Simple Pendulum, Week 7, Lab 7

1 Title of the Experiment: Simple Pendulum Student’s name: Naeim Naeimi Section SLN: PHY122-76056 TA’s Name: Jude Pereira, Yash Patil , Ayush Kumar Singh Week of the experiment: Week 7, Lab 7

2 Objectives: In this lab we are learning about Simple Pendulum. This experiment teaches us about simple harmonics and how oscillations vary with different factors (length, amplitude, gravity, and mass) Experimental Data: PART 1. Period T vs Amplitude θ Diameter of the bob: 20cm Density of the bob: 8470 kg/m 3 Amplitude (°) Total Time (10 swings, s) Period (s) 5 28.31 2.831 10 28.32 2.832 15 28.38 2.838 20 28.58 2.858 25 28.56 2.856 Relationship observed: As θ increases, period “stays the same.” Fig. 1

4 PART 3: Period T vs Length Pendulum Length (cm) Total Time (10 swings, s) Period (s) 60 15.61 1.561 100 20.10 2.010 140 23.93 2.393 180 27.06 2.706 220 29.92 2.992 Relationship observed: As L increases, period “increases.” Fig. 3 PART 4: How does the period depend on Gravity? Period on Asteroid Brian: 7.558 Relationship observed: As g increases, period “decreases.”

5 Data Analysis (10 points): PART 1.  Show sample calculation for volume and mass of the bob for one of the materials. o Volume of a sphere = ସ ଷ 𝜋 r 3 o Mass of sphere w density = ସ ଷ 𝜋 r 3 * 𝜌 o Density of Brass = 8470 kg/m 3 o Diameter of bob = 20 cm = 0.2 m  M = ସ ଷ 𝜋 r 3 * 𝜌 = ସ ଷ 𝜋 (0.1) 3 * 8470 kg/m 3 = ସ ଷ 𝜋 * 0.001 m 3 * 8470 kg/m 3 = ସ ଷ * 0.0031459 m 3 * 8470 kg/m 3 = 35.479 kg PART 3.  From the slope calculate gravitational acceleration (g) and its uncertainty for Earth. o Period T = 2 𝜋 √ ௟ ௚ o T 2 = 4 𝜋 2 ௟ ௚ o T 2 = ସ (గ)^ଶ ௚ 𝑙 o Slope = Y = mX + b o Y = T 2 o X = l o m = ସ (గ)^ଶ ௚ o g = ସ (గ)^ଶ ௠  g = ସ (గ)^ଶ ସ.଴଻଼ = ସ∗ଽ.଼଺ଽ଺ ସ.଴଻଼ = ଷଽ.ସ଻଼ସ ସ.଴଻଼ = 9.6808 m/s 2

7 Results (3 points) Experimental g Theoretical g % discrepancy Earth 9.68 m/s 2 9.81 m/s 2 1.325% Asteroid Brian 0.0069 m/s 2 0.007 m/s 2 1.429% Discussion and Conclusion (10 points): In this experiment we learned about a simple pendulum and how different factors influence them. The beginning of our tests was to verify the factor of angle/amplitude on the system. We learned that the angle really did not have any effect on the system. At 5 ̊ there was a period of 2.83 seconds per cycle where at 25 ̊ we had a period of 2.85 seconds per cycle, and it is difficult to tell if that variance is because of the change in angle or human error. Going by the equations we don’t have an angle given. The main reason behind that is the increased angle will give an increased distance to travel which will increase the period, but we also increase the period, thus they balance each other out. The next set of tests was to show the effect of mass on our system. We maintained a constant 10 ̊ and we found out that with a 2.3kg mass or an 80.8kg mass we maintained a 2.83 second period per cycle. Again, going by the equations we don’t have a mass given. The main reason behind this effect because the larger the mass the greater the moment of inertia, thus the greater force needed to move the object and the greater the mass the greater the gravitational force on the object will be, so they cancel each other out. The next set of tests is where things started to affect our period per cycle. The first one that had an effect was adjusting the length on our pendulum. Using the equation T = 2 𝜋 ඥl/g , which shows that as we increase the length time per cycle will increase. We approved this with multiple measurements showing at 60cm we had a period of 1.56 seconds and at 220cm we had a period of 2.99. If we calculate these out for 60cm we get: T = 2 𝜋 ට ଴.଺ ଽ.଼ଵ = 1.55, we have a difference of 0.01 seconds, a discrepancy of 0.64% and T = 2 𝜋 ට ଶ.ଶ ଽ.଼ଵ = 3.11, we have a difference of 0.12 seconds, a discrepancy of 3.85%. For more precise data with less discrepancies, we would need to run more tests. Per experiments, the longer the pendulum the larger the time per cycle gets. The reason behind it is because increasing the length of the pendulum increases the distance the bob must travel to complete a cycle. The final tests showed the effect of gravity on our system. We tested gravity based on earths’ gravity, the moon, and the asteroid Brian. With a very short pendulum, 10cm/0.01m, we had a hard time measuring the period of the system on earth and the moon. The main reason is that because the length was so small, the period was too short, T = 2 𝜋 ට ଴.଴ଵ ଽ.଼ଵ = 0.20 seconds per cycle on earth. When we went to Brian, we found out that we were able to take measurements of the cycle, which came out to 7.558 seconds per period. We used this to calculate the gravity on Brian, T = ସ గ^ଶ ௟ ௚ → T = 7.558, g = 0.0069m/s 2 , when we checked that against the theoretical value, we found a difference of 0.0001m/s 2 the discrepancy of 1.429%. The reason the gravity plays such an important role in our

8 calculations is that because when we increase our gravity, we also increase the force on the object so the greater the acceleration, thus the period decreases and when we lower the gravity, we have a lower acceleration. In conclusion, simple pendulum is primarily, in a closed system, only affected by the length and gravity. When we increase the length, we will have a larger period and when we increase the gravity, we have an inverse effect, and we shorten the period. The discrepancies in our measurements and calculations were due to human error of hitting a timer and stopping it at the correct spot as well as measuring how long the string was for the pendulum. In a lab setting you might also get discrepancies due to drag from air, as well as having mistaken measurement for the string and timer as well.