Lab Report #3

Report for Experiment #3 Uniform Circular Motion Serena Le Lab Partner: Nathan Lim TA: Jinzheng Li 06 February 2024 Abstract This lab to tested and verified that the equation for centripetal acceleration, as given by a c = v 2 r , does indeed satisfy the equation for Newton’s second law. Using a Sargent-Welch centripetal force apparatus, we observed a hanging bob in centripetal motion and varied the magnitude of centripetal force exerted on the bob through tension by varying the strengths of the springs used to supply the force. In doing so, we observed a linear relationship between the centripetal tension force and the tangential velocity squared, which is equivalent to the mass of the bob divided by the radius of the circle. We obtained an experimental value of this relationship to be 2.95 kg / m± 0.018 kg / m , which falls withing the range of uncertainty of theoretical value of this relationship, given by 2.84 kg / m± 0.009 kg / m .

Introduction In this experiment, we explored circular motion and centripetal acceleration. When an object is moving in a circle, it is constantly changing direction, therefore constantly accelerating. In the case where the object is moving at the same speed tangential to the circle, the scalar component of the velocity stays the same, but, again, the change in direction indicates an acceleration. This acceleration, called centripetal acceleration, is different from linear acceleration in that the object is not speeding up or slowing down; only the direction of motion changes. This acceleration can be expressed as: a c = v 2 r , (I1) where v is the velocity tangent to the circle and r is the radius of the circle. Applying Newtons second law, we get: ΣF = ma c = m v 2 r . (I2) This force is called the centripetal force and is equivalent to the net force of an object in motion. We also used a stopwatch to calculate the period of motion, or the time it takes to complete one full circular revolution. The equation for period is given by: Ͳ = time ( s ) ¿ of revolutions (I3) Using a stopwatch, we encountered uncertainty mainly due to human reaction time. To calculate the error, we used standard deviation σ , and standard error in the mean, δ t , as given by: σ = √ ( t ¿¿ 1 − t ) 2 +( t ¿¿ 2 − t ) 2 + … +( t ¿¿ n − t ) 2 n ¿¿¿ (I4) δ t = σ √ n , (I5) where t is the average and n is the number of trials. For errors where the uncertainty could be quantified, we used the error propagation formula: δz = z∙ √ ( δx x ) 2 + ( δy y ) 2 . (I6) In this experiment, we use a Sargent-Welch centripetal force apparatus to measure the force exerted on a hanging bob. This force, the centripetal force, is supplied by an elastic via a tension force, which we measure using a bucket with additional masses. Throughout this experiment, we vary the quantities in Eq. (I2), such as force and velocity, to test whether the equation for centripetal force does satisfy Newton’s second law.

We then measured the distance r , from the tip of the pointer to the center of the rotating post, using a ruler. This distance is the radius of the circle and remains constant throughout this investigation. We then calculated the circumference using the equation: C = 2 πr , (1) and its uncertainty using the equation: δC = | 2 π | δr , (2) These measurements and their uncertainties are recorded in Table 1. Table 1: Measured mass of bob, radius, and calculated circumference in SI units with respective errors. mass of bob (kg) δm (kg) radius (m) δr (m) circumference (m) δC 0.4584 0.00005 0.1615 0.0005 1.014734427 0.00314159 3 We then began data collection. We started by attaching one end of a rubber band to the bob with a paper clip and securing the other end to the post. This pulled the tip of the bob away from the pointer and in towards the post. The force required to pull the bob back to the pointer while the apparatus is stationary is equivalent to the centripetal force of the bob in circular motion. To quantify this force, a bucket was attached to the bob with a string and paper clips and hung over the pulley wheel at the edge of the base. Small masses were then added until the tip of the bob realigned with the pointer. Once realigned, the bucket, string, and paperclip were detached from the bob and placed on a scale to find the mass, and the data was recorded in Excel, as shown in Table 2. To find the error in mass, δm , we reattached the bucket and added more masses until the tip of the bob moved about 5 millimeters past the pointer. The bucket was then reattached and massed on the scale. The difference between the first and second bucket masses was denoted as δm . Using Newton’s second law, we derived the tension force required to realign the bob to be equal to the gravitational force of the bucket and string, given by: Σ F y = T − F g = m a y = 0 T = F g = mg. (3) (4) We calculated the force using Eq (3) and its error using Eq (2) in Excel. Returning to the setup, the bucket was removed from the bob so that the bob hung from the arm and was pulled in towards the post by the rubber band. One person then began to rotate the apparatus using the grip at the bottom of the post until the tip of bob began to realign with the pointer while in motion. Once the two were realigned, the person rotating the apparatus let go so that arm swung freely, and the other person began to record the time using a stopwatch. The time began when the bob crossed the pointer and stopped after two full revolutions. This timing process was repeated a total of three times. We repeated this process of supplying tension, realign the bob with a bucket and mass, recording the time of revolution, and making all calculations 5 more times for a total of 6 trials. Each time, the tension force was adjusted. In Trial 2, two rubber bands were used, and in Trials 3-6, springs of varying strengths were used to supply tension. Also, in Trials 3-6, we recorded

