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PHYS 1010
Date: _____1/13/26______ Name: ___________Pebble Yaffe__ Assignment: Week 4: Design Kinematics Experiment_____ Instructor: Schenk Physics 1010 Lab 04 Worksheet
1. QUESTION AND HYPOTHESIS
How does the angle of inclination ( θ ) affect the acceleration ( a y ) of an iOLab cart, and does
this acceleration follow a predictable trigonometric trend established by kinematic theory? Hypothesis: If the angle of inclination is increased, then the acceleration of the iOLab cart will increase proportionally to the sine of the angle. Based on the principles of two-dimensional kinematics, gravity acts downward and can be resolved into vector components relative to the ramp. The
component of gravitational acceleration acting parallel to the slope is defined as g sin θ.
Therefore, I predict that a plot of measured acceleration ( a y ) vs. sin θ will yield a linear
relationship with a slope approximately equal to the acceleration due to gravity ( g ≈ 9.8 m/ s^2 ).
2. EXPERIMENTAL DESIGN
Experimental setup and variables: The experiment is structured to investigate the relationship between the incline of a plane and the resulting linear acceleration. The independent variable is
the angle of inclination ( θ ). The dependent variable is the average acceleration ( a y ) measured
along the y-axis of the iOLab cart.
I am utilizing a long and flat wooden board ( L =2.08915 m) that will be supported at one to
create variable inclined planes. To investigate the relationship between the incline angle and
acceleration, the board’s angle of inclination ( θ ) is adjusted incrementally to five distinct angles:
10º, 15º, 20º, 25º, and 30º. A protractor is used to set up and verify these angles at the ramp’s
pivot point. For each angle, I will also measure the vertical height ( h ) with a meter stick from
the floor to the underside of the board at the end of the elevated side. This provides a
secondary, definitive geometric data point to calculate sin^ θ^ =
h
L
and verify the protractor’s accuracy.
Procedures and data integrity: For each angle, the experiment will be tested in triplicate (three trials) to ensure consistency and minimize random. To maintain consistency, a fixed
displacement path is marked on the board ( ∆ d =1.85 m) to ensure the length of travel for the
iOLab cart remains constant for each trial. Using the iOLab’s wheel encoder, the acceleration (
a y ) is measured directly along the path for each trial. The average acceleration for each height
will be used for primary analysis, while the velocity vs. time graph will serve as a secondary verification tool to ensure a constant slope during the data selection window. Theoretical comparison and friction analysis: The design includes calculating the theoretical
acceleration for each trial using the formula a theoretical = g sin θ. This model assumes a
frictionless environment in which gravity is the sole force acting on the cart. By building the direct comparison into the design, I am establishing a baseline that allows for the quantification of the “frictional deficit,” and allow for a direct comparison between experimental data and established kinematic theory. This will provide a basis for calculating percent error and evaluating the impact of friction later in the discussion. Reasoning behind the design:
- Direct and reliable data: iOLab’s wheel encoder provides direct and consistent
acceleration data, allowing for a robust measurement of a y along the y-axis, which
relates to the vector component of gravity parallel to the ramp.
- Geometric cross referencing: By measuring both the angle ( θ ) with a protractor and
height ( h ) with a meter stick, I can use the relationship sin θ =
h
L
to verify accuracy. If any discrepancy occurs, the height measurement serves as the definitive geometric
check to calculate the actual sin θ for the final data analysis.
- Vector Component Isolation: Selecting specific angles allows for a controlled study of how the sine of the angle affects the gravitational vector component.
- Variable control: Using a single wooden board, a start and end position, a fixed 1.85 m
path, and a “release from rest” methos (^ v^0 =^0 ) ensures that environmental variables
and initial impulses are controlled or eliminated.
- Safety: Place a soft object (e.g., pillow or blanket) immediately after the marked ending position to safely receive the iOLab cart.
- Establish incline height and verify geometry: Support one end of the board using stable supports (e.g., a stack of books, a small box, a chair) to reach first target angle (10º).
a. Use a meter stick to measure the vertical height ( h ) from the floor to the underside
of the board at the very end of the elevated side. b. Record both the target angle and the measured height. This height, combined with
the constant L , provides the definitive geometric sine (sin θ =
h
L
) for data verification.
- Conduct trials: a. Place the iOLab cart at the marked starting position with the y-axis pointing down the ramp (indicated by the arrow on the iOLab cart). This ensures that the acceleration values recorded by the wheel sensors are positive. b. Open the iOLab application and select the “Wheel” sensor. Make sure “Acceleration” and “Velocity” are selected and deselect “Position.”
c. Begin recording data and release the cart from rest simultaneously ( v 0 = 0 m/s).
