106 – Rolling motion on inclined ramps

This experiment explores how mechanical energy, motion, and geometry interact to influence the behavior of a rolling object. By studying the motion of a marble on inclined planes, students will investigate how variations in ramp angle and height affect acceleration, velocity, and ultimately the maximum height reached after launch.

Rolling motion is a fundamental concept in physics that combines both translational and rotational dynamics. Unlike objects that slide, rolling objects distribute their energy between linear motion and rotational motion, resulting in different acceleration and speed relationships. This laboratory highlights the importance of these differences and introduces the role of the factor 10/7, which accounts for rotational inertia in rolling objects.

In this activity, two ramps with different angles and heights are compared. Although both ramps have the same descending length, their inclination angles differ, leading to variations in acceleration and final speed at the base of the ramp. These differences influence how much energy is available for the marble to climb the ascending ramp and reach its maximum height. By applying theoretical models and verifying them experimentally, students will determine which ramp configuration is more effective.

This laboratory also emphasizes the comparison between theoretical predictions and experimental data. Students will calculate speeds and heights using energy conservation and kinematic equations, then validate these results using sensors. Any discrepancies between predicted and observed values will be analyzed in terms of real-world factors such as friction and air resistance.

Through this investigation, students will develop a deeper understanding of motion on inclined planes, the conservation of mechanical energy, and the relationship between physical parameters and experimental outcomes. This lab bridges theoretical physics concepts with practical experimentation, reinforcing both analytical reasoning and scientific methodology.

Educational Goals

Understanding rolling motion and energy transformation
Develop an understanding of how rolling motion differs from sliding motion by analyzing how energy is distributed between translational and rotational components. Learn how gravitational potential energy is converted into kinetic energy and how this affects the speed of the marble.

Application of physical models and equations
Apply key physics equations, including conservation of mechanical energy and kinematic relationships, to predict the speed of the marble at the base of the ramp and the maximum height reached after ascent. Understand the origin and significance of the rolling motion factor 10/7.

Analysis of the influence of ramp geometry
Examine how variations in ramp angle and height affect acceleration, final velocity, and energy transfer. Determine how these parameters influence the marble’s ability to reach a higher position after launch.

Experimental verification and data comparison
Use sensors to measure actual speeds and heights and compare these experimental values with theoretical predictions. Evaluate the agreement between theory and experiment and identify trends across different ramp configurations.

Development of analytical reasoning
Interpret discrepancies between calculated and measured results by considering real-world factors such as friction, rolling resistance, and air resistance. Strengthen the ability to justify conclusions using evidence.

Use of laboratory equipment and measurement tools
Gain experience using stopwatch sensors and measurement systems to collect data accurately. Learn how proper setup and calibration affect the reliability of results.

Protocol

Introduction

1. We want to estimate the speed at the bottom of the descending slope of each ramp in order to estimate which ramp will give the marble the highest takeoff.
2. Two different ramps are used for this laboratory :

• Ramp 1 is 0.259 m high and has an inclination angle of 15°
• Ramp 2 is 0.342 m high and has an inclination angle of 20°

3. Each of the two ramps can be broken down into two portions : a descending portion, and an ascending portion.
4. The two ramps differ with respect to the angle and the height of their descending portion. The length of the descending portion of both ramps is 1 m.
5. The ascending portion of each ramp is otherwise identical.

Manipulations A

Considering the parameters provided above and knowing that the acceleration due to gravity is 9.8 m/s, estimate the speed at the bottom of each descending ramp (in m/s).

1. Position one of the stopwatch sensors on the small board placed at the end of the descending slope of the ramps (white location). Position the 2nd sensor on the other small board placed at the end of the ramps (black location).
2. Position the marble at the top of ramp 1 on the gray horizontal line.
3. Press button A in order to obtain the speed of the marble at the end of the descending slope of ramp 1.
4. Observe the demonstration.
5. The demonstration data are entered in the results table. Consult the data obtained in order to compare them to your calculations.
6. Repeat steps 2 to 5 using ramp 2.

Question A

Knowing the speed at the bottom of each descending ramp, considering the following parameters :

• Length of ascending ramps: 0.17 m
• Angle of ascending ramps: 45°

What will be the maximum height reached by each marble (in m)?

Manipulations B

1. Position the marble at the top of ramp 1 on the gray horizontal line.
2. Press button B in order to obtain the maximum height reached by the marble after taking off from ramp 1.
3. Observe the demonstration.
4. The demonstration data are entered in the results table. Consult the data obtained in order to compare them to your calculations.
5. Repeat steps 1 to 4 using ramp 2.

Questions B

1. Which ramp will give the marble its maximum height?
2. If the calculations are different from the experimental data, what factors explain these differences?

