BY: Tahmid Zaman.Group: Javeria, NidaTable of Contents1.0 Purpose 32.0 Materials 33.0 Diagram of Apparatus 34.0 Procedure 45.0 Results 55.1 Position -Time Data 55.2 Isolated Charts 65.3 Position – Time Graph 65.4 Average velocity – Average Time Graph 76.0 Calculations 76.1 Theoretical 76.2 Experimental 97.0 Coefficient of Friction 108.0 Percentage Difference 119.0 Source of Errors 1110.0 Discussions 1211.0 Conclusions 1312.0 Appendix i Symbols 13Purpose:The purpose of this lab was to calculate coefficient of rolling friction of a cart along a ramp using the theoretical acceleration using Newton’s 2nd Law and experimental acceleration using ticker-tape data analysis.Materials:* Ticker-tape timer* Ticker-tape* Transparent tape* 0.839kg Cart with three wheels* 0.500kg weight* 0.200kg weight* A ramp approximately 2 m in length with a pulley attached at the end* Ruler – 1m* Two desks (labeled 1,2 according to the diagram)* 1 Approximately 2m long stringDiagram of Apparatus:Procedure:1. The apparatus was set up as shown on Figure 1.2. Firstly, the string was attached to the front of the cart and the other end of the string looped and suspended over the pulley.3. A long piece of ticker-tape was attached to the back end of the cart and pulled through the ticker-tape timer.4. The 200g or 0.200 kg weight was hooked on to the loop of the string, which was the falling end of the string.5. The 500g or 0.500 kg weight was placed on top of the 839g or 0.839 kg cart.6. Then everything was set with the 200g weight suspended and the cart held in position.7. Then, the ticker-tape time was turned on and the cart was released.8. The cart was stopped at the end of the ramp short of falling over.9. The data on the ticker-tape timer was then laid out and measured with 6 dots (0.1 second) per data point.ResultsPosition-Time Data for a cart rolling along a ramp:Dot #TimePosition ?d?d?ttav (s)vav (cm/s)00.00.00.800.10.05860.10.81.800.10.1518120.22.62.700.10.2527180.35.33.700.10.3537240.49.04.700.10.4547300.513.75.700.10.5557360.619.46.600.10.6566420.726.07.600.10.7576480.833.68.400.10.8584540.942.09.500.10.9595601.051.510.300.11.05103661.161.811.500.11.15115721.273.312.400.11.25124781.385.713.500.11.35135841.499.214.400.11.45144901.5113.615.200.11.55152961.6128.815.200.11.651521021.7144.08.200.11.75821081.8152.22.500.11.85251141.9154.7Isolated Charts:Time (s)Position ?d0.00.00.10.80.22.60.35.30.49.00.513.70.619.40.726.00.833.60.942.01.051.51.161.81.273.31.385.71.499.21.5113.61.6128.81.7144.01.8152.21.9154.7tav (s)vav (cm/s)vav (m/s)0.0580.80.15181.80.25272.70.35373.70.45474.70.55575.70.65666.60.75767.60.85848.40.95959.51.0510310.31.1511511.51.2512412.41.3513513.51.4514414.41.5515215.21.6515215.21.75828.21.85252.5Position-Time Graph:Average Velocity – Average Time Graph:Calculations:Theoretical:Weight (0.200 kg)Cart (1.339 kg)Let “w” represent the Weight that was droppedm w = 0.200 kga = g = 9.80 m/sï¿½Fnet = maReplacing acceleration with gravity as it is fallingFnet w = m w g – FTFT = m w g – Fnet wFnet = m w aLet “c” represent the Cartm c = 0. 500 kg + 0. 839 kgm c = 1. 339 kgFnet = maFnet = FappFnet = FTm c a = FTIf FT is constant then:m c a = m w g – Fnet wSubstituting Fnet w with mwa as Fnet = m w am c a = m w g – m w aRearranging:m w g = m c a + m w aCommon factoringm w g = a (m c + m w )Therefore isolating for a:m w ga = ——————–(m c + m w )Solving for a:(0. 200 kg)(9.80 N/kg)a = ——————————-1. 339 kg + 0. 200 kg1. 96 Na = ——————–1. 539 kga = 1. 27355 m/sï¿½a = 1. 274 m/sï¿½Experimental:* The graphs were accepted to the 1.5s as the cart was stopped after the respective time.Acceleration of system = 1.274 m/sï¿½Ff r = ï¿½Fr . FNï¿½Fr = Ff r / FNUsing line of best fit from the V avg – t avg Graph*:The experimental acceleration of the system:Using two random points – 1.25s point and the 0.35s point (as they fall on the line):?VavgM x y = a = ————-?tavg12.4 – 3.7a = ————-1.25 – 0.35a = 0.967 m/sï¿½Coefficient of Friction:Theoretical acceleration: 1.274 m/sï¿½Experimental acceleration: 0.967 m/sï¿½Ff r = ï¿½Fr . FNï¿½Fr = Ff r / FNFnet = m . aFnet also can be expressed as:Fnet = Fapp – FfSubstituting the net force, the force applied on the cart and the force of friction:(m system a) system = m c a c – ï¿½Fr . FNFN = F g = mgRearranging and substituting for Fnormal:(m system a) system – m c a cï¿½Fr = ————————————–m c gSolving for ï¿½:(1.539 kg x 0.967 m/sï¿½) – (1.339 kg x 0.967 m/sï¿½)ï¿½Fr = —————————————————————–1.339 N x 9.80 m/sï¿½1.488213 N – 1.294813 Nï¿½Fr = ——————————————-13.1222 N0.190083 Nï¿½Fr = ——————————–13.1222 Nï¿½Fr = 0.014485604ï¿½Fr = 