Chiu I Hsuan Physics HL Year 1 Period 2 October, 20, 2010 The Angle of the Ramp Affects the Acceleration of the Ball Rolling Down the RampIntroduction: When a ball is placed on an angled ramp, it accelerates down the ramp. The situation issimilar to skiing down a hill with no applied forces. The acceleration of the ball in affect of theinclination angle of the ramp is investigated. A force influences an object to undergo acceleration; there are mainly two types of forces, appliedforce and conservative force. A common conservative force is gravity which is the main force involved inthis research. The equation of force in relation with acceleration is described as below. F=MA [Equation 1]Where F is force, M is mass and A is the acceleration. Y axis FNormal Fgx X axis Fgy Fgravity ѲFigure 1: Shows a force diagram indicating possible forces acting on the ball in the situation. According to Figure1, the net force acting on the ball in X axis is Fgx. Fgx = Sin (θ)Fg. Ifcombined with equation 1, the result equation is shown below. (Hecht)
A = g*Sin (θ) [Equation 2] Where A is acceleration, g is the gravity and θ is the angle of the ramp to the ground.Equation 2 indicates a proportional fit between A and Sin (θ), thus A is proportional to Sin (θ).The result graph is expected to be a direct linear proportional fit which passes through the origin.Design:Research Questions: How does the inclination angle of the ramp affect the acceleration of the ball rollingdown the ramp?Variables: The independent variable is the angle of the ramp to the ground, and the dependentvariable is the acceleration of the ball rolling down the ramp. The range of the independentvariable is 10.83degrees to 49.75degrees. The controlled factors are the temperature of the room,dimensions of the ball used and the width of the U ramp. The temperature can affect datacollection for motion detector, because the motion detector requires the speed of sound in the airin order to calculate the change in position over time. The temperature of the room was keptconstant by turning on the air conditioner. The same ball was used during the research, thereforethe size and the mass of the ball is kept constant. Since the wider the U ramp is, the lower theacceleration, the width of the U ramp was kept constant. Diameter of the Squash Ball: 5 centimeters Width of the U Ramp: 1.5 centimeterFigure 2: Showing the cross section view of the squash ball rolling down the U ramp.
Motion Detector Squash BallWooden Block U Ramp Figure 3: Demonstrates the set up of the investigation. HypotenuseHeight Length Figure 4: Naming of the ramp set up. Angle of the Ramp to the Ground (θ) Procedure: A motion detector was attached to a U ramp with tape. The temperature of the LoggerPro set up for the motion detector was 26Celsius and the sample collection rate was 40samples per second. Wooden blocks were stacked under to vary the height of the ramp. The motion detector was faced along the ramp, and after the data collection started; the squash ball was set and released on the ramp, 20centimeter away from the motion detector. Data was collected and saved. The setting of the derivative calculations was set to five data points. Then the hypotenuse and the length were measured to calculate the angle of the ramp. The intervals of the angles of the ramp to the ground are 10.83, 14.79, 13.13, 17.97, 21.65, 28.12, and 49.75degrees, changed by varying the number of wooden blocks under the ramp.
Data Collecting and Processing: Angle of the Ramp and the Average Acceleration Acceleration (±.09m/s2) Angle of the SineLength of the Height of the Ramp(θ) Angle AverageRamp(±.001meter) Ramp(±.001meter) (±.2 ) (±.003) Trial 1 Trial 2 Trial 3 Acceleration .713 0.99 1.01 1.07 1.02 .134 10.83 .188 .713 1.31 1.30 1.30 1.30 .162 13.13 .227 .713 1.66 1.47 1.51 1.55 .182 14.79 .255 .713 1.84 1.82 1.85 1.84 .220 17.97 .309 .713 2.16 2.19 2.19 2.18 .263 21.65 .369 .713 2.83 2.84 2.75 2.81 .336 28.12 .471 .642 4.92 4.84 4.80 4.85 .490 49.75 .763 Table 1: Presents the actual measurements of the ramp and the average acceleration for each angle. The average acceleration is calculated by half of range of 14.79 .Data from angle 49.75 is used for sample calculation. The length and height of the ramp are shown in Figure 4. Sample Calculation: Sample Graph:
Figure 5: Demonstrating a raw data graph . The last two plots of theacceleration trend were not included in the fit because the derivative calculation was set to use 5data points to calculate the middle point. Therefore the last two was left out of the fit to preventunsuitable data.Angle of the Ramp (θ):Sin (θ) = (Height / Hypotenuse) θ = Sin-1 (Height / Hypotenuse)Sin (θ) = (.490 / .642) θ = Sin-1 (.490 / .642) θ = 49.75Uncertainty of the Angle of the Ramp (θ):Average: θ = Sin-1 (Height / Hypotenuse)Maximum: θ = Sin-1 (Height + Uncertainty / Hypotenuse - Uncertainty)Minimum: θ = Sin-1 (Height - Uncertainty / Hypotenuse + Uncertainty)Uncertainty: (Maximum – Minimum) / 2Average: θ = Sin-1 (.490 / .642) θ = 49.75Maximum: θ = Sin-1 (.490 +.001 / .642 - .001) θ = 49.995Minimum: θ = Sin-1 (.490 -.001 / .642 + .001) θ = 49.508Uncertainty: (49.995 – 49.508) / 2 = ±.2435
