Secondly, it is difficult for us to decide the moment that the board is at rest. As there are many factors that can cause the board to move such as the movement of the air. If we mark the intersection points on the paper when the board still moves, the inaccurate points will be marked on the paper. Then the inaccurate lines and intercept point are drawn on the paper. As a result, an incorrect center of gravity of the irregularly shaped board is found. To avoid such error, we should close all the windows and turn off the all fans and air conditioners to minimize the effect of movement of air on the boards. Then we have to wait for the boards to become at rest. If the board continues to move, we choose the moment that the motion of the board is minimum to mark the intersection points.
Besides, the surface of the boards and the edges of the meter rule are not smooth enough. If we still use such apparatus, we cannot draw the correct straight that can connect the two intersection points. As a result, incorrect intercept point and hence the center of gravity are drawn. To get rid of these errors, we should stick the paper to the board carefully with the cellulose tape. We should use the cellulose tape to stick edges of the board to the paper to ensure the smoothness of the paper and the boards. For the meter rule, we choose a better ruler with smooth edges to replace the old one in order to draw the accurate and precise lines on the paper. To ensure that the surfaces of the boards are smooth surface, we can use sand paper to make the surface of the board smoother.
On top of that, the holes A, B and C on the paper are very small, it is very difficult for us to draw line from the intersection points that can pass through the center of the holes. As the lines drawn cannot pass through center of the holes, we will draw an inaccurate intercept point and hence center of gravity. To do away such error, we should make the hole bigger by using the pin. As the holes become bigger, it is easier for us to find out the center. However, we should not make the hole become too large, otherwise, inaccurate lines that connect the intersection points are drawn.
In this experiment, there are some assumptions we had made. Firstly, we assume that there is no air movement during the experiment. If there is air movement, it will cause the board to move. It results in the incorrect position marked of intersection points on the paper and hence incorrect center of gravity marked on the paper. Secondly, we assume that the mass of the board is evenly distributed over the whole board. If the mass is not evenly distributed, it will affect the actual position of the center of the gravity. At last, we assume that there are uniform gravitation fields and the pin with the cork is held horizontally.
The experimental errors can be divided into two errors, systematic errors and random errors. The smoothness of the surfaces of the board, paper and the edges of the meter rule are belonged to the systematic errors. The motion of the boards, the size of the holes and the time that the board is at rest are the random errors. An experiment with small systematic and random errors is more precise.
Questions
2. NO, the center of the gravity is not necessary insides the body for example, donuts
3. The experimental results of the center of gravity of the irregular shapes are close to that of the theoretical values. It proves that the intercept point method can find out the center of gravity. However, for the L-shaped board, the center of gravity of it should be outside the body theoretically. But in the experiment, it is inside the board body. I think that the reason is the size and the shape of the board. The L-shaped board is much wider than normal one:
Normal one experiment
As a result, the center of gravity of the L-shaped board used in the experiment is inside the board body.
Conclusion
By finding out the intercept point of the lines that across the board between the intersection points, we can find out the center of the gravity of the boards.
The centers of the gravity of the three irregularly shaped boards are shown in the result section. In addition, we further find out the center of gravity of regularly shaped and compound body. Then we can compare the experimental results between them.
References
Wikipedia (center of gravity)
http://en.wikipedia.org/wiki/center_of_gravity
Experiment 2B: Measurement of the gravitational acceleration (g) using a simple motion
Objectives:
To determine the local acceleration of free fall g using a simple pendulum
Experimental Design
Apparatus:
Experimental set-up:
Description of design:
In this experiment, we will measure the local gravitational acceleration due to the earth. In order to investigate the objective of the experiment, we should set up all the apparatus as above diagram. The optical pin with cork is fixed at the stand by using the plasticine as the reference point and the equilibrium point. We measure the time taken for the pendulum to complete on rotation by using the stopwatch. From the period and length of the thread, we can calculate the gravitational acceleration. In addition, we can find out g from the slope of the graph to be plotted.
Theory:
To determine the gravitation acceleration due to the earth in the laboratory, we use the simple pendulum set-up to find it out.
The pendulum used performs simple harmonic motion(SHM) which the acceleration is directly proportional to the displacement but always in opposite direction. The forces acting on the bob are the tension in the thread, T (radially inward) and the weight of the bob, mg (vertically downward). In this system, the pendulum also performs circular motion, as the tension does not involve in the speed change, the tension provides the centripetal force. The net force of the system is the tangential component of the weight of the bob.
F = ma, F = -mg sinθ → ma = -mg sinθ → a = -g sinθ
In order to investigate g, the angle formed should be smaller than 10o , sinθis almost equal to θ and then displacement(x) is equal to the length *θ(l *θ). As a result,
a = -g sinθ= -gθ = -(g/l)x = -ω2x
Displacement (x) is the distance of the bob from the equilibrium of the oscillation. From the expression, the acceleration is directly proportional to the displacement but always in opposite direction. Therefore, it is SHM if the angle is smaller than 10.o
For simple pendulum, the period of the motion can be expressed as:
ω2 = g/l → T = 2π/ω = ------(1)
From the expression, we can know that the period of the simple pendulum depends on the length of the thread and gravitation acceleration only if the angle is smaller than 10.o. The longer is the thread or the smaller is the gravitation acceleration, the longer is the period. Therefore, in this experiment, we vary the length of the thread and the period of SHM to investigate g by the above equations.