Anticipated Outcomes

Compute the speed at the bottom of each ramp (in m/s) with the following parameters:

• Descending length ramp 1 Δx_d1 = 1.0 m
• Descending ramp 1 angle θ_d1 = 15°
• Descending length ramp 1 Δx_d2 = 1.0 m
• Descending ramp 2 angle θ_d1 = 20°
• The acceleration due to gravity g = 9.8 m/s²

We use the Conservation of mechanical energy for a rolling object on an incline equation to compute the speed at the bottom of each ramp:

v = √(10/7 * g * Δx * sin θ)

• Speed descending ramp 1 = v_d1 = 1.91 m/s
• Speed descending ramp 2 = v_d2 = 2.19 m/s

Compute the height of the ball on each ramp (in m) with the following parameters:

• Speed descending ramp 1 = v_d1 = 1.91 m/s
• Speed descending ramp 2 = v_d2 = 2.19 m/s
• Descending ramps angle θ_a = 45°
• Ascending ramps length Δx_a = 0.17 m

We firstly use Rolling motion acceleration equation to find the acceleration (in m/s²) on the ascending ramp:

a = 5/7 * g * sin θ

This equation tells you how quickly a rolling marble speeds up or slows down on a ramp.

• The sin θ part comes from gravity pulling the marble down the slope
• The 5/7 comes from the fact that the marble is rolling, not sliding
• The negative sign means the acceleration is opposite the direction of motion

a_a = -4.95 m/s²

We then compute the speed at the exit of the ascending ramp (in m/s) using the Third kinematic equation:

v_a = √(v_d² + 2 * a * Δx_a)

For each ramp, the equations will be:

• Ramp 1: v_a1 = √(v_d1² + 2 * a * Δx_a) = 1.40 m/s
• Ramp 2: v_a2 = √(v_d2² + 2 * a * Δx_a) = 1.76 m/s

Furthermore, we compute the vertical speed (in m/s) of the ball when exiting ramp using the sin of the ascending ramp (sin 45°):

• Ramp 1: v_a1y = v_a1 * sin 45° = 0.98 m/s
• Ramp 2: v_a2y = v_a2 * sin 45° = 1.25 m/s

We then can finally compute the maximum height of the ball (in m) using a derivative of the Third kinematic equation, considering:

• at maximum height h the vertical velocity is 0
• v = v_ay
• a = -g
• height at launch h₀ = m

then

h = v_ay² / 2 * a + h₀

The results are:

• Ramp 1: h₁ = v_a1y² / 2 * a + h₀ = 0.17 m
• Ramp 2: h₂ = v_a2y² / 2 * a + h₀ = 0.20 m

The following table aggregate results:

	Ramp 1	Ramp 2
Descending ramp length	1.0 m	1.0 m
Descending ramp angle	15°	20°
Ascending ramp length	0.17 m	0.17 m
Ascending ramp angle	45°	45°
Speed end of descending ramp	1.91 m/s	2.19 m/s
Speed end of ascending ramp	1.40 m/s	1.76 m/s
Vertical speed end of ascending ramp	0.99 m/s	1.25 m/s
Maximum height of the ball	0.17 m	0.20 m

 

The expected results of this laboratory are based on the application of energy conservation principles and kinematic equations for rolling motion. By comparing the two ramps, the analysis focuses on how differences in inclination angle affect the marble’s speed at the base of the ramp and, consequently, the maximum height reached after ascending.

Both ramps have the same descending length (1.0 m), but Ramp 2 has a greater inclination angle (20°) than Ramp 1 (15°). Since the component of gravitational force acting along the ramp is proportional to sin θ, a steeper angle results in greater acceleration. As a result, the marble on Ramp 2 gains more kinetic energy during its descent, leading to a higher speed at the bottom of the ramp. This is confirmed by the calculated speeds: approximately 1.91 m/s for Ramp 1 and 2.19 m/s for Ramp 2.

Once the marble reaches the ascending ramp, part of its kinetic energy is converted back into gravitational potential energy. Because both ascending ramps are identical (same length and angle), the only factor influencing the maximum height reached is the initial speed at the base of the ascent. Since Ramp 2 provides a higher initial speed, it is expected that the marble will reach a greater height compared to Ramp 1.

The calculated maximum heights reflect this relationship, with Ramp 1 reaching approximately 0.17 m and Ramp 2 reaching approximately 0.20 m. This confirms that a greater initial kinetic energy leads to a higher vertical displacement, consistent with the conservation of mechanical energy.

However, in real experimental conditions, measured values may differ slightly from theoretical predictions. These discrepancies can be explained by several factors. Friction between the marble and the ramp surface reduces the mechanical energy available for motion, resulting in lower speeds and heights than expected. Additionally, air resistance opposes motion, particularly during the ascent phase, further reducing the maximum height reached.

