Analysis of relationship between angle of
inclined plane and the acceleration of an object
descending the inclined plane
Name: Zalkefl Ali
Due Date: Feb 16, 2023
Group Members: Daniel McReynolds, Ghaith Janabi, Jack Arata Amadeo,
Kathy Li, Lucian Khalid
Purpose:
The reason this experiment is being conducted is so that the relationship between the angle of the
inclined plane, the ramp and the object, the SMART cart can be compared to determine how the
acceleration of an object is changed.
Background Info:
To begin the experiment, the PASCO smart carts are required. PASCO smart carts are carts that
can be connected to the Sparkvue software, which is a software program that enables the user to
input their results and save them as a file. The carts connect via bluetooth and send information
such as velocity, acceleration, position and forces through the built-in sensors. One of the factors
that accelerate the smart cart down the ramp is gravity. Gravity is a vector quantity that can be
used to measure the force of the earth acting upon an object, which is -9.8m/s^2. A vector
quantity is a value that has both magnitude and a direction to it. Another factor that can be
measured is acceleration. Acceleration is a vector quantity that measures how fast the object’s
velocity changes over time. Additionally, velocity is a vector quantity that measures the rate of
change in position of an object. These vector quantities can be represented by different types of
vector graphs; velocity-time graphs, position-time graphs and acceleration-time graphs. The
reason the cart travels down the ramp is because of the imbalance between forces. As said
before, there is the force of gravity that constantly acts upon an object that is in free fall and a
normal force, which acts in a perpendicular direction of the force of gravity. When objects are
stationary and parallel to the ground, the normal force and gravity force are both balanced and go
in their respective directions. Since the object is at an incline, this logic cannot apply as there are
more forces acting upon the object. The gravitational force acting on an object is split into 2
different forces; a force that is parallel to the object and one that is perpendicular to the object in
motion. The perpendicular force of the object is balanced out by the normal force that acts on the
object. Since there are 3 forces, the parallel force is the only force acting upon the object, causing
the object to move at an incline. This is because there is an imbalance between the forces,
causing the object to accelerate faster as the angle of incline becomes steeper. Simply, the
parallel force of the inclined object can be represented by the equation 𝐹𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 = 𝑚𝑔𝑠𝑖𝑛(θ).
Assuming there are no other forces such as friction or any force changing the direction of the
force, that equation can be used to determine the parallel force. If the acceleration of the object at
an incline be calculated, assuming no other forces are at play, the equation 𝑎 = 𝑔𝑠𝑖𝑛(θ) can be
used to calculate the acceleration.
Materials/Apparatus:
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PASCO Smart carts
Ramp (Aluminium ramp, 1.25m by 30cm)
Retort stand with large base and adjustable clip
Meter Sticks
Calculator
Notebook
Pencil
Ipad with Sparkvue software
Measuring Tape
Procedure:
1. The SMART cart and Ipad were checked to make sure that the cart was connected and the
software was functioning..
2. Zalkefl was responsible for releasing the cart from the top of the ramp. Lucian was responsible
for catching the cart at the end of the ramp. Kathy and Ghaith were in charge of measuring the
ramp’s length, width and height. Daniel and Jack were in charge of pressing the start button on
the software upon release of the cart.
3. The smart cart was ensured by Zalkefl to be facing the right direction before being released as
the button was clicked.
4. Lucian caught the cart and Daniel stopped the software.
5. The ramp was then adjusted to an angle using the retort stand and the length of the ramp along
with the height and overall length were recorded to determine the angles that were being
experimented (2.5, 8, 14, 21, 41).
6. Steps 1-5 were repeated for all 5 angles that were being experimented.
7. The data was saved by Daniel and emailed to the other group members while the rest of the
members helped to clean the area.
8. Based on the results from the trials, each group member created the graphs and determined the
acceleration of each angle.
9.Additionally, after the experiment, each group member took the weight measurement of the
smart cart, which weighed 0.25kg.
Data
Table 1.1 - Angle 2.5
Time (seconds)
Velocity (m/s)
1.2
0.060
1.3
0.102
1.4
0.137
1.5
0.169
1.6
0.199
1.7
0.237
Table 1.2- Angle 8
Time(s)
Velocity(m/s)
2.1
0.729
2.2
0.853
2.3
0.981
2.4
1.103
2.5
1.230
2.6
1.355
Table 1.3- Angle 14
Time(s)
Velocity(m/s)
2.7
0.008
2.8
0.173
2.9
0.405
3.0
0.642
3.1
0.872
3.2
1.105
Table 1.4- Angle 21
Time(s) Velocity(m/s)
2.6
0.031
2.7
0.297
2.8
0.641
2.9
0.985
3.0
1.330
3.1
1.673
Table 1.5- Angle 41
Time(s) Velocity(m/s)
1.9
0.301
2.0
0.931
2.1
1.567
2.2
2.197
2.3
2.829
2.4
3.445
Relationship between Angle and Acceleration:
Table 2.1- Angle vs Acceleration
Angle(degrees) Acceleration(m/s^2)
2.5
0.345
8
1.252
14
2.234
21
3.329
41
6.298
Acceleration is directly proportional to the angle of incline as when the incline increases, the
acceleration increases proportionally. More specifically, the proportionality constant between
acceleration and the angle of incline is 0.154.
Calculations Continued
Finding the angle of the triangles
a=5.3cm ± 0.2cm
b=121.5cm ± 0.1cm
c=121.9m ± 0.1cm
Required: Angle of A
𝑡𝑎𝑛Θ = 𝑠𝑖𝑛/𝑐𝑜𝑠
−1
𝑜
𝑜
𝑡𝑎𝑛 (5. 3/121. 5) = 2. 4977 = 2. 5 = Angle of A
Exact same steps were repeated to find the other trial incline angles.
