Lab circular motion - lab write-up PDF

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1 Circular Motion SPH 4U Thursday, October 12, 2017 Introduction Objects are always under some type of force, be it the force of gravity, a normal force, a tension force, or an applied force. To explain movement and how forces influence objects in different ways, Newton’s three Laws of Motion can be studied. Newton’s first Law of Motion explains the concept of inertia, which simply states that an object at rest will remain at rest while an object in uniform motion will stay in motion unless acted on by another outside force. Newton’s second Law of Motion explains that when an object experiences outside forces that do not cancel one another out, the object has an acceleration. The equation, F=ma , can be used to find the object's acceleration based off of its’ net force and mass. Finally, Newton’s third Law of Motion explains action and reaction forces. Every action force has an equal reaction force on another object in the opposite direction. By following these three laws, many different types of motion can be explained, including circular motion. An object in uniform circular motion is always in acceleration as its direction is constantly changing despite the speed remaining the same. By looking at Newton’s first law, the object wants to follow a straight path, but is being pulled in towards the centre of the circle, causing it to turn direction. This force that is pulling the object in towards the centre of the circle is called the centripetal force, Fc. Due to this relation, the instantaneous velocity of the object is always perpendicular to the centripetal force, or direction of acceleration, as seen in Fig.1. Due to this constant change of velocity at different points on the circular path, the equation for centripetal acceleration can 2 be found: ac =v  /r. By consulting Newton’s second law, F=ma , the 

appropriate version for circular motion would be F c =ma  c , which in turn 2  /r. Furthermore, by studying the motion of the object becomes Fc =mv 

around the circumference of its circle of motion, the equation v=2 𝛑r/T  can be found which can   2 2 be substituted into both centripetal acceleration and force: a c =4 𝛑2r/T  and Fc =4 𝛑2 rm/T  . The  

period of the turn can also be substituted with frequency as T=1/f . All these equations can help

2 find different aspects of the uniform circular motion, such as the radius, frequency, mass of the object, centripetal force and acceleration. When looking at uniform circular motion, it can be studied through an inertial frame of reference and a noninertial frame of reference. An inertial frame of reference is from the Earth’s perspective, where inertia does exist and causes the object to want to travel in a straight line as discussed before. However, a noninertial frame of reference is in the object’s perspective. The object is not moving in this frame, yet it feels a constant force in one direction (the centripetal force in the inertial frame). To explain this imbalance of force and lack of movement, a fictitious force was created, as inertia cannot exist in this reference. This fictitious force, called a centrifugal force, acts opposite to the centripetal force, balancing it out and resulting in no movement. By understanding the different features to uniform circular motion and using the derived formulas, the relationships and influences can be studied of one feature on another. The effect of different sizes of radius, different magnitudes of acceleration, and different ranges of mass can be compared to see how the object’s motion changes in each instance. By understanding how one aspect of motion can affect another aspect, uniform circular motion can be manipulated to benefit many fields including science and sports. Purpose To find how the magnitude of the force, the radius of a circular path, and an object’s mass affects the frequency of the revolution of an object in uniform circular motion. Hypothesis I believe that with a greater force, the frequency of the revolving object in uniform circular motion will increase. If the centripetal force of tension increases, the acceleration of the object should increase as well, thus speeding up the revolving object. Now, the object should be able to make more rounds in a second than before, resulting in a higher frequency. This can also   Mg/4 𝛑2rm , where “Mg” stands for the magnitude of the be seen in the modified equation f  2=

force. Clearly, if this value increases, frequency increases as well. For changes in the radius of the circular path, I believe that a larger radius will result in a lower frequency. With a higher radius, the circular path that the object must travel becomes