In order to investigate gravitation acceleration, we can reach objective by mathematical method and graphical method.
For mathematical method, we can make use of equation (1) to calculate the gravitational acceleration and then calculate the average g of the seven sets of data.
For graphical method,
Therefore, we should plot a graph with T2 against l and find out the slope of the graph. Hence we can find out the gravitation acceleration.
Procedures
- The clamp was fixed on the stand.
- The optical pin with cork was fixed on the middle point of the stand by using plasticine.
- The stand is fixed on the table firmly by using the G-clamp
- The length of the plumb line was measured and recorded by the meter rule.
- The plumb line was fixed between the two wooden blocks and then the blocks were fixed on the clamp.
- The pendulum was then moved a few cm to one side and then was released.
- The time taken for twenty periods was recorded by using stopwatch.
- Step 7 was repeated for two more times.
- The average value was calculated and recorded.
- Step 4 to 9 were repeated except varying the length of the plumb line.
- All the results were recorded.
Precautions
To ensure the accurate and precise data is obtained, we should be aware of the following precautions:
In the experiment, we have to use the optical pin with cork to be the reference and equilibrium point to count the period of the SHM of the bob. The pin is very sharp; it may hurt us. To ensure safety of students, the optical pin should be always pointed away from the students to avoid any accident.
Secondly, in the experiment, the bob is under SHM, it will swing side by side with large amplitude. It may hit the students who are near to the experimental set-up. In addition, the bob may also the objects which are placed near the edges of the bench and then cause accidents. To prevent accident occurs, all students should keep away from the experimental set-up and the set-up should be well cleared from the edge of the bench.
To improve the accuracy of the results, we should ensure that the bob should be moved in the same plane. Otherwise, the bob will only perform circular motion and not SHM. Then the equations used to find the period and the gravitation acceleration cannot be used at this movement. Hence we use the wrong data to calculate the period and gravitation acceleration, we will get the incorrect results from the equations.
Moreover, we should ensure the angle to the vertical is smaller than 10o . Otherwise, the displacement of the bon is not equal to the sinθ and then the bob is not under SHM. The equations used all cannot be applicable in the calculation process. If we still use wrong data in the calculation, incorrect gravitation acceleration will be obtained.
Results & Calculations
The values of period and the gravitation acceleration in the experiment can be calculated in both mathematical and graphical method:
Result Table:
In the experiment, we had made some assumptions:
- The bob is moving in the same plane.
- The angle to vertical is smaller than 10 degree.
- No air resistance
- The centers of mass and gravity are concentrated at the bob.
For the mathematical method:
The variable d = 0.15 m
T2 = (4π2l) / g
g = (4π2l) / T2
Set 1:
g = (4π2l) / T2 = (4π2* (0.81+0.15)) / 2.012 = 9.38 ms-2(cor. to 3sig fig)
Set 2:
g = (4π2l) / T2 = (4π2*(0.695+0.15)) / 1.902 = 9.24 ms-2 (cor. to 3sig fig)
Set 3:
g = (4π2l) / T2 = (4π2* (0.61+0.15)) / 1.792 = 9.36 ms-2 (cor. to 3sig fig)
Set 4:
g = (4π2l) / T2 = (4π2* (0.565+0.15)) / 1.692 = 9.88 ms-2 (cor. to 3sig fig)
Set 5:
g = (4π2l) / T2 = (4π2* (0.53+0.15)) / 1.672 = 9.63 ms-2 (cor. to 3sig fig)
Set 6:
g = (4π2l) / T2 = (4π2*(0.48+0.15)) / 1.642 = 9.25 ms-2 (cor. to 3sig fig)
Set 7:
g = (4π2l) / T2 = (4π2* (0.40+0.15)) / 1.552 = 9.04 ms-2 (cor. to 3sig fig)
Average g = (9.38 + 9.24 + 9.36 + 9.88 + 9.63 + 9.25 + 9.04) / 7
= 9.40 ms-2 (cor. to 3sig fig)
Maximum error in set 1 = 9.38(0.001/0.96 + 2*(0.1/40.2))
= 0.06 ms-2 (cor. to 2 d.p.)
Maximum error in set 2 = 9.24(0.001/0.845 + 2*(0.1/37.9))
= 0.06 ms-2 (cor. to 2 d.p.)
Maximum error in set 3 = 9.36(0.001/0.76 + 2*(0.1/35.9))
= 0.06 ms-2 (cor. to 2 d.p.)
Maximum error in set 4 = 9.88(0.001/0.58 + 2*(0.1/33.8))
= 0.08 ms-2 (cor. to 2 d.p.)