0.015Percentage Difference:| R1 – R2 |Percentage Difference = |——————–| x 100 %| (R1 + R2)/2 || Theoretical – Experimental |Percentage Difference = |—————————————| x 100 %| (Theoretical + Experimental)/2 || 1.274 – 0.967 |Percentage Difference = |————————-| x 100 %| (1.274 + 0.967)/2 || 0.307 |Percentage Difference = |—————-| x 100 %| 1.1205 |Percentage Difference = | 0.27398 | x 100 %Percentage Difference = 27.4 %Sources of Error:During the experimental process, there were a number of factors that might have caused the variances in results. The elimination of friction for calculating the theoretical acceleration is the first source of error. Another error that may be caused a difference in acceleration was the friction of the pulley wheel and the string on the pulley that was unaccounted for; it would be quite insignificant, but it might make a difference on the hundredth or thousandth digit. The measuring tool – the ticker-tape timer it self might have been an error as the tape behind might have caused more friction due to air and passing through the timer.DiscussionThis experiment was carried out to calculate the coefficient of rolling friction of a cart along a ramp using the theoretical acceleration derived using Newton’s 2nd Law and the experimental results from ticker-tape analysis. The theoretical acceleration of 1.274 m/s2 was obtained using Newton’s 2nd Law on the tension on the string. The theoretical acceleration could also have been calculated by using the mass and time from the system as a whole. The experimental acceleration was calculated from the analyzed ticker-tape data where the displacement – time graph showed a steady acceleration. The average velocity – average time graph further proved how straight the data points were that most of the points could be used as the slope. However, there were deviations like the line of best fit not meeting zero and sudden change in velocity after the 1.5th second. Perhaps the line of best fit wasn’t exactly on zero due to human error where the cart was slipped a little bit and then let go. Another reason might be that the dots were measured right from the first one where the ticker-tape timer was turned on before the cart was released; therefore the slight difference of time between the start of the timer and the release of the cart caused the graph to go off zero. Also, the sudden change in velocity on both the average velocity – average time and position – time graphs after the 1.5th second could be explained due to the stopping and rapid negative acceleration near the end of the ramp. Therefore, the data was only valid till the 1.5th second as the cart was stopped.After acquiring the experimental acceleration, the ï¿½ of the rolling friction was calculated using the ï¿½Fr = Ff r / FN equation where the force of friction is the net force of the system subtracted by force applied on the cart. So, rearranging the equation allowed solving for the ï¿½ of friction of 0.015. This amount was certainly accurate as observing the motion of the cart and the displacement covered in the short space of time was fast. This showed that the coefficient of friction must have been very low even before the calculations were made.The percentage difference between the theoretical and experimental acceleration was a mere 27.4 %. However, the through the observation of the motion of the cart, the velocity was relatively high and it took a mere 1.9 s to travel about 1.5 m, which includes the deceleration. The graphs show that the cart had an acceleration of nearly 1 m/s ï¿½ as opposed to the theoretical 1.274 m/sï¿½, which is very close therefore in agreement with the coefficient of friction and the force of friction.This experiment was not nearly as perfect as one could hope, but the surface of the ramp and the friction of the wheels were fairly low as a mass close to 13 % of the cart could exert such a high force and accelerate the at such a high rate. Even though the surface was very smooth, it came with some visible bumps and dents that could have caused increased friction. So, a smoother surface would allow more accuracy. All in all this experiment was a success as the surface was very with a very low coefficient of friction.Conclusion:After the experiment being carried out, the coefficient of friction was deduced at 0.015 derived from the experimental acceleration at 0.967 m/sï¿½. The theoretical acceleration at 1.274 m/sï¿½ was 27.4% more than that of the experimental acceleration.

# The Physics of a Cart Rolling along a Ramp Essay

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