Ensure no initial push is given. d. Stop recording when the device hits the soft object. e. Highlight the data on the iOLab’s interface, ending right before the steep change in the graph (i.e. crash into the soft object) and export the .csv Excel file and label it accordingly to keep the data organized. f. Reset and repeat this procedure for a total of three trials.
- Vary the incline: Adjust the support structure to increase the ramp’s incline to the next target angle and repeat steps 4 and 5 for the remaining target angles: 15º, 20º, 25º, and 30º.
Measure and record the vertical height ( h ) for every change in incline.
- Master Data Sheet Organization: Open a new Excel sheet (Master Data Sheet) to compile the data from the experiment in a Master Data Table with the following columns: Target Angle
Measured Height (
h )
Acceleration from height
( g × h / L )
Trial 1
a y (
m/ s
Trial 2
a y (
m/ s
Trial 3
a y (
m/ s
Average
a y (
m/ s
Theoretical
a y (
a = g × sin θ )
Percent Error (%) 10º
Master Data Sheet Organization: Open a new Excel sheet (Master Data Sheet) to compile the data from the experiment in a Master Data Table with the following columns:
8. Data extraction and validation: Fill in the measured height ( h ) for each angle previously
recorded. a. Starting from data set 1, open the first trial’s Excel sheet. b. Create a velocity vs. time scatter plot. The trend line should be relatively linear to confirm constant acceleration. i. Delete data points from the start or end that deviate from this linear trend (e.g., release noise or impact. c. Use the AVERAGE function on the acceleration data and record it in your Master
Data Sheet and record it as the trial’s^ a^ y.
d. Repeat for each trial.
- Calculations and graphical analysis: a. Calculate the
h
L
ratio for each height and record it in the “Experimental Incline Ratio” column. This serves as the independent variable (x-axis) for the final graph where x =
h
L
b. Calculate theoretical acceleration for each height using a =9.8 × (
h L
) and record
under “Acceleration from height” column.
c. Calculate the theoretical acceleration for each target angle using a =9.8 × sin θ
and record the results in the “Theoretical a y ( a = g × sin θ )” column.
d. Calculate the percent error between the average measured a y and the theoretical
a using the formula:^ %Error=
Average a y −Theoretical a
Theoretical a
× 100 and
record the results in the “Percent Error (%)” column.
The analysis was designed to evaluate the two-dimensional vector decomposition of gravity by isolating the component of acceleration acting parallel to the surface of the inclined plane.
According to Newton’s Second Law and kinematic theory, the acceleration ( a y ) of an object on
a frictionless incline is defined as^ a^ y =^ g^ sin^ θ^ (angle theoretical model).
To maintain high data integrity and fulfill the objective of evaluating different measurement methodologies, the analysis was executed through the following stages: Data reduction was performed using Excel for averaging and variance reduction: For each of
the five tested inclination angles ( θ = 10 ∘^ , 15 ∘^ , 20 ∘^ , 25 ∘^ , and 30 ∘^ ), three independent
measurements of acceleration (𝑎𝑦) were recorded. The AVERAGE function was applied across
the trials to yield a single, more statistically robust value for the mean acceleration ( a¯^ y ),
minimizing the impact of random fluctuations. Geometric and angular cross-referencing: The independent variable, the angle of inclination ( θ ), was verified through two independent channels to mitigate setup error.
- Direct angular measurement: The nominal angles (10º, 15º, 20º, 25º, and 30º) were initially set via a pivot-point protractor.
- Geometric Ratio: To establish a secondary, definitive check, the vertical height ( h ) was
measured at the elevated end of the board ( L = 2.08915 m). The experimental sine of
the angle was then calculated using the trigonometric ratio (geometry-derived theoretical model):
sin θ =
h
L
- This provided a definitive geometric data point that corrected for any manual inaccuracies in the protractor alignment. Synthesis of theoretical baselines: The data were compared against two distinct theoretical models to identify the source of potential deviations:
- Theoretical (sin θ ): The idealized acceleration predicted by the target angles (
a = g sin θ ).
- Acceleration from height: The refined theoretical acceleration based on the actual
measured geometry ( a = g × h / L ).
Statistical reduction of kinematic data: For each incline, the iOLab’s wheel encoder recorded
raw acceleration data over the fixed displacement path ( ∆ d =¿ 1.85 m). These data points
were filtered to ensure the cart had reached a state of constant acceleration and then averaged across triplicate trials. This averaging minimized the impact of random errors, such as slight variations in the "release from rest" technique. Regression and the gravitational constant: To test the hypothesis of a linear relationship, a
scatter plot was generated representing the average measured acceleration ( a y ) as a function
of the Incline Ratio ( h / L ). A linear regression analysis was applied:
a y = m
h
L )
+ b
In this model, the slope (m) corresponds to the experimental magnitude of gravity ( g ). The y-
intercept (b) was analyzed to determine the presence of systematic offsets, such as sensor calibration bias or the "frictional deficit" identified in the experimental design.
