Lab 1 report (PCS211)

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Toronto Metropolitan University *

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211

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Physics

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Apr 3, 2024

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TORONTO METROPOLITAN UNIVERSITY FACULTY OF ENGINEERING DEPARTMENT OF PHYSICS Motion of Carts and Projectiles PCS 211: Physics: Mechanics | Section 43 | Group 37 Vallis King Hang Luong(501214102) Jal Gandhi (501263806) Course: PCS211 Section: 43 Instructor: Professor Rabello TA name: Professor Diana Ha Date of experiment: Sep 11th, 2023 Date of submission: Sep 17th, 2023

1 Introduction The objective of this lab is the identification of a block of unknown material through comparing the density to a table of known values. The objective of this lab is to prepare ourselves with the basic scientific approach to completing a physics lab and to familiarize ourselves with uncertainty calculations. The objective of this lab is to determine the volume of the block using 3 separate single measurement methods. This is achieved using a meter stick, Vernier Calipers and a graduated cylinder to calculate and measure the volume. By determining the density of the unknown material, it is possible to accurately compare and find out the type of metal.

2 Theory Background Defining Volume and Mass Volume is the amount of space an object occupies. It is a derived quantity and is measured in . Mass is the total amount of matter and is measured in kg or g. ?? 3 Defining Density Density is the measurement of how compact molecules are in a material. Consider the comparison of density between metals: Figure 2.1: Table depicting density values for different metals The table shows that there is a different density value corresponding to each different type of metal, meaning we can determine the type of metal through the specific density value.

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3 Now, consider the relationship of density, mass and volume: Figure 2.1.2: Diagram of the relationship of density, Volume and Mass As shown, density is proportional to mass and inversely proportional to volume. (1) ? = ? ? Hence, the unit for density is . 𝑔/?? 3 Defining Uncertainties Uncertainty identifies as the amount of variation in an already measured value. There is a need to provide uncertainties for most calculations as errors are likely to happen in labs and experiments. The standard deviation formula is used to indicate how much on average measurements differ from each other. A high standard deviation means low precision and vice versa. The formulas that we will use in this lab will comprise mostly the addition/subtraction uncertainty formula ( Equation 2) , the multiplication/division formula ( Equation 3), the mean formula ( Equation 4) and the standard deviation formula( Equation 5) . (2) △? = △? 2 + △? 2

4 (3) △? ? = ( △? ? ) 2 + ( △? ? ) 2 (4) µ = Σ? 𝑖 𝑁 (5) σ = ∑(? 𝑖 −µ) 2 𝑁 Procedure Figure 3.1: Experimental Apparatus used Unknown Material Vernier Calipers Triple Beam Balance Graduated Cylinder Metre Stick Beaker Note: the Beaker was only used to pour water for the Cylinder Set Up of the experiment Figure 3.2 correctly shows the placement of tools for one of the tests(water displacement test) conducted to measure the volume of the material.

5 Figure 3.2: Drawing depicting the water displacement test The mass of the material is calculated using a triple beam balance. The volume of the material is calculated in 3 different methods. It is calculated by measuring the dimensions of the material using Vernier Calipers, measuring the dimensions of the material using the meter stick and using the water displacement test. The water displacement test was done multiple times to ensure a more accurate reading. A slow descent of the material to the measuring cylinder is advised to provide an accurate measurement and equipment safety. The Vernier Calipers were tested for zero errors and resting on a flat surface before conducting the measurement. All equipment uncertainties were written down and accounted for before measuring.

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6 How the experiment was conducted The Vernier Calipers were calibrated to ensure scientific accuracy. Measuring the dimensions first is advised before finding out the mass and volume of the unknown material. Using the meter stick first provides a rough estimation of what the Vernier Calipers Measurements are going to be. Using the Vernier Calipers will affirm the dimensions in a more accurate manner. Finding the mass comes next as we do not want to get any water on the material which might affect the triple beam balance measurements. Carefully place the material and change the sliders until the white line near the measuring units line up with the stationary white line on the other end of the measuring plate. 50mL of water was used as the starting volume in the measuring cylinder and the material was then carefully placed inside to measure the displacement. 3 runs were recorded to ensure scientific accuracy and decrease uncertainty in our results. Results and Calculations Finding volume using the dimensions Finding Volume. To find the volume, the mean formula was used in the water displacement test. The measurements and results also have uncertainties accounted for to maintain scientific accuracy. The Equation 6 was used to calculate the volume: (6) ?????? = 𝐿??𝑔?ℎ · ?𝑖??ℎ · 𝐻?𝑖𝑔ℎ? The table below shows the different volumes calculated through using different methods to measure the volume or dimensions:

