Lab report 7-Lab 131_Centripetal Force-Nujhat Fariha Khan (1)

Physics Laboratory Report(111A) Title (5 points) Lab number and Title: Lab Report 7- Lab 131:Centripetal Force Name: Nujhat Fariha Khan Date of Experiment: _03_ _/_ 05_ _/_ 2024 ____ Date of Report Submission: _ 03 __/ _19 __/_ 2024 ____ Course & Section Number: PHYS 102A, 102 Instructor’s Name: Homa Yazdi Karimi Partners’ Names: Harsha Dasari, Eric Zhu and Shaylah Lee 1. INTRODUCTION 1.1 OBJECTIVES In this laboratory, the objective was to verify the expression of centripetal force. 1.2 THEORETICAL BACKGROUND Pendulum swing movement is an example of non-uniform rotational motion with a non-constant centripetal force F c . The pendulum bob moves with a non-constant speed v along a circle of constant radius r (Here this radius means the pendulum length L.). The velocity vector tangent to the circle is always perpendicular to the circle radius. There are two forces (string tension T and gravity mg, when neglecting air drag) acting on the pendulum bob of mass m when it is in movement. Newton's Second Law can be written in two directions which include along the string and perpendicular to the string. The equations related to these concepts include- (1) ∑ F=T-mg cos(θ) = ma r = F c (along the string) (2) -mg sin(θ) = ma p = F p (perpendicular to the string) When the pendulum is at the lowest point, the angle θ is zero and the equations become: (3) T-mg = ma r = F c (along the string) (4) 0 = ma p = F p (perpendicular to the string) At the bottom of the pendulum's swing the centripetal force F c , is determined by equation (3). The centripetal force is along the string and always points to the center (suspension point). The tension has to overcome the gravity component along the string (mg.cosθ) to provide the centripetal

force for the pendulum's rotational motion. The tension T reaches maximum at the lowest point (θ=0 ° ) when the oscillating bob is vertically below the suspension point. The centripetal force at the lowest point of the swing can be found by using the linear speed (v), rotational radius (r) and the pendulum mass (m) as shown in equation (5). (5) F c =ma c =mv 2 /r In this experiment, the force sensor helps to measure the string tension T vs. time when a pendulum is swinging and a photogate to measure the linear speed v when the pendulum passes through the lowest point of the swing, You can compare the centripetal force found from the tension T at the lowest point and the gravity of the pendulum to one that is found from linear speed and rotational radius. 2. EXPERIMENTAL PROCEDURE To start this experiment, the following pieces of equipment were used which included Lab Computer with Capstone software installed, 850 Universal Interface, Force sensor, Photogate, Right Angle Clamp (x2), Long Steel Rod (located near the rod), Short Aluminum rod, Pendulums with string. The experiment was set up by positioning the photogate so that the pendulum bob at rest blocks the photogate’s beam. The center of mass of the pendulum bob should be at the same height as the photogate’s beam. The force sensor was connected to “PASPort 1” on the 850 Universal Interface and the photogate to “Digital Inputs 1” and the computer was logged in. The AC adapter power cord was connected to an electrical outlet under the lab table. The power push, located on the left corner of the interface was pressed. The green LED indicator below the power push button should light up. The “Lab 131 Centripetal Force” file in “Physics 102A Lab Experiments” folder on Desktop was opened. The “Zero” button located on the force sensor was pressed without hanging the pendulum. This made the Force Sensor set to zero since there should be no force on it at this point. Then the mass of the pendulum was measured. The length of the pendulum was measured from the bottom of the Force Sensor’s hook to the center of mass of the pendulum bob. The length of the pendulum was recorded in the Data Table. The pendulum was hung on the hook of the force sensor. The pendulum bob was pulled to the side and was released for swinging. The recording was started by pressing the “RECORD” button on the bottom of the screen. The data of Force and linear speed vs.time was plotted. It was important to remember that the pendulum was hit by the photogate. Once the measurement was done, the maximum force, the tension T at the lowest point was deduced by using “Add a coordinate tool” located on top of the graph and the linear speed. Those data were recorded in the data table. This experiment was done for each of the aluminum pendulum and copper pendulum at three different starting points or heights.The masses were measured for each of the aluminum pendulum and copper pendulum and the diameters of each of the pendulums were measured using a digital vernier caliper. The tension T at the lowest point was deduced by using “Add a coordinate tool” located on top of the graph and the linear speed for for each of the aluminum pendulum and copper pendulum and the centripetal forces of each of those pendulums,along the string and perpendicular to the string were found.

