Does the
Distribution of Mass Affect How Fast an Object Traverses a Valley?
Research
If a ball is rolling down an inclined plane, the mass does not matter and the size does not matter in determining how fast it will get to the bottom. All that matters is the way the mass is distributed, i.e., whether the balls are hollow or solid. Galileo did an experiment similar to, but not the same as, this one. He rolled two solid balls of different sizes down and up an inclined plane. He demonstrated that the mass and size of the balls has no affect on how fast they go downward. That worked because they were both solid balls. If one of them was hollow, the solid ball would have gotten to the bottom first.
Energy is the ability to do work. Work has a different meaning in science than it does in regular terms. The scientific definition of work is force times displacement. If an object does not move, no work has been done. If an object moves and returns to the place it started, no work has been done.
There are two types of energy that
are related to this experiment. The
first is gravitational potential energy, which is a measure of how far an
object is able to fall. The other is
kinetic energy. Kinetic Energy is the amount of motion an
object has.
The law of conservation of energy states that the amount of energy is
the same at all times. Energy can be
transfered from one object to another, but cannot be created or destroyed.
When a ball is at the top of a valley, it has gravitational potential energy. When the ball starts rolling down the valley, it starts to pick up kinetic energy at the same time that it starts to lose gravitational potential energy. Some of the kinetic energy goes into rotating the ball, and some goes into moving the ball forward. The ball with mass closer to the center will go down to the bottom faster because it takes less energy to rotate and more energy is available to move forward. Ignoring friction, and because of the conservation of energy, the balls will have just as much kinetic energy before they go into the valley as when they leave. This is because they have the same amount of gravitational potential energy and therefore they must have the same amount of kinetic energy as well. In reality, they will be slower because of friction. We know they will have the same speed on both sides of the valley, ignoring friction, but we don’t know which will get to the other side first.
Hypothesis
If I test the distribution of mass by rolling two cylinders, one solid and one hollow down and up a smooth valley that has equal height on both sides, then I think that the solid object will get to the other side first.
Materials
(original) :
·
Wooden planks,
30cm wide , 1.2 cm thick in the following sizes
·
2 planks 70 cm
long
·
3 planks 10 cm
long
·
1 plank 30 cm
long
·
Cylinders
·
1 hollow metal
cylinder. 11.4 cm diameter, 6.351cm hole in middle, 5.08cm long
·
1 solid metal
cylinder.6.351 cm diameter, 3.176 cm long
·
3 m long airline
hose
·
2 Pasco Photo
gate timers. These will measure both the
speed of an object and the time it passes a particular point.
·
computer with USB
connection and Pasco Data Studio installed
·
8 pieces of wood
3.65x4.21x35 cm (also known as 2x4)
·
2 pieces of wood
3.65x4.21x50 cm (also known as 2x4)
·
10 small C clamps
·
5 small hinges
·
20 wood screws, 3
cm long
·
Wooden shims
·
2m of vinyl
siding
·
Duct tape
Procedure
(original)
To run the experiment, do the
following:
1)
Assemble the
planks. Attach hinges to the planks as
shown in the diagram below. Hinges are
attached on the acute angles.
2)
Screw the 8
wooden supports to the corners of the two 10 cm planks to create legs.
3)
The two planks
with the legs attached must be at exactly the same height. To be sure, tape the airline hose on both
sides of the valley and fill it with water until the water is level with one of
the planks. Make sure it is also level
with the other plank. If it is not,
adjust the supports using wooden shims until both sides are level with the
water.
4)
Attach the vinyl
siding using C clamps at each hinge. The
siding will make the entire path smooth so the cylinders will not bounce or
slide.
5)
Attach Photogate
timers to the edge of the valley 1 cm from the edge.
6)
Attach the Photogate
timers to the computer.
7)
Place the solid
cylinder on the launch ramp
8)
Start the timer
and release the cylinder
9)
Record the
velocity and time the cylinder passed each timer
10)Repeat
steps 7-9 using the hollow cylinder
11)Repeat
steps 7-10 releasing the cylinders from different heights on the launch
ramp. This will result in different
velocities as the cylinder enters the valley.
12)Repeat
steps 7-11 five times
Materials ( modified)
·
1 box of rubber
bands
·
Computer with USB
port
·
2 Photogate
timers and Datastudio software
·
Table saw with
24.5cm diameter blade
·
6m of wood 4.21cm x35 cm (also known as 2x4)
·
1 metal L beams 3.5 cm x 3.5 cm x 350 cm
·
1 metal L beams
3.5cm x 5 cm x 350 cm
·
30 #12 wood
screws
·
Cylinders
o
1 hollow metal
cylinder. 11.4 cm diameter, 6.351cm hole in middle, 5.08cm long
o
1 solid metal
cylinder.6.351 cm diameter, 3.176 cm long
·
Duct tape
·
2 boards with the
dimensions: 20cm by 25½cm by 2cm
·
Cinder blocks
·
4 clamps
·
Cushion
·
Metal Bar
Procedure (modified)
To build the ramp:
1.
Cut a 244cm
section of the 2 by 4.
2.
Cut a 75.5cm
section of the 2 by 4.
3.
Cut a 44.5cm
section in the 2 by 4.
4.
Cut a 14cm wide
by 2.5cm deep semicircular hole in the 44.5 section of the 2 by 4. This is the base.
5.
Cut a 3½cm by
10cm rectangular hole in the 20cm by 25½cm by 2cm board. This will be the landing where the timer is
clamped
6.