Sine Angle:Sin (θ) = (Height / Hypotenuse)Sin (θ) = (.490 / .642) = .7632Uncertainty of the Sine Angle:Average: Sin (θ) = (Height / Hypotenuse)Maximum: Sin (θ) = (Height + Uncertainty / Hypotenuse - Uncertainty)Minimum: Sin (θ) = (Height- Uncertainty / Hypotenuse + Uncertainty)Uncertainty: (Maximum – Minimum) / 2Average: Sin (θ) = (.490 / .642) Sin (θ) = .7632Maximum: Sin (θ) = (.490 + .001 / .642 - .001) Sin (θ) = .7659Minimum: Sin (θ) = (.490 - .001 / .642 + .001) Sin (θ) = .7605Uncertainty: (.7659 – .7605) / 2 = ±.0027
Result Graphs:Inclination Angle of the Ramp vs. Average Acceleration of the Ball Rolling Down the RampFigure 6: Indicating a linear proportional fit between angle of the ramp and the averageacceleration.Figure 7: Demonstrating a high-low fit of the ramp angle vs. average acceleration graph. Theuncertainty of the slope is ±.003, and the uncertainty of the y intercept is ±.04m/s/s
Sine of the Inclination Angle of the Ramp vs. Average Acceleration of the Ball RollingDown the RampFigure 8: Indicting a proportional fit between sine of the angle and the average acceleration.Note that the proportional fit does not go through all the plots.Figure 9: Demonstrating a high-low fit of the sine angle vs. average acceleration graph. Theuncertainty of the slope is ±.3, and the uncertainty of the y intercept is ±.1m/s/s.
Conclusion: Am equation is extracted as below from angle vs. acceleration result graph. Acceleration = (.097±.003m/s2 ) Angle + (.06±.04m/s2) [Equation 3] The equation demonstrates that the angle of the inclined ramp has a direct linearrelationship to the acceleration of the ball rolling down the ramp. No direct theory was used tosupport this fit. This equation can be used to answer the research question. The following equation is presented from a Sin (angle) vs. acceleration graph, derivedgraph from angle vs. acceleration graph in order to compare the data with the theory mentionedin Equation 2. Acceleration = (6.1±.3m/s2)*Sin (Angle) + (0.0±.1m/s2) [Equation 4] The slope of the equation refers to the acceleration of gravity in Equation 2. But aconsiderable amount of energy was used in the rolling motion of the ball, therefore causes adecrease in the acceleration. The theory does support Equation 4, but the value of the slope hasno precise evidence that indicates to the gravity. Because the trend was supported by the theory,the data collected was also supported. Since Equation 3 and 4 came from the same set of data,only derived, the data of acceleration vs. angle was also back upped by the theory. The level of confidence is medium, because no theoretical support was given to the fit ofEquation 3, only the quality of the data collected was supported. The uncertainty of Equation 3 isfairly small compare to its values, therefore gives a high level of confidence in data collection.The fit in Equation 4 does not pass through all the data points, in effect; the fit of Equation 4 hasa low level of confidence. The experiment result is applicable when a ball of the same nature and dimensions isused on a U ramp with the same width in a rolling motion. Although not strongly supported, theresult of this investigation indicates a proportional relationship between the sine angle of theramp and the acceleration of the ball down the ramp in any similar situations. But the resultcannot be used when the ramp is inclined approximately over 50degrees.
Evaluation: The method of the experiment has a few defects that might affect the result; such assuming therolling rate of the ball stays constant throughout all the angles, short data collecting time, and the possibleeffect of air resistance that might built up in higher angles. The less the ball rolls when travelling down the U ramp, the higher the acceleration, because it doesnot use as much energy in the rolling motion. This can be improved by using a small block of metal sothat it slides instead of rolling. Not enough sample points were collected during trials, because the ramp was too short and when theangle higher, the average velocity of the ball increases, therefore causing difficulties in collectingqualitative data. This can be improved by using a longer ramp, thus increasing the sample collecting time. , air friction might cause When a ball is rolling down a relatively high angle, in this case 49.75an effect of decreasing the ball’s acceleration, because a hollow, light squash ball was used in theexperiment. The error can be eliminated by using a denser object, such as a small block of metalas mentioned above. An extension of this investigation would be researching the amount of force used in the rollingmotion of the ball. Thus the acceleration of a none rolling object is required to be collected and comparedto the acceleration of a rolling ball at each angle.