Work and Energy Lab Report 5

Work and Energy Abstract: The purpose of this experiment was to verify the conservation of energy, the work-theorem, and the modified work-energy theorem. Throughout the experiment, the cart had to cover a set distance on an air-track and the time was measured using a photogate timer. The theorems were verified since the % difference were below 5% and the % difference between work and energy were 2%, which told us that they are linked. Introduction: The purpose of this experiment was to verify the conservation of energy principle as well as the work energy theorem. Alongside these two concepts, the work with a conservative force as well as a non-conservative force was observed. The two forces differ in regards to their dependence on the path they take, where a conservative force regardless of the path it takes will produce the same amount of work. In contrast, a non-conservative force depends on the path it takes like friction for example. The principle of conservation of energy states the energy can neither be created nor destroyed but only transformed from one form to another. Work is done when a force acts upon an object, given that the force contributes somehow to the displacement of the object. Energy is the ability to do that work. Once one form of energy transforms to another that can attributed to work being done by the force acting upon the object. Thus, to put the work-energy theorem in simpler terms; whenever work is done, energy is transformed. We test the principles and see whether or not they are similar to one another theoretically and experimentally. Within mechanical systems, the object is usually isolated from any external non-conservative force such as friction and the only usual relevant forces of energy are kinetic and gravitational energy. In this experiment, that would then mean that the sum of kinetic energy and potential energy would be the total energy calculated. Individual potential and kinetic energies do not need to remain constant in accordance to the principle of conservation energy. However, the total energy must be constant throughout the process. Within this experiment, the motion of a cart on top of an air track was the main object of focus. A silver mass was added to the cart to increase the mass of it. An aluminum block as placed right under one of the legs of the track to incline the plane to further examine the cart under the influence of gravity. A pullet was also attached to the air track for a separate part of the experiment with a string that was attached to the cart and ran onto the pullet leading to the hanging mass, which were the paperclips. A photogate timer was positioned at a specific location to measure the cart’s time to travel through the photogate timer. We measured the distance the cart traveled and height of the aluminum block using a ruler that was on the air track itself and weighed the cart itself, paperclips, and silver mass using a mass balance. The final velocity was calculated after three trails using formula 1, where L is the length of the cart and the Δt is the average of the three trials. (1) V f= L Δt After calculating the final velocity of the cart, the energy of the cart while it was moving (kinetic energy) was calculated using formula 2. (2) KE= 1 2 M ( Vf ) 2 The gravitational potential energy was calculated using formula 3, which is the energy the cart has whilst in the gravitational field, having the potential to move. The variables were assigned as follows; M is the mass of the cart, g is the acceleration due to gravity, d id the displacement of the cart from the center to its final destination, H is the thickness of the aluminum block (which was used to incline the air track), and D is the distance between the legs of the air track. (3) GPE = Mgd ( H ) D If any force acts upon the object, then work is being done on the object. Work is the actual conversation of one form of energy to another. The work done is constant and was calculated using formula 4. The force acting on the object is represented by F, d is the magnitude of the object’s displacement and θ is the angle between the force and displacement vectors. (4) W =Fdcos ( θ ) The force acting against the object is the force tension. This was calculated using formula 5, where M is the mass of the cart, g is the acceleration due to gravity, and m is the hanging mass. (5) Ft= mMg M +m The metal block is used to create an inclined plane, thus calculating the tension through equation 6. (6) Ft= H mMg (1+ ) D M +m The relationship of work and energy are shown through equation 7. The work-energy theorem is then further modified and takes into account gravitational potential energy and is of much better use when an object is on an inclined surface, shown by formula 8. Formula 7 is the work-energy theorem and formula 8 is the modified work-energy theorem. (7) Wnet =ΔKE (8) W = ΔKE + ΔPE The percent difference was then calculated