Repeat step 5
again for the other board.
7.
Cut two 244cm
sections from the 3.5cm x 5cm L beam.
8.
Cut two 75.5cm
sections from the3.5cm x 3.5 cm L beam.
9.
Cut two sections
of the 2 by 4 with the following dimensions (not drawn to scale).
10. Attach the 244cm 2x4 and the 75.5cm 2x4 to the base as
shown in the diagram
11. Screw both sections of the 244cm L beams on the 244cm
2 by 4 to make a channel. The diagram
below shows the sideways view.
12. Screw both sections of the 75.5cm L beams on the
75.5cm 2 by 4 the same as in step 10.
13. Screw the 35
cm supports under the 244 cm channel 80cm from the base
14. Screw the 62 cm supports under the 244 cm channel 136cm
from the base
15. Screw the 92
cm supports under the 244 cm channel 209cm from the base
16. Screw the 35 cm supports under the 75.5 cm channel
70cm from the base
17. Screw the landings onto the platform so that both are
exactly the same height. Clamp the
timers onto the landings. The cylinders
must be able to pass through the timers without touching them.
18. The cylinders sometimes wobble on the way up out of
the valley. Place two 2x4 blocks on the
outside of the channel and hold them in place with cinder blocks.
19. Install DataStudio software on a computer that has a
USB port.
20. Plug the Photogate timer in the USB port
21. The timer will detect when the cylinder reaches the timer
and when it exits the time. Calculate
the distance that the cylinder travels between these two events. Record the location of the cylinder when the
timer detects the cylinder. Then record
the location of the cylinder when the timer detects it leaving. The difference between these two locations is
the distance.
22. Repeat step 20 for each timer.
To run the experiment:
1.
Place 3 rubber
bands around each cylinder. This
prevents the cylinder from slipping.
2.
Place a cushion
on the ground beyond the landing. This
will catch the cylinder as it falls off of the landing after it goes through
the second timer.
3.
Click the start
button on the DataStudio program
4.
Hold the solid
cylinder in place with a metal bar 70cm up the launch ramp from the timer. Quickly remove the bar and the cylinder will
roll down the channel. The timers will
record the time that the cylinder enters and leaves each timer.
5.
Repeat step 4
with the hollow cylinder from the same distance.
6.
Click the stop
button on the Datastudio program and save the data in a file.
7.
Repeat steps 3
through 6 5 times.
8.
Repeat steps 3
through 7 using distances of 80cm, 90cm, 100cm, and 110cm .
Data
Results
The
original procedure did not work. As I
built the ramp and did some tests, it became clear that the original experiment
would not work properly. Here are the
problems I had.
·
The originally
planned hinges would cause too much lateral movement. I used wedges to make the sides of the ramp and
used a block of wood at the bottom with a semicircular arc cut into it.
·
The wooden
sidings I put on were too low and were too wide. Instead I used metal “L” beams in as siding.
And used some wooden blocks as channels.
·
The launch ramp I
had originally planned was much too short.
I put in a much longer launch ramp instead.
·
The cylinder slid
up on one side because there wasn’t enough static friction between the ramp and
the cylinders. I put on three rubber
bands on each cylinder to increase the friction.
·
The cylinder
bounced on the landing between the launch ramp and the start of the
valley. I eliminated the landing and
moved the L beams so that the Photogate would still work.
·
The wooden blocks
I put in on the side were slowing the hollow cylinder down. I used 2 cinderblocks to hold the wooden blocks
in place a little farther apart.
·
The Photogate
timer detected the hollow cylinder as 2 separate cylinders because of the hole
in the middle. I put duct tape on the
hole so the timer would detect it only once.
I ran several tests, releasing the cylinders from different distances up the launch ramp and found that any distance up the launch ramp less than 70cm would not work because the hollow cylinder would not make it all the way across the valley. At 70cm, sometimes the hollow cylinder would make it up the other side, and sometimes it would not. I used distances of 70cm, 80cm, 90 cm, 100 cm, and 110cm.
I ran the experiment many times before I realized that the wooden channel was too tight for the hollow cylinder to go through easily, which meant that those trials were not valid, and I started over. I solved the problem by holding the channel with two cinder blocks, instead of using clamps. After this, there was no problem.
Sometimes a cylinder would have a little lateral movement, causing some of the readings to be wrong. Other times the cylinder hit the timing device, and as a result, the time between gates was not known. I excluded these trials from the analysis.
I measured the distance the cylinder travels through the initial timer. This was 21.1cm. This is used in calculating the initial velocity. The calculation is velocity = .211m/time in gate.
The graph below shows that he solid cylinder gains more speed on the launch ramp. This is because the solid cylinder has more mass near the center.
The graph below show the average time it took for the cylinders to cross the valley for each launch distance. The launch distance is the distance up the ramp where the cylinders are released. This shows
that the solid cylinder is faster across the valley, but that is because the solid cylinder gains more speed on the launch ramp
The graph below shows that the solid curve is below the hollow
curve. This means that even when the
velocity is the same at the top of the ramp, the solid cylinder is still faster
across the ramp. This proves my
hypothesis.
Conclusion
My hypothesis was proven to be correct. With the same initial velocity as it enters
the valley, the solid cylinder had a faster time across the valley.
Bibliography
1. Huffman, Arthur, The
Cartoon Guide to Physics, 1990, U.S.A, 73-82.
2. Didonia,
Acknowledgement
I would like to thank Mr.
Lucien Sanner for making the cylinders in his machine shop