Table 2. m (kg) Points T at θ= 0 ° (N) F c = T-mg (N) Linear Velocity (m/s) F c = mv 2 /r (N) % diff 1 0.3522 0.11602 1.7627 0.11010 5.1 0.0241 2 0.3816 0.14542 1.9563 0.13564 6.7 3 0.4697 0.23352 2.5181 0.22472 3.8 1 0.9393 0.25918 1.4965 0.22856 11.8 0.0694 2 1.1741 0.49398 2.1051 0.45227 8.4 3 1.3796 0.69948 2.4871 0.63130 9.7 For the aluminum pendulum- At Point 1:- T at θ= 0 ° is , 0.3522 N To find F c = T-mg, F c = T-mg F c = 0.3522 N-(0.0241kg x 9.8m/s 2 ) F c = 0.11602 N (5 significant figures) At Point 1:- Linear velocity (m/s) is, 1.7627 To find F c =mv 2 /r, F c =mv 2 /r F c = [0.0241kg x (1.7627 m/s) 2 ]/ 0.68m F c = 0.11010 N (5 significant figures) % diff. [(F c = T-mg) - (F c = mv 2 /r)/ (F c = T-mg) ] x 100% = [(0.11602 N-0.11010 N)/ 0.11602 N] x 100% = 5.1% (2 significant figures) At Point 2:- T at θ= 0 ° is , 0.3816 N To find F c = T-mg, F c = T-mg F c = 0.3816 N-(0.0241kg x 9.8m/s 2 ) F c = 0.14542 N (5 significant figures) At Point 2:- Linear velocity (m/s) is, 1.9563 To find F c =mv 2 /r, F c =mv 2 /r

F c = [0.0241kg x (1.9563 m/s) 2 ]/ 0.68m F c = 0.13564 N (5 significant figures) % diff. [(F c = T-mg) - (F c = mv 2 /r)/ (F c = T-mg) ] x 100% = [(0.14542 N-0.13564 N)/ 0.14542 N] x 100% = 6.7% (2 significant figures) At Point 3:- T at θ= 0 ° is , 0.4697 N To find F c = T-mg, F c = T-mg F c = 0.4697 N -(0.0241kg x 9.8m/s 2 ) F c = 0.23352 N (5 significant figures) At Point 3:- Linear velocity (m/s) is, 2.5181 To find F c =mv 2 /r, F c =mv 2 /r F c = [0.0241kg x (2.5181m/s) 2 ]/ 0.68m F c =0.22472 N (5 significant figures) % diff. [(F c = T-mg) - (F c = mv 2 /r)/ (F c = T-mg) ] x 100% = [(0.23352 N-0.22472 N)/ 0.23352 N] x 100% = 3.8% (2 significant figures) For the copper pendulum- At Point 1:- T at θ= 0 ° is , 0.9393 N To find F c = T-mg, F c = T-mg F c = 0.9393 N-(0.0694kg x 9.8m/s 2 ) F c = 0.25918 N (5 significant figures) At Point 1:- Linear velocity (m/s) is, 1.4965 To find F c =mv 2 /r, F c =mv 2 /r F c = [0.0694kg x (1.4965m/s) 2 ]/ 0.68m F c =0.22856 N (5 significant figures) % diff. [(F c = T-mg) - (F c = mv 2 /r)/ (F c = T-mg) ] x 100% = [(0.25918 N-0.22856 N)/ 0.25918 N] x 100%