using formula 9 to decide how accurate the experimental results were to the actual results. (9) ¿ x 100% 1 (calculated−experimental) 2 Percentdifference=¿ ¿ calculated−experimental∨ (9) ¿ x 100% 1 (calculated −experimental ) 2 Percentdifference=¿ ¿ calculated −experimental∨ Results: The total experiment was done in three overall parts. Part A consisted of tests regarding the conservation of total mechanical energy. The thickness of a metal block was measured which came out to be 0.025 m (H). The distance measured from the center of the cart until the end of a leg on the air track was 1.6 m (d). The length of the cart was measured to be 0.124 m (L). The velocity was taken of both with the cart with no mass on it and with the weight of a silver mass on it (M). Cart Trials 1 2 3 Time (s) 0.1411 0.1413 0.1412 Cart with 1 2 3 0.143 0.1418 0.1403 Mass Δt M (kg) d (m) Vf (m/s) Kef (J) GPEi (J) % Dif 0.141 0.174 1.6 0.8782 0.0673 0.0685 1.7 2 5 0.141 0.224 1.6 0.8751 0.086 0.0881 2.4 7 5 Table 1. “Data of the conservation of total mechanical energy”. It did not seem to affect the velocity of the cart once the extra weight was added on. There was a larger margin of difference with the added weight on the second cart. There was no significant difference between the time it took the cart with or without the weight to travel across the allotted distance. Velocity was calculated using formula 1. Vf (cart without added mass) = Vf (cart without added mass) = 0.1240m 0.1412s 0.8782 m s Using formula 2, I found kinetic energy. ( KE (cart without added mass) = 1( m 0.1745kg ) 0.8782 2 s KE (cart without added mass) = 0.673J 2 ) Using formula 3, I found the gravitational potential energy. ( ) GPE (without mass added mass) = ( 0.1745kg ) 9.81 m2 ( 1.600m )( 0.025m ) 1.00m s GPE (without mass added mass) = 0.068J Finally, the percent difference was calculated using formula 9. ¿ x 100% 1 (.0685J +0.0673J ) 2 Percentdifference=¿ ¿ 0.0685J −0.0673J ∨ Percentdifference=1.747% For Part B, we worked with the work-energy theorem. The aluminum block was removed from the leg, thus leaving the air track leveled. A string was attached to the cart and then a set amount of paperclips was attached at the end of the string as the hanging weight. The distance set for the cart to travel was 0.600m and the photogate was positioned at the end of that distance. There were two overarching trails, which differed in the cart having a weight attached to it, or not. And under each of those overarching trials, the hanging mass either consisted of either 3 paperclips or 6 paperclips. There were two separate trials, one with either 3 or 6 paperclips and with or without a silver mass atop the cart. The mass of the 3 paperclips totaled 0.003 kg, and the mass of the 6 paperclips totaled 0.006 kg. The silver mass came out to 0.0502 kg. Per each separate trial, there were 3 runs done and the average of them was taken and shown in Table 2. Run s Time (s) 1 0.2988 2 0.2947 3 0.2954 1 0.214 2 0.2146 3 0.215 1 0.3438 2 0.3431 3 0.3437 1 0.2383 3 Paper Clips 6 Paper Clips 3 Paper Clips w/ mass 6 Paper Clips w/ mass 2 0.2381 3 0.2386 Δt M (kg) m (kg) d (m) 0.296 3 0.194 4 0.003 0.214 5 0.194 4 0.006 0.6 0.578 0.032 5 0.343 5 0.244 4 0.003 0.6 0.361 0.157 0.238 3 0.244 4 0.006 0.6 0.5203 0.033 1 0.6 Vf (m/s) 0.4185 KEf (J) 0.017 FT (N) 0.029 % Dif. 0.017 2.1 4 0.057 0.029 0.057 Table 2. “Data for work-energy theorem”. We saw a trend that without the mass, the cart would have a lower time to reach the set distance, and it would go even quicker when introduced to 6 paperclips rather than 3 paperclips. Formula 5 was used in order to find the force of tension acting on the cart. FT (3 paperclips) = W (J) ( 0.003kg)(0.1944kg )( 9.81 m ) s2 0.1944kg+ 0.003kg FT (3 paperclips) = 0.029 N Part C was similar to Part B, but here we were focused on the modified work-energy theorem. 0.034 3 0.174 0.034 5 5.3 9.1 4.1 The densities for the ruler and Vernier caliper were 2972 (kg/m3) and 2762 (kg/m3), respectively. Results for the caliper were much closer to the actual value of 2698.9 (kg/m3) because of the lower uncertainty values. For density 1 and density 2, the uncertainties with the caliper were ± 26.61 (kg/m3) and ±31.78 (kg/m3), respectively. The uncertainties for the ruler were ±86.54 (kg/m3) for density 1 and ±167.2 (kg/m3) for density 2. The uncertainties were determined as “independent and random” (density 1) as well as “dependent and random” (density 2). Table 1: Table of Values Length Length l (m) δl (m) Ruler 0.037 0.0005 Calipe 0.0382 0.0000 r 4 2 Mass m (kg) Mass δm (kg) Width w (m) 0.025 Width δw (m) 0.0005 Height h (m) 0.038 0.02544 0.00002 Actual Density (kg/m3) 2972.97297 3 2762.83560 1 0.03888 Density 1 ρ±δρ (kg/m3) 86.539471 4 26.605002 4 Volume (m3) 0.0000351 5 0.0000378 2 Height δh (m) 0.0005 0.00002 Density 2 ρ±δρ (kg/m3) Ruler 0.1045 0.001 167.202337 Calipe r 0.1045 0.001 31.4768682 Table 1 – Density Calculations The density of