the time of 10 revolutions instead of 2 for a more accurate calculation of the period. All measured and calculated data is shown in Table 2 below. Table 2: Measured quantities of mass, number of revolutions, and time and calculated quantities of force and error. m(g) δm (g) m (kg) δm (kg) Force (N) δF (N) # of rev t1 (s) t2 (s) t3 (s) 92.2 17 0.092 2 0.017 0.90448 2 0.16677 2 3.1 3.45 3.38 240.1 33 0.240 1 0.033 2.35538 1 0.32373 2 2.27 2.2 2.21 64.5 22.9 0.0645 0.0229 0.632745 0.224649 2 4.33 4.51 4.28 278.6 35.3 0.2786 0.0353 2.733066 0.346293 10 9.74 9.77 9.54 404.2 56.1 0.4042 0.0561 3.965202 0.550341 10 8.43 8.67 8.4 599 29.7 0.599 0.0297 5.87619 0.291357 10 7.13 7.2 7.25 We then began data analysis for each trial. To find the error in time δt , we used standard deviation and standard error in the mean as given by Eq (I4) and Eq (I5). We then found the average time t , using the average function in Excel, which was then used to find measurement to find the period, Ͳ , using Eq (I3). To find the error in period, δ Ͳ , we used the equation: δ Ͳ = δ t ¿ of revolutions (5) This information was then used to find the tangential velocity of the bob in circular motion. Tangential velocity, v , was calculated using the equation: v = C Ͳ , (6) where C is the circumference of the circle as calculated at the beginning of the investigation. To find error in velocity, δv , we used the error propagation formula from Eq (I6). However, as stated in Eq (I2), centripetal force is relational to v 2 , not v , so we calculated v 2 , and its error, δ ( v ¿¿ 2 ) ¿ , using the equation: δ ( v 2 )= 2 v ∙δv . (7) All calculated quantities of period, velocity, and associated errors are shown in Table 3 below. Table 3: All calculated quantities of average time, period, velocity, velocity squared, and respective errors . t avg (s) δt (s) (s) Ͳ δ (s) Ͳ v (m/s) δv (m/s) v 2 (m/s) 2 δ(v 2 ) (m/s) 2 3.3100 0.1069 1.6550 0.0535 0.6131 0.0199 0.3759 0.0244 2.2267 0.0219 1.1133 0.0109 0.9114 0.0094 0.8307 0.0171 4.3733 0.0698 2.1867 0.0349 0.4641 0.0075 0.2153 0.0070 9.6833 0.0722 0.9683 0.0072 1.0479 0.0085 1.0981 0.0177 8.5000 0.0854 0.8500 0.0085 1.1938 0.0126 1.4252 0.0300 7.1933 0.0348 0.7193 0.0035 1.4107 0.0081 1.9900 0.0229 This data was then used to create one plot of F vs. v, as shown in Figure 2, and one plot of F vs. v 2 , as shown in Figure 3.

squared v 2 resulting in a linear trend. The plot is fit with a linear line of best fit. The slope of this can be given by F c v 2 = T v 2 = m bucket g v 2 . Using Eq (I2), we can solve for this slope m bucket g v 2 : Σ F c = m bob v 2 r = m bucket g m bucket g v 2 = m bob r . (8) (9) This gives a theoretical value of the slope to be m bob r = 0.4584 0.1615 = 2.84 kg / m . The error in this theoretical value was calculated using the error propagation formula Eq (I6) to get an error of 0.009 kg / m . This gives a theoretical value of 2.84 kg / m± 0.009 kg / m . Using the IPL Straight Line Fit Calculator to find the observed slope and error of the line of best fit, we got 2.95 kg / m± 0.018 kg / m . This falls within the error range of uncertainty given by the theoretical value and error. Conclusion In conclusion, this lab served to test and verify that the equation for centripetal acceleration as given in Eq (I1) does indeed satisfy the equation for Newton’s second law as given in Eq (I2). In using a Sargent-Welch centripetal force apparatus, we were able to observe a hanging bob in centripetal motion and vary the magnitude of centripetal force exerted on the bob by the tension. By varying the strengths of the springs used to supply the force, we were able to observe a linear relationship between the centripetal tension force and the tangential velocity squared, which is equivalent to the mass of the bob divided by the radius of the circle. We obtained an experimental value of this relationship to be 2.95 kg / m± 0.018 kg / m , which falls withing the range of uncertainty of theoretical value of this relationship, given by 2.84 kg / m± 0.009 kg / m . While this does mean our results satisfy our objective, we still encountered various sources of error. Air resistance, for example, while small, was not completely negligible in our setup. Running our experiment in a vacuum sealed environment or an environment with less air would be a solution. Much of the error with time came from random errors due to human reaction. This could be lessened through statistical methods such as recording the time of more revolutions to get a more accurate calculation of period or by using a camera software to record the time. Additionally, potential friction in the pole could affect the velocity of the bob and slow down the motion. Adding lubricant could reduce this error. In the end, we took various steps and used various error propagation methods to reduce and account for the various sources of instrumental, random, and systematic errors. This resulted in an observed value of the relationship between the centripetal force and velocity squared that was within the range of uncertainty of the theoretical value.

Questions 1. What are the units of the slope of your F vs. v 2 plot? The units of slope are N ( m s ) 2 = kg m s 2 m 2 s 2 = kg m . 2. The lines that you draw through the data on your F vs. v 2 plot should pass through the origin. Give a simple reason for this. The line passing through the origin means the centripetal force is 0 when the tangential velocity is 0. In short, there is no centripetal force when the object is not moving. 3. A marble rolls along the inside of a semicircular track made of copper tubing that is lying on a level table that is parallel to the floor (Fig. 5.6). As the marble leaves the track at point 0, does it follow path A, path B, or path C? Explain your answer in terms of the concepts developed in this experiment. Figure 5.6 The marble would travel along path B. This is because once the track ends, there is no normal force from the track exerted onto the marble to supply the centripetal force. Assuming there is no wind and that the floor is not at an angle, there is no external force acting in or against the direction of the marble motion, meaning the marble would continue moving in a straight line at the tangential velocity it was traveling before it left the track.