7 Figure 4.1: Table of results with measurements from meter stick and Vernier Calipers Vernier Caliper Length (cm) Width (cm) Height (cm) Volume (cm^3) 4.12 ± 0.005 cm 0.86 ± 0.005 cm 1.7 ± 0.005 cm 6.02 ± 0.06 cm^3 Meter stick Length (cm) Width (cm) Height (cm) Volume (cm^3) 4.3 ± 0.05 cm 1 ± 0.05 cm 1.3 ± 0.05 cm 5.59 ± 0.56 cm^3 Figure 4.2: Table of results with the water displacement test Initial Graduated Cylinder Volume (mL) Volume of Graduated Cylinder W/ Unknown Material (mL) Volume (mL) 50 ± 1 mL 57 ± 1 mL 7 ± 1 mL 50 ± 1 mL 56 ± 1 mL 6 ± 1 mL 50 ± 1 mL 56 ± 1 mL 6 ± 1 mL Average 6.33 ± 1 mL Finding Density Finding Density. To find density, the Equation 1 is used: (7) ? = 16.8𝑔 6.02 ± 0.06?? 3 = 2. 79𝑔/?? 3

8 There is a different density provided for each different measuring method used. Below is the table showing the results obtained from the measurements: Figure 4.3: Table showing different densities from the different volumes Tests Calculations Density Vernier Caliper P = 16.8 g / 6.02 ± 0.06 cm^3 2.79 g/cm^3 Meter stick P = 16.8 g / 5.59 ± 0.56 cm^3 3.00 g/cm^3 Water displacement test P = 16.8 g / 6.33 ± 1 mL 2.65 g/mL Average 2.79 g/cm^3 + 3.00 g/cm^3 + 2.65 g/mL / 3 = 2.81 g/cm^3 Comparison of Results with Table of known materials It is important to select the most suitable or accurate value of the density calculated in Figure 4.2 closest to a known material in Figure 2.1 . We have found that the unknown material with the density of 2.81 g/cm^3 bears the closest resemblance to the metal Aluminum. Using the mean formula to calculate the density gives a result of 2.81 g/cm^3, which is not closer to the density of Aluminium on the table than all 3 separate densities that were calculated, the closest to the exact value in this case is the result using the graduated cylinder to calculate volume. This shows that the average density is not beneficial for this situation, so we can conclude that determining the average density is not valid for determining the unknown metal from the 3 measurement methods.

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9 (8) ??𝑔 = ????𝑖?? 1 + ????𝑖?? 2 + ????𝑖?? 3 3 Discussion and Conclusion Concrete Values and Conclusion The main goal of the lab was to calculate the density of the unknown material and determine the material using the density by comparing it to a table of known values. A total of 3 different results using different equipment were set up to provide the best comparison and conclusion. The 3 calculated densities of the material are 2.79 g/cm^3, 3.00 g/cm^3 and 2.65 g/mL respectively and the unknown material was concluded to be Aluminum. Calculating the standard deviation using Equation 5, = 2.81g/cm^3 µ = (2.79 𝑔/??^3 + 3.00 𝑔/??^3 + 2.65 𝑔/?𝐿) 3 σ = ∑(8.44−2.81) 2 3 = 3. 25𝑔/??^3 We see that we have a high standard deviation which indicates low precision. Looking at the 3 different densities with their percentage errors in Figure 5.1, Figure 5.1: Table of the 3 different densities with percentage error Tests Calculations

10 Vernier Caliper (2.79 g/cm^3 - 2.70g/cm^3 / 2.70 g/cm^3) x 100% = 3.33% Meter stick (3.00g/cm^3 - 2.70g/cm^3 / 2.70g/cm^3) x 100% = 11.11% Water displacement test (2.65 g/mL - 2.70g/cm^3 / 2.70 g/cm^3) x 100% = 1.85% (9) δ = υ 𝐴 −υ ? υ ? | | | | | | · 100% A high percent error would show that we have bad accuracy and vice versa. Looking at the percent errors, there is mostly a small percent error for each method (except for the meter stick), which shows good accuracy for those 2 methods. In conclusion, we have good accuracy and low precision in this lab. Looking at the reason why our water displacement method had the most accuracy, it is pretty obvious to find that it is the only test to have undergone multiple trials. To improve our precision further onwards, it is important to do multiple trials of each method instead of only the water displacement method. Errors and Uncertainties Some calculations were done on paper and calculators, hence there are rounding uncertainties. A lot of random errors might also have occurred. A misread of the volume using the measuring cylinder or Vernier Calipers might result in an estimated 1mL difference in the result or 0.1 cm difference in dimensions. An unnoticeable zero error not

11 found on the Vernier Calipers might also result in a 0.1cm difference in dimensions. Considering these factors, by repeating the measurements and the experiment, a more accurate result could have been determined and that might lead to a different conclusion for our unknown material. There is a fundamental difference between percent errors and uncertainties; the error is actually the difference between the calculated value and the desired value, while uncertainty is an estimate of the range of the difference. Hence, the percentage error is vital in maintaining scientific accuracy. References Toronto Metropolitan University Laboratory 1 “Lab 1 - Uncertainty” Accessed: Sep 12, 2023.

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Lab 1 report (PCS211)

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