3 larger. With the same speed, the revolving object now completes less cycles in a second than   Mg/4𝛑 2rm before, decreasing frequency. This can also be seen in the equation f  2 = , where “r ”

stands for the radius of the circular path. Even mathematically, if this value increases, frequency decreases. Lastly, I believe that an increase in the revolving object’s mass will make the frequency lower. A heavier object requires more force than a lighter object to keep it revolving. However, in this experiment, the magnitude of the force stays fixed. Thus, the acceleration must decrease to fit Newton’s second Law of Motion. Now, the revolving object has a lower speed and completes less cycles in one second, decreasing frequency. This can also be seen in the equation   Mg/4𝛑2 rm f 2 = , where “m ” stands for the object’s mass. Similar to the radius, if the mass

increases, the frequency must decrease. Equipment/Materials ● Safety Goggles

● 50g, 100g, 200g weights

● Electronic Scale

● Metre stick

● 3 rubber stoppers with centre holes

● 1.5 m fishing line

● Hollow Tube

● Masking Tape

Observations Table 1. Dependence of frequency of rotations on hanging mass M (g)

Δt1 (s)

Δt2 (s)

Δt3 (s)

Δtavg (s)

f 2(Hz2) experiment

f 2(Hz2) calculated

50

8.13

8.93

8.92

8.66

1.33

1.22

100

6.55

6.86

6.90

6.77

2.18

2.43

200

4.99

4.83

4.85

4.89

4.18

4.87

Table 2. Dependence of frequency of rotations on radius r (cm)

Δt1 (s)

Δt2 (s)

Δt3 (s)

Δtavg (s)

f 2(Hz2) experiment

f 2(Hz2) calculated

45

6.02

6.99

6.22

6.41

2.43

2.70

4

53

7.23

6.70

6.49

6.81

2.16

2.29

60

7.35

7.70

7.45

7.50

1.78

2.03

Table 3. Dependence of frequency of rotations on rotating mass m (g)

Δt1 (s)

Δt2 (s)

Δt3 (s)

Δtavg (s)

f 2(Hz2) experiment

f 2(Hz2) calculated

9.11

4.95

4.52

4.83

4.77

4.40

5.15

14.19

5.08

5.05

5.77

5.30

3.56

3.30

20.43

7.23

6.70

6.49

6.81

2.16

2.29

Analysis Using the information above in the charts, three separate graphs can be constructed: Frequency vs. Tension Force The following table was used to derive the tension force from the hanging mass: M (kg)

Tension Force (N) M x 9.81 m/s2

0.00500

0.0491

0.100

0.981

0.200

1.96

5

This graph shows the relationship between the frequency of the revolving object and the magnitude of the force. The red data is the experimental square of the frequency for each magnitude of force, while the blue data is the calculated square of the frequency using the   Mg/4 𝛑 2rm . Using Google Sheets, the line of best fit was found for each set of data. equation f  2=

As seen in the graph, the lines of best fit are linear and the data points seem to be in-line, showing no discrepancies in observations. These lines of best fit show that as the tension force increases, so does the frequency of the revolving object. The experimental data and calculated data yield slightly different slopes, showing how human experiment can differ from calculations.    can be consulted. “f 2 ” is already Mg/4 𝛑2 rm To find the origin of these slopes, the equation f  2=

graphed as the y-axis. The tension force, also calculated as “Mg ” , is the x-axis. By rearranging    . Thus, the slope must be 1/4 𝛑2rm , which the equation into y=mx  form, we get f 2 =(1/4 𝛑2 rm)Mg

should equal 2.09 or 2.48. This value can be checked for as all the variables in the slope are fixed. The radius for this experiment was 0.5m, and the revolving mass was 0.02043kg:  m=1/4 𝛑2rm  m=1/4𝛑2 (0.5)(0.02043)

m=2.48   does equal the calculated data’s slope of 2.48. As the experimental data is The slope of 1/4𝛑 2rm

quite similar, it is safe to assume that the slope of that graph must be the same value as well, just

6    is shown to be reliable and Mg/4𝛑 2rm offset due to human errors. Additionally, the equation f  2=

correct as it accurately shows the relationship between the frequency and the force of tension. Frequency vs. Radius The following table was used to derive one over radius from the radius: Radius (m)