4. ANALYSIS and DISCUSSION In the experiment, the maximum tension T at θ= 0 ° , at the lowest point, was determined for each of the three different heights for both the aluminum and copper pendulums and the centripetal force for both the aluminum and copper pendulums along the string was deduced by the equation, T-mg = ma r = F c (along the string) or F c = T-mg (N). Also, the centripetal force for both of the aluminum and copper pendulum at the lowest point of the swing was found by using the detected linear velocity (v) through the graph on the computer screen, rotational radius (r) and the pendulum mass (m) and the equation used was F c =ma c =mv 2 /r or F c = mv 2 /r (N). For the aluminum pendulum, at point 1, the experiment resulted in a measured value of 0.11602 N for F c = T-mg (N) and a predicted value of 0.11010 N for F c = mv 2 /r (N) with a % difference of 5.1. At point 2, the experiment resulted in a measured value of 0.14542 N for F c = T-mg (N) and a predicted value of 0.13564 N for F c = mv 2 /r (N) with a % difference of 6.7. At point 3, the experiment resulted in a measured value of 0.23352 N for F c = T-mg (N) and a predicted value of 0.22472 N for F c = mv 2 /r (N) with a % difference of 3.8. For the copper pendulum, at point 1, the experiment resulted in a measured value of 0.25918 N for F c = T-mg (N) and a predicted value of 0.22856 N for F c = mv 2 /r (N) with a % difference of 11.8. At point 2, the experiment resulted in a measured value of 0.49398 N for F c = T-mg (N) and a predicted value of 0.45227 N for F c = mv 2 /r (N) with a % difference of 8.4. At point 3, the experiment resulted in a measured value of 0.69948 N for F c = T-mg (N) and a predicted value of 0.63130 N for F c = mv 2 /r (N) with a % difference of 9.7. The errors might be due to the air resistance and friction not being considered and the slight variations in the temperature of the surroundings which might have impacted the elasticity of the string during the experiment. The results are reasonable with error analysis, as slight errors or differences were found when comparing the values of F c = T-mg (N) which is measured and F c = mv 2 /r (N) which is predicted for three different heights in case of aluminum and copper pendulums. The results met the objectives of the lab. The lab's objective was to verify the expression of the centripetal force. To support the discussion, in the experiment, at three different heights, we were required to find the centripetal force for both of the aluminum and copper pendulums along the string through the tension at the lowest point, T at θ= 0 ° and masses and gravities of each of the pendulums. This was verified by using the equation, T-mg = ma r = F c (along the string) or F c = T-mg (N). We were also required to find the centripetal force for both of the aluminum and copper pendulums at the lowest point, T at θ= 0 ° of the swing was found by using the detected linear velocity (v) through the graph on the computer screen, rotational radius (r) and the masses of the pendulums (m). This was verified by using the equation F c =ma c =mv 2 /r or F c = mv 2 /r (N). Questions in the lab manual 1. The amplitude of tension becomes smaller with time. What is the reason for that? Over time, air resistance and other forces like gravity act on the pendulum, eventually causing the amplitude of the tension to decrease with time.

2. How different is your measured value of centripetal force (equation (3)) from the calculated value of centripetal force using equation (5)? The measured value of centripetal force (equation (3)) is slightly different from the calculated (predicted) value of centripetal force using equation (5). For instance, for the aluminum pendulum, at point 1, the experiment resulted in a measured value of 0.11602 N for F c = T-mg (N) and a calculated (predicted) value of 0.11010 N for F c = mv 2 /r (N) with a % difference of 5.1. For the copper pendulum, at point 2, the experiment resulted in a measured value of 0.49398 N for F c = T-mg (N) and a calculated (predicted) value of 0.45227 N for F c = mv 2 /r (N) with a % difference of 8.4. 3. How would the plot of the Force look if the experiment were made in a vacuum? Due to the absence of air resistance, the force's plot would appear to be more consistent and uniform with each swing in a vacuum. 4. Do your results verify the expression of centripetal force? Yes, our results verify the expression of centripetal force. 5. CONCLUSIONS From this experiment, I gained knowledge about how the centripetal force along the string at three different heights is deduced from the tension, T at the lowest point, θ= 0 ° and the masses of each of the aluminum and copper pendulum through the equation, T-mg = ma r = F c (along the string) or F c = T-mg (N). I also gained an understanding how the centripetal force for both of the aluminum and copper pendulums at the lowest point of the swing is found at three different heights by using the detected linear velocity (v) through the graph on the computer screen, rotational radius (r) and the masses of the pendulums (m), through the equation, F c =ma c =mv 2 /r or F c = mv 2 /r (N). I also learned that L=length of the pendulum from hook to the center of mass of each of the pendulum is equal to radius (r) in the equation, F c =ma c =mv 2 /r or F c = mv 2 /r (N) about which I got confused earlier. One question was raised in my mind while experimenting. This includes what would happen to the centripetal force if each of the aluminum and copper pendulums did not have the same length from the hook to the center of mass of the pendulums individually.