the given aluminum block was found by weighing it on an electronic scale and using both a metric ruler and Vernier caliper to measure it in meters. The density of the aluminum block using the calipers was closer to the actual value of aluminum density. The uncertainties for density 1 were determined based on the following equation: ¿ l∨¿ δl ¿ ¿ w∨¿ δw ¿ ¿ ¿ h∨¿ δh ¿ ¿ (1.) ¿ m∨¿ δm ¿ 1 /2 ( ¿ ¿2 ] ¿ ¿ ( ¿ ¿2 +¿ ¿ ¿ ¿ ρ∨¿=¿ δρ ¿ 2 2 2 δρ 0.0005m 0.0005m 0.0005m 0.001kg = + + + | 0.037m| | 0.0250m| | 0.0380m| | 0.1045kg| kg 2973 3 m | | [( δρ )( )( ) ( 1 2 2 2 1 2 ] ×2973 kg m3 3 δρ =86.5kg/ m The uncertainties for density 2 were determined based on the following equation: 2972× | 2973 = [ ( 0.0135m ) + ( .0200m ) + ( .0132m ) + (.00957m ) )] 2 | 2 2 ¿l ∨¿ δl ¿ ¿ w∨¿ δw ¿ ¿ ¿ h∨¿ δh ¿ ¿ ¿ m∨¿ δm ¿ 1 /2 2 (¿ ¿ ] (2.) ¿ ¿ ( ¿ ¿2 +¿ ¿ ¿ ¿ ρ∨¿=¿ δρ ¿ ( |[ )( ( | [ ) ( 2 2 )( 2 ) ( )] 2 1 /2 δρ = 0.0005m + 0.0005m + 0.0005m + 0.0005kg | 0.037m| | 0.0250m | | 0.0380m| | 0.1045kg| kg 2972 3 m kg 2973 m3 2 2 2 δρ = 0.0005m + 0.0005m + 0.0005m + 0.0005kg × | | | | | | |0.1045kg| 0.037m 0.0250m 0.0380m kg 2763 3 m | | )( )( ) ] ×2973 kgm 2 δρ =167kg/ m3 Density of the aluminum block dimensions was determined for the Vernier caliper based on the equation that follows: p= mass ( kg ) (3.) 3 volume(m ) p= .1045(kg) 3.515×10−5 ( m3 ) p=2973 kg m3 3 Also, the volume was determined for the aluminum block based on the following equation. The example is based off the data for the metric ruler, but the same equation was used for the Vernier caliper V= Length * width * height (4.) V = ( 0.03824(m)) × (.02544 ( m) ) × ( 0.03888( m ) −5 3 V= (3.78×10 )(m ) The second part of this experiment consisted of dropping a ball from a two-meter height a total of ten times. The data for the 10 trials can be seen in Table 2. The average time it took the ball to hit the ground was taken, which came out to about 0.55 seconds. The height of 2 meters was then used to calculate the theoretical value, which came out to 0.64 s which is seen in (5.) These two numbers were also then used to calculate the percent of error that resulted in 14.1% error which is seen in (6.) Trial # Time to drop (s) 1 2 3 4 5 6 7 8 9 10 Average tavg±δt 0.54 0.54 0.52 0.59 0.59 0.51 0.58 0.56 0.56 0.52 0.55 0.55±0.04 δt 0.04 Table 2 – “Time for ball to drop 2 meters.” The table shows timings of a tennis ball dropping from a height of 2 meters and me timing each drop. The times vary due to reaction time being different each time or placement of the drop itself not occurring from exactly the same height as the other trails. Overall timings look fairly consistent. (5.) Theoretical Yield x́ = ∑x n Theoretical Value = √ 2y g Theoretical Value = √ 2( 2m) 9.8m/ s2 = 0.64s Percent Error (6.) Percent Error = (TheoreticalValue− ActualValue) TheoreticalValue Percent Error = (0.64s−0.55s) 0.64s * 100% * 100% Percent Error = 14.1% Once all the times were inputted as well as the average into the table, an uncertainty was calculated from those value, found below in figure 7. It was found that the uncertainty was about 0.04s. Uncertainties for the ball drop timings were determined using the following equation: t (¿¿ large−t avg)+(t avg−t small) (7.) 2 δt=¿ δt= ( .59 ( s )−.55(s ) ) +(.55 (s )− 51(s )) 2 δt=.04(s) Error Analysis In the ball dropping experiment, there may have been several errors that were involved such as the stick not being exactly 2 m tall due to curvature of the stick. Due to this, it may have been slightly above or slightly below 2 m. Also, another error introduced could have been the hand placement. The ball could have been placed at different points above or on the top of the meter stick, which can alter the overall amount of time it takes to hit the ground. Each data point may have been affected depending on where the ball was placed, since not every trail could have started in the exact same position. Furthermore, the timing may have been affected based on the reaction time of the ball dropping to the ground and stopping the timer on time. There could have been a delay or the timer could have been stopped before the ball reached the ground. The experimental error was 14% from the actual value of . 64 s. In the introductory experiment, I had a reaction time of 0.46 ± 0.11 s. This delay of reaction could have played a part at which time I actually pressed the start on the stopwatch to where I had to react to both the ball hitting the ground and me hitting the stop on the stopwatch. For the measuring the dimensions of the aluminum block, there may have been errors in the block not being properly placed in the caliper or the caliper sliding a bit after taking it off one of the sides. There could have also been some inconsistencies with using the ruler, since it did have a higher uncertainty threshold. It may have not been properly placed at the beginning of the edge of measurement, or it could have been misread while taking the measurement.