Lab Report #3

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Northeastern University *

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Physics

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Apr 3, 2024

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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.

Investigation 1 Throughout this experiment, we utilized an instrument called a Sargent-Welch centripetal force apparatus. On one side of the base of the apparatus stands a post, which supports the arm from which the bob is suspended from by a string, and counterweight that makes sure the arm is completely parallel with the base and perpendicular to the post. At the middle height of the post is a loop from which a spring or elastic band can be attached to the bob. In the middle of the base is a pointer that, when the bob hangs freely without any attached springs or rubber bands, lines up with the pointer at the bottom of the bob. At the end opposite the post is a pulley wheel from which a bucket can hand freely over while attached to the bob. See Fig 1 for an image of the setup. Figure 1: Labeled image of a Sargent-Welch centripetal force apparatus at rest. This apparatus is arranged to measure the force exerted by the elastic. The rubber band (or spring) has been stretched by adding washers to the bucket, increasing its mass M until the bob is aligned with the pointer. We began by detaching the bob and measuring its mass using a scale and taking the instrumental error, which we assumed was half the smallest increment on the scale. These measurements are recorded in Table 1. The bob was then reattached, and we began leveling the base to eliminate systematic and random error, adjusting the leg height at each corner until the arm stopped moving and remained stationary. The pointer on the base was moved to a center point between its two extremes and secured into place by tightening the screws. The arm was adjusted until the tip of the bob aligned with the tip of the pointer. We made sure that the bob hung freely a few millimeters above the tip of the pointer and secured the arm in place by tightening the screw into the post. The counterweight was adjusted on the opposite end of the arm to balance it so that it didn’t droop to one side and screwed into place as well.

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

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