1/Radius (1/r)

0.45

2.2

0.53

1.9

0.60

1.7

This graph shows the relationship between the frequency of the revolving object and the radius of the circle. The red data is the experimental square of the frequency while the blue data is the calculated square of the frequency. The line of best fit was found for each set of data and the data points are in the same range, showing minimal discrepancies in observations. These lines of best fit show that as the inverse of the radius increases, so does the frequency of the revolving object. Thus, in reality, when the actual radius is increasing, the frequency is actually decreasing. Furthermore, the two slopes of the lines are very similar, showing that the experiment resulted in correct data. The origin of these slopes can be found by consulting the   Mg/4 𝛑 2rm equation f  2=  in the same manner as before. By following the same steps, the slope

7  , which should be close to the values of 1.1 or 1.22. This value can be should equal Mg/4𝛑 2m

checked for as all the variables in the slope are fixed. The hanging mass for this experiment was 0.100kg, and the revolving mass was 0.02043kg:  m=Mg/4𝛑2m  m=(0.1)(9.81)/4𝛑 2(0.02043)

m=1.22 2

The slope of Mg/4𝛑 m  does equal the calculated data’s slope of 1.22, which is very similar to the   Mg/4𝛑 2rm experimental slope as well. Additionally, the equation f  2=  is shown to be reliable and

correct as it accurately shows the relationship between the frequency and the radius of the circle. Frequency vs. Object’s Mass The following table was used to derive the inverse of the object’s mass: Object’s Mass (kg)

1/Mass (1/m)

0.00911

110

0.01419

70.47

0.02043

48.95

This graph shows the relationship between the frequency of the revolving object and the object’s mass. The red data is the experimental square of the frequency while the blue data is the

8 calculated square of the frequency. The lines of best fit were found for the data, and the data points are in the same range, showing no outliers in observations. These lines of best fit show that as the inverse of the object’s mass increases (meaning the actual object’s mass decreases), the frequency of the revolving object increases. The two slopes of the separate lines are very similar, a difference only caused by experimental errors. To find the origin of these rates of    can be consulted. By following the same steps as before, the Mg/4𝛑2 rm change, the equation f  2=

slope came to be Mg/4𝛑 2r , which should be close to 0.0469 or 0.0414. This value can be checked for as all the variables in the slope are fixed. The hanging mass for this experiment was 0.100kg, and the radius was 0.53m: m=Mg/4𝛑2 r  m=(0.1)(9.81)/4𝛑 2 (0.53)

m=0.0469 The slope of Mg/4𝛑 2r  does equal the calculated data’s slope of 0.0469, which is very similar to    is shown to be reliable Mg/4𝛑2 rm the experimental slope as well. Additionally, the equation f  2=

and correct as it accurately shows the relationship between the frequency and the revolving object’s mass. All the graphs have the point (0,0) in their data to improve their accuracy and reliability. This point occurs because if there is no tension force, no radius, or no revolving object, there is nothing that is going in uniform circular motion. Thus, the frequency must be zero, making the square of the frequency zero as well. Hence, the point (0,0) is included in all of the graphs as it adds a physical aspect of realism to the data collected. Results When the tension force increases due to the hanging mass, the frequency increases as  well, just as was reasoned in the hypothesis. The rate of change for this relation is 1/4 𝛑2rm  when

the radius and revolving mass are fixed. This means that for every Newton increase in the  . tension force (about 0.1kg), the square of the frequency will change by 1/4 𝛑2 rm

As was hypothesized, when the radius decreases, the frequency of the object increases or  , assuming that the revolving mass and vice versa. The rate of change for this relation is Mg/4 𝛑 2 m

9 hanging mass are fixed. This means that for every decrease in radius by a factor of 1/metre, the  square of the frequency increases by Mg/4 𝛑 2m .