Calculating Acceleration:
Using the slope function, the slope of velocities within the time interval of 0.6 seconds was
calculated.
Example: At angle 5, the velocity at 1.3 seconds was 0.102 and at 1.7 seconds it was at 0.237.
𝑚 = (𝑦2 − 𝑦1)/ (𝑥2 − 𝑥1)
𝑚 = (0. 237 − 0. 102)/(1. 7 − 1. 3)
2
𝑚 = 0. 34𝑚/𝑠 [𝑑𝑜𝑤𝑛] (𝑤𝑖𝑡ℎ 𝑠𝑖𝑔 𝑓𝑖𝑔𝑠)
Calculating accepted value of acceleration:
𝑎 = 9. 8𝑠𝑖𝑛(𝑙𝑜𝑤𝑒𝑠𝑡 𝑟𝑎𝑚𝑝 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
𝑎 = 9. 8𝑠𝑖𝑛(2. 5)
2
𝑎 = 0. 43𝑚/𝑠 [𝑑𝑜𝑤𝑛] (𝑤𝑖𝑡ℎ 𝑠𝑖𝑔 𝑓𝑖𝑔𝑠)
Error Analysis
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 =
|𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙|
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 =
|0.427469996𝑚/𝑠 − 0.3375𝑚/𝑠 |
2
2
2
0.427469996𝑚/𝑠
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = 21. 04%
× 100%
× 100%
The percentage error for the lowest angle (2.5 degrees) was 21.04%. There are a few factors that
have impacted the experiment. The biggest factor to the percent error is friction. Friction reduced
the velocity of the cart and impacted the results of the experiment. The cart weighs .25kg, and
gravity is 9.8N/kg. The calculations of the error can be seen here. Since the smart cart is on an
inclined plane and there is no friction, the equation would be written as such. This would also be
considered the accepted value since there is no force acting on the cart in this equation.
𝐹𝑛𝑒𝑡 = 𝑠𝑖𝑛 (θ) 𝑚𝑔
𝐹𝑛𝑒𝑡 = 𝑠𝑖𝑛(2. 5)(. 25𝑘𝑔)(9. 8𝑁/𝑘𝑔)
𝐹𝑛𝑒𝑡 = 0. 106𝑁 (𝑤𝑖𝑡ℎ 𝑠𝑖𝑔 𝑓𝑖𝑔𝑠)
Accepted value of forces.
The equation that would be used to calculate friction is 𝐹𝑓 = 𝑐𝑜𝑠(Θ)(µ𝑘)𝑚𝑔. This would be
used because the smart cart is moving at an incline. Since the smart cart was on an aluminum
ramp and the wheels are mild steel, the kinetic friction coefficient can be assumed to be 0.004.
𝐹𝑓 = 𝑐𝑜𝑠(2. 5)(0. 004)(0. 25𝑘𝑔)(9. 8𝑁/𝑘𝑔)
−3
𝐹𝑓 = 9. 79 * 10 𝑁
𝐹𝑛𝑒𝑡 = 𝑠𝑖𝑛(θ)𝑚𝑔 − 𝐹𝑓
𝐹𝑛𝑒𝑡 = 0. 0962𝑁
Experimental value of forces.
Percentage error:
−3
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 =
|0.106𝑁 − 8.96*10 𝑁|
0.106
× 100%
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = 9. 23%
9.23% was the error of impact due to friction. This percentage error could be lowered had a
different material been used as the ramp. Since it was an aluminum ramp, a different material
ramp like oak could have been used for the ramp in order to reduce the amount of friction and
the percent error. Additionally, other factors that impacted the percentage error was human error.
When Zalkefl had placed the cart onto the ramp, he did not place it properly within the grooves
of the ramp, which caused the cart to sway and move in other directions besides down. This
sway in the other direction would have caused the cart’s acceleration to decrease, impacting the
percentage error. A fix for this error would be to have carefully examined that the cart was
placed into the grooves of the ramp correctly, which would reduce the chance of the cart
swaying, which would have lowered the percent error. Another factor that impacted the
percentage error was the dust on the ramp. The reason for this is by dust being on the surface of
the ramp, it impacts the percentage error as it further contributes to the friction coefficient. For
comparison, on a similar surface with sand, the sand would contribute to the friction coefficient
due to the particulates having a rougher surface and shape. This is very similar to the dust resting
on the ramp. To fix this, the ramp should have been thoroughly cleaned and wiped down to have
a clean surface to use.
Conclusion
In conclusion, as the ramp angle increases, the acceleration of the cart increases in a positive
linear way. Due to various factors such as friction, human error and dust, the expected results
were a bit off from the actual value of acceleration there should be. Despite this, with the
evidence provided, as the angle of incline increases, the acceleration increases.
References
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16, 2023, from https://www.britannica.com/science/acceleration
Inclined planes. The Physics Classroom. (n.d.). Retrieved February 16, 2023, from
https://www.physicsclassroom.com/class/vectors/Lesson-3/Inclined-Planes
Kirvan, P. (2022, December 7). What is a vector? WhatIs.com. Retrieved February 16, 2023,
from
https://www.techtarget.com/whatis/definition/vector#:~:text=A%20vector%20is%20a%20q
uantity,force%2C%20electromagnetic%20fields%20and%20weight.
Smart cart (red) • me-1240. PASCO scientific. (n.d.). Retrieved February 16, 2023, from
https://www.pasco.com/products/sensors/wireless/me-1240
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https://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity