oppose motion. Friction is usually caused by the
roughness of the surfaces and interactions at the molecular level. The coefficient of friction for a
pair of surfaces is a measure of how strong the force of friction between them is. For example, the
coefficient of friction between rubber-soled shoes and linoleum is much higher than the coefficient
of friction between an ice-skater's blade and the ice.
1.
Click Reset. For Ramp 1, select a Rubber block on a Wood ramp with an Angle
of 50°. For Ramp 2, select a Wood block on a Wood ramp, also with an Angle of
50°.
a.
Which block do you think will encounter more friction as it slides?
b.
Which block do you think will slide faster and reach the bottom first?
c.
Click Play to test your predictions. Were you correct?
d.
Select the ENERGY tab. In addition to the previous two bars, there
is now a red bar representing the energy lost due to friction. Which
block lost more energy to friction?
e.
The energy that was not lost to friction was changed to kinetic
energy. Which block had more kinetic energy and was going faster when
it reached the bottom of the ramp?
f.
Turn on Show coefficient of friction for both ramps. Which
combination of materials had a higher coefficient of friction?
2.
Click Reset. Remove the block from the Ramp 2, and change the Ramp 1 Angle to
30°.
a.
b.
Press Play and describe what happens.
You should have found that the object did not slide at all. The ramp
is not steep enough to overcome the frictional force, so no movement
occurs. Adjust the Angle until the plane is just barely steep enough for
the block to slide down.
c.
What is the greatest angle the ramp can have and still have no
movement?
d.
Use a calculator to find the sine, cosine, and tangent of this angle.
Which of these three trigonometric value is similar to the coefficient of
friction for a Rubber block on a Wood ramp?
e.
Select different materials, and see if the relationship you found
persists.
f.
Challenge: Based on what you have seen, describe how you would
experimentally determine the coefficient of friction for a combination of
materials.
Rolling Objects
(Note to teachers and students: This Gizmo was designed as a follow-up to the Inclined Plane Sliding Gizmo. We recommend you begin with that Gizmo before trying this activity.)
Imagine the following scene from a blockbuster spy movie: Our hero skis down a mountain, fleeing
ruthless enemies. The bad guys release a giant snowball to roll down the slope and crush the hero.
Will he make it down the mountain, or will the giant snowball catch him first?
To answer this question, you need to consider the physics of rolling and sliding objects.
Comparing Sliding to Rolling
In this activity you will observe the differences between sliding and rolling motion.
1.
In the Ramp 1 section of the Controls pane, check that a Ring of Steel on Ice is
selected, and that the Angle is 20°. (To quickly set a slider to a value, type the
value in the field to the right of the slider and hit Enter.) For Ramp 2, check that a
Block of Steel on Ice will slide down a 20° ramp.
a.
Which object do you think will reach the bottom first? (Or will they
reach the bottom at the same time?) Press Play (
) to test your
conjecture.
b.
Click Reset (
), and change the material of each object to
Rubber. Which object will reach the bottom first in this case? Press
Play to see.
c.
Change the Angle of Ramp 1 and Ramp 2 to 60°. Which object will
reach the bottom first now? Press Play to test your conjecture.
2.
Which object reaches the bottom more quickly depends on the materials and the
angle of the ramp. Create a table similar to the one below.
a.
Fill in the first row with the results of your first three experiments.
(For example, if the block was the first to reach the bottom, write
"block.")
b.
Run the additional experiments to complete the table.
c.
What two trends do you see in the table? In which situations was the
block more likely to win?
3.
In a race with a ring, the block was favored when the incline was steep or the
coefficient of friction was low. To see why, consider the two types of kinetic energy
that a rolling object has. Translational kinetic energy (T KE ) is the energy of
moving in a straight line, while rotational kinetic energy (RKE ) is the energy of
spinning. Click Reset. For both ramps choose Wood for the object material and
ramp material. Set both ramps to an Angle of 50°. Select the ENERGY tab and
click Play.
a.
In this experiment how much rotational kinetic energy does the block
have? Why?
b.
Which object loses more energy due to friction?
c.
Which object ends up with greater translational kinetic energy?
Which object won the race?
d.
Even though the ring loses less energy to friction, it also ends up
with less translational energy. How is this possible? Where did the rest
of the energy go?
4.
As an object moves down a slope, its potential energy decreases. Some of this
energy is lost to friction, while the rest is converted to kinetic energy. If the object is
rolling, it will have both translational kinetic energy and rotational kinetic energy.
Run several experiments using different materials and different angles.
a.
Is it always the case that the rolling object loses less energy to
friction?
b.
What can you say about the translational kinetic energy of the object
that wins the race?
Rotational Kinetic Energy
In this activity, you will investigate how rolling objects differ in how much rotational kinetic energy
they have. Note: In physics a disk is a "filled-in" ring (like a plate), a ball is a solid, while a sphere
is hollow.
1.
Set the material of each object to Steel and the material of each ramp to Ice. This
will minimize the effects of friction.
a.
Choose a Ring for Ramp 1 and a Disk for Ramp 2. Set the Angle
of each ramp to 30°. With this setup do you think a ring or a disk will
win the race? Click Play to check your answer.
b.
Repeat the above experiment using different rolling objects (Rings,
Disks, Balls and Spheres) until you can order the four objects from
fastest to slowest. What is this order?
c.
Repeat the experiment with a different angle. Does the order remain
the same?
d.
What do the two faster objects have in common? What do the two
slower objects have in common? (Hint: Refer to the definitions above.)
2.
Set the Angle of Ramp 1 to 60° and the Angle of Ramp 2 to 20. For both ramps,
set up the experiment to have a Ring of Steel roll down a plane of Ice. Select the
ENERGY tab, and turn on Show values. Click Play.
a.
At the end of the simulation, how much of the Ramp 1 ring's energy
is expressed as translational kinetic energy? How much is rotational?
b.
c.
What is the energy breakdown for the ring on Ramp 2?
Try other angles. Do the percentages of translational and rotational
energy change?
3.
Run similar experiments with other objects (Disks, Balls, and Spheres).
a.
What are the characteristic translational and rotational energies for
disks, balls, and spheres at the bottom of the ramp?
b.
Is there a relationship between how quickly an object goes down the
plane and its energy breakdown?
4.
If you have ever pushed your friends on a playground merry-go-round, you may
have noticed it takes much more effort to push the merry-go-round when your
friends are on the edge than when they are in the center. People at the edge travel
faster than people in the center, so they have greater rotational energy (and require
more energy to accelerate).
a.
Of the objects in the Gizmo, which is most similar to a merry-goround with people at the edge? Does this object also have the greatest
rotational energy?
b.
Which object is most similar to a merry-go-round with people at the
center?
c.
Explain why some objects take longer to roll down the ramp than
others.
5.
Scientists assign a value to objects, called the moment of inertia (I), that
represents the object's resistance to rotating. The moment of inertia for many
objects is given by the formula
I = k mr 2
Where m is mass, r is the radius, and k is a number between 0 and 1 based on the
shape of the object. The lower the value of k, the lower the value of I, which means
it takes less energy to rotate the object.
a.
For rolling objects, the rotational kinetic energy is equal to k times
the translational kinetic energy.
RKE = k • T KE
Use this fact and the Gizmo to estimate the value of k for each of the
objects.
b.
What simple fractions are these values close to? If possible, discuss
your results with your classmates and teacher.