Lastly, when the mass of the rotating object decreases, the frequency of the object increases or vice versa. The rate of change for this relation is Mg/4 𝛑 2r  when the hanging mass and radius are fixed. This means that for every decrease in the revolving mass by a factor of 1/kg, the square of the frequency increases by Mg/4 𝛑2r . This result makes the hypothesis completely correct as the change in frequency was exactly what was reasoned. Discussion The frequency was found to increase when the magnitude of the tension increased, when the radius decreased, or when the mass of the object in motion decreased. The rate of change for each factor was slightly different between the experimental and the calculated. Overall, the percent deviation of the experimental slope from the calculated slope only ranged from about 10-16% depending on the trail. Considering friction, measuring methods, and other errors, this percent deviation is very small. These errors can be easily explained when reflecting back on the procedure and materials. A major factor affecting the collected data would be how the time was kept. As the experiment relied heavily on human reaction to count ten revolutions and instantly hit the stopwatch, there is a large area for human error. This method of using a stopwatch can never be accurate as it will always result in time recordings that are slightly off. The time recording was the main factor needed to get the experimental frequency for each trial, making it a huge influencer on how the frequency changed. Another source of discrepancies is the friction between the fishing line and the tube on the side where the object revolved. No matter how smooth the two surfaces are, there is bound to be a slight amount of friction that will change the calculated results. Furthermore, the issue of maintaining the right radius greatly affected the results. As the tape was put a few centimeters below the tube, it was difficult to get the same radius in all of the trials. The tape would also start sliding on the fishing line, changing the measurement all together. This inconsistency in radius affected the acceleration and speed of the object, hence changing the collected data. Likewise, the radius was also affected by the angle of the fishing line. When the rubber stopper was swung around, the circular path was almost never

10 completely horizontal as it is difficult to get it to that perfect 90°. Due to this inability to get the string and tension force to be horizontal, both the range and the magnitude of centripetal force were affected. As seen in Fig. 2, both the centripetal force and actual radius where components of the full amount that was documented. Due to this difference in actual values and documented values, the experimental and calculated results became even more different. Furthermore, as the frequency of the object increased, it became more difficult to keep track of the revolutions and time. With a higher frequency comes a shorter radius or faster speed of the rubber stopper. With either change, it is harder for the human eye to individually see each revolution and the exact moment when ten cycles have been completed. Due to this lack of skill and concentration, the time readings are less reliable and have been proven to be very susceptible to human error. Multiple steps were taken in an attempt to minimize the effects of the errors present in the procedure and many future improvements can be made as well. Since recording the time with a stopwatch held many chances for human error, the procedure required three separate recordings. This way, the final time used in the calculations was an average of three trials, making it a little more accurate to the actual time. Another way that the possible errors in the lab were minimized was through practicing how to spin the rubber stopper. The tension in the string needed to be horizontal for the best results. So, the group member in charge of swinging the stopper did a few trials before the actual lab to get the string as horizontal as possible. Due to these extra steps taken, the final result was a bit more accurate than it would have been otherwise. An improvement for future use is to consider techniques to achieve the most accurate recordings possible. One example is to watch the wrist of the person swinging the rubber stopper, not the object. In higher frequencies, it becomes difficult to see when the object completes ten

11 revolutions as it is spinning very fast. In these instances, it will be easier to look at the person’s wrist and watch for the subtle movements that it makes to pull on the object. This way, a more accurate reading of time for ten revolutions can be taken. Furthermore, in place of the masking tape that kept moving on the string, it would be better to use a marker to make a mark instead. This way, a more exact radius measurement can be taken and the rubber stopper can be swung with no worry about the tape moving or getting into the tube. The only difficulty with this solution is that multiple radius markings cannot be made on the string as it can get confusing. Nonetheless, these are all valuable next steps that can be considered to get more accurate results in future activities. Conclusion In conclusion, the purpose of this lab was to find how the magnitude of the force, the radius, and an object’s mass affects the frequency of the object revolving in uniform circular motion. This purpose was achieved as the frequency was found to increase when the force increased, the radius decreased, and the ob...