7. RESULTS Linear relationship and experimental validation: The primary experimental results confirm a strong linear relationship between the sine of the incline and the measured acceleration of the iOLab cart, validating the core hypothesis derived from two-dimensional kinematics. The independent variable was defined using the geometric
ratio sin θ = h / L (where L =2.08915 m ), which allowed for a direct correlation between
the physical ramp dimensions and the resulting motion. Regression analysis and gravitational acceleration: The resulting data, processed from triplicate trials at 10º, 15º, 20º, 25º, and 30º, yielded a linear regression defined by the equation:
a y =9.9008(
h
L )
In this model, the slope represents the experimentally measured acceleration due to gravity ( g
), which was determined to be a y =9.9008 m/s^2 ± 0.052m/s^2. The margin of error reflects
the 90% confidence interval derived from the standard error of the regression. This is compared
The most significant systematic error identified in this experiment was the deviation between the
theoretical acceleration derived from the geometric ratio ( h / L ) and the actual measured
acceleration. The analysis revealed that the geometric model consistently under-predicted the
acceleration. Since the board length ( L ) was a fixed constant (2.08915 m) throughout the
experiment, this discrepancy indicates a systematic underestimation of the vertical height ( h ).
Manual measurement of the vertical displacement is prone to human error and difficulty in locating the exact center of mass relative to the pivot, resulting in recorded height values that were slightly smaller than the true physical elevation driving the cart.
The linear regression revealed a positive y-intercept +0.1792 m/s^2. This offset is critical
because it contradicts the expected frictional deficit. Typically, friction acts as a retarding force that would reduce acceleration (creating a negative intercept). The observation of a positive intercept suggests that a calibration bias in the sensor or a slight non-zero inclination of the floor (leveling error) was significant enough to overcome the frictional forces. While friction was undoubtedly present in the wheel bearings, the systematic "push" from the calibration/leveling error masked its effect in the data. Random errors were present in the form of slight fluctuations between the three trials conducted for each angle. These fluctuations could stem from inconsistent release techniques of the cart, minor variations in the surface of the wooden board at different points along the 1.85 m path, or external vibrations affecting the sensor during data collection. To minimize the impact of these random fluctuations, the experimental design utilized a fixed displacement path and averaged
the results three trials. The high correlation coefficient ( R 2 ≈ 1 ) of the linear regression
suggests that while random errors were present, they were sufficiently controlled, leaving a clean linear trend that validated the kinematic relationship.
8. DISCUSSION and CONCLUSIONS This experiment convincingly demonstrated the kinematic principles of motion on an inclined plane, specifically the vector resolution of gravitational acceleration. The results confirmed that the acceleration of an object on a frictionless incline is not determined by its mass but is strictly a function of the angle of inclination. By isolating the parallel component of gravity (
a y = g sin θ ) ,the data showed that the cart’s acceleration increased linearly with the sine of
the angle, validating the fundamental trigonometric relationship predicted by kinematic theory. These results are highly reasonable when evaluated against the established physics principles and the associated error analysis. The experimental magnitude of gravity calculated was determined to be 9.90 ± 0.052 m/s^2 at a 90% confidence interval, which deviates from the accepted value of 9.80 m/s^2 by only 1.03%. This low percent error indicates that the iOLab sensor successfully isolated the kinematic variable despite the limitations of an at-home setup. The positive y-intercept of 0.1792 m/s^2 is also reasonable in this context. While friction typically causes a negative intercept, the positive value suggests that the floor used for the experiment had a slight non-zero inclination, or the sensor retained a minor calibration bias. This constant was significant enough to mask the rolling resistance. However, these errors are small enough that they do not undermine the central finding: the motion followed the predicted physical laws with high precision. To further improve the precision of these results, several changes to the experimental design could be implemented to mitigate identified sources of error:
- Instead of relying on manual measurement, a digital inclinometer could be used to verify
the angle ( θ ) independent of the board height.
- Increasing the sample size by conducting five trials instead of three at each angle would further reduce the impact of random errors. - Implementing a mechanical release trigger rather than a manual release would eliminate variations in initial velocity, ensuring that the cart begins each trial from a true state of rest. - To address the positive intercept (+0.1792 m/s^2 ), the starting surface should be verified with a bubble level before experimentation to ensure it is perpendicular to the gravity vector, eliminating the systematic offset caused by the floor's inherent slope.