Another source of variation may come from measurement uncertainties, such as sensor precision, alignment of the ramps, or slight differences in the marble’s release conditions. Small variations in these factors can influence the recorded speed and height values.

Overall, the expected results demonstrate a clear relationship between ramp angle, acceleration, speed, and maximum height. The analysis confirms that Ramp 2 should produce a greater maximum height, as it allows the marble to gain more kinetic energy during descent. Any deviations observed in the experimental results should be interpreted in light of real-world energy losses and measurement limitations, reinforcing the importance of critical analysis in experimental physics.

Summary of Assignment by Grade Range

Grade 9–10 (Introductory Level)

At the introductory level, this laboratory serves as a first structured exposure to motion on inclined planes and the relationship between height, speed, and energy. The primary focus is on developing observational skills, understanding basic physical relationships, and becoming familiar with laboratory procedures and safety practices.

Students explore how a marble behaves when released from ramps with different angles and heights. They observe that steeper ramps result in faster motion and that this increased speed allows the marble to reach a greater height after ascending. At this stage, the emphasis is placed on qualitative understanding rather than detailed mathematical analysis. Students are encouraged to describe what they see, compare the two ramps, and identify which ramp allows the marble to reach a higher position.

Teacher guidance is essential, as students are introduced to fundamental concepts such as gravity, motion, and energy in an intuitive way. Basic calculations may be introduced with support, but the primary goal is to build confidence in recognizing patterns and making logical connections between cause and effect.

Safety remains a central component. Students learn to handle equipment carefully, follow instructions, and record observations accurately. They also begin to understand the importance of consistency in experimental procedures.

Learning outcomes at this level include:

• Recognizing how ramp angle affects motion
• Describing differences in speed and height qualitatively
• Identifying the relationship between motion and energy in simple terms
• Following laboratory procedures safely and accurately
• Recording observations clearly in a structured format

Grade 11 (Intermediate Level)

At the intermediate level, the laboratory shifts toward a more quantitative and analytical approach. Students are expected to apply physics equations to calculate the speed of the marble at the bottom of the ramp and the maximum height reached after ascent. The concept of energy transformation—from gravitational potential energy to kinetic energy and back—is explored in greater depth.

Students independently use formulas derived from the conservation of mechanical energy and kinematic equations to predict results. They calculate speeds using the rolling motion equation and determine maximum height using vertical motion relationships. The separation between theoretical prediction and experimental verification becomes a key component of the activity.

Experimental data collected using sensors are compared with calculated values. Students are required to analyze discrepancies between these values and explain them using scientific reasoning. At this level, they begin to understand the impact of non-ideal factors, such as friction between the marble and the ramp and air resistance, which reduce the total mechanical energy available.

The development of critical thinking skills is emphasized. Students must justify their conclusions, explain trends observed in the data, and demonstrate an understanding of how different variables influence the outcome.

Learning outcomes at this level include:

• Applying equations to calculate speed and height
• Interpreting the relationship between energy and motion quantitatively
• Comparing theoretical and experimental results
• Explaining discrepancies using concepts such as friction and air resistance
• Demonstrating independence in laboratory procedures and analysis

Grade 12 / College Level (Advanced Level)

At the advanced level, the laboratory becomes an exercise in model validation, critical evaluation, and scientific reasoning. Students are expected not only to perform calculations accurately but also to understand the assumptions behind the physical models used.

Students analyze the rolling motion equation in detail, including the significance of the 10/7 factor, which accounts for rotational inertia. They evaluate how energy is distributed between translational and rotational components and how this affects acceleration and final speed. The laboratory becomes an opportunity to explore the limitations of simplified models and the importance of considering real-world effects.

A deeper error analysis is conducted. Students identify both systematic and random sources of error, including measurement uncertainties, sensor limitations, variations in ramp alignment, and inconsistencies in marble release. They assess how these factors influence results and discuss the reliability and reproducibility of their data.

At this level, students are also expected to communicate their findings clearly and professionally. They must present logical arguments supported by data, justify conclusions, and relate their results to broader physical principles.

Learning outcomes at this level include:

• Evaluating the validity of physical models and their assumptions
• Performing detailed error analysis and uncertainty evaluation
• Understanding the role of rotational motion in energy distribution
• Justifying conclusions using quantitative and qualitative evidence
• Communicating scientific results in a clear and structured manner

Laboratory essentials

Instruments

• Wooden ramps (0.342 m at 20°, 0.259 m at 15°)
• Rulers
• Marble
• Photodiodes & electronic timer
• 2 buttons wired to the timer

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