Maximum error in set 5 = 9.63(0.001/0.68 + 2*(0.1/33.4))
= 0.07 ms-2 (cor. to 2 d.p.)
Maximum error in set 6 = 9.25(0.001/0.63 + 2*(0.1/32.7))
= 0.08 ms-2 (cor. to 2 d.p.)
Maximum error in set 7 = 9.04(0.001/0.63 + 2*(0.1/30.9))
= 0.07 ms-2 (cor. to 2 d.p.)
Maximum error in average value = 0.07 * 2 + 0.08* 2 + 0.06*3
= 0.42 ms-2 (cor. to 2 d.p.)
The gravitation acceleration calculated = (9.40 ± 0.42) ms-2
Maximum percentage error
=(0.42/ 9.40) x 100%
= 4.47 %
For graphical method,
From the express, we deduce that :
Y= mx + c
m = 4π2 / g
g = 4π2 / m
From the graph,
The coordinate of the centroid C: (0.73, 3.08)
The slope (m) = (3.08-0.35) / (0.73-0) = 3.7397
g = 4π2 / m = g = 4π2 / 3.7397 = 10.56 ms-2(cor. to 4sig fig)
Let the slopes of the two good- fit lines be m1 and m2.
m1= (0.75 – 3.08) / (0.10 - 0.73) = 3.6984
m2= (0 – 3.08) / (0.14 - 0.73) = 5.2203
Δm = [(3.6984-3.7397) + (5.2203-3.7397)] /2
= 1.44 (cor to 2 d.p.)
Δg = 1.44 ms-2
The gravitational acceleration obtained from the graph = (10.56 ± 1.44) ms-2
Maximum percentage error
= (1.44/10.56) x 100%
= 13.6 %
Discussion
Accuracy & improvements
In this experiment, we had made several experimental errors but we can improve the experiment by the following improvements.
For a start, the bob do not move in the same plane. If the bob don’t move in the same plane, the bob will not perform SHM and it only performs circular motion. The data we obtained in the experiment are all wrong and the equations mentioned in the theory part cannot be used in the calculation process. If we use the wrong data and unsuitable equations, the period and the gravitation acceleration we obtained will be wrong. To avoid such serious error, we ensure the bob is on the same plane as vertical before the experiment starts. If we find that the bob does not move in the same plane, we should stop and redo the experiment again until the bob moves in the same plane.
Secondly, the angle to vertical sometimes is larger than 10 degree. If the angle is larger than 10 degree, the displacement is not equal to sinΘ, hence the motion of the bob does not obey to the definition of SHM. As a result, the bob does not perform SHM., the data and the equations are not suitable in the calculation process. The gravitational acceleration calculated will be incorrect. To prevent such error, we should prepare a projector. When we move up the bob, another student should use the projector to measure the angle to ensure it is smaller than 10 degree.
Besides, the thread used in the experiment may be slightly elastic. During the experiment, it may extend to certain length which is longer than the measured one before the experiment. It directly affects the period of the SHM as the period depends on the length of the thread. As a result, inaccurate data is obtained from the experiment. Hence the calculated gravitation acceleration is inaccurate. To get rid of such error, we choose a less elastic or non-elastic string instead of using the old one.
On top of that, there is reaction time error. It leads to inaccurate period measured by using the stopwatch. To do away this error, we can increase the number of rotations we have to measure to minimize the reaction time error. Moreover, there are also zero errors of the stopwatch and the meter rule. This results in the inaccurate measurement of the length of the thread and the period of SHM. These two factors directly affect the gravitational acceleration calculated as the g only depends on the period and the length of the string.
Finally, there must be the effect of air resistance on the motion of the bob. The air resistance will oppose the motion of the bob and increase the time for one period.
It results in the inaccurate data obtained and the final result of the gravitational acceleration calculated. To minimize the effect of air resistance, we should close all the windows and turn off all fans and air-conditioners in order to minimize air movement.
In this experiment, there are some assumptions we had made. Firstly, we assume that the bob is moving in the same plane during the experiment. Otherwise, the bob will not perform SHM and it only performs circular motion. The data we obtained in the experiment are all wrong and the equations mentioned in the theory part cannot be used in the calculation process. If we use the wrong data and unsuitable equations, the period and the gravitation acceleration we obtained will be wrong. Secondly, we also assume that the angle to vertical is smaller than 10 degree. If the angle is larger than 10 degree, the displacement is not equal to sinΘ, hence the motion of the bob does not obey to the definition of SHM. As a result, the bob does not perform SHM., the data and the equations are not suitable in the calculation process. The gravitational acceleration calculated will be incorrect. The third assumption is that there is no air resistance. The air resistance will oppose the motion of the bob and increase the time for one period. It results in the inaccurate data obtained and the final result of the gravitational acceleration calculated. However, the effect of air resistance on the bob cannot be totally removed. So we assume that there is no air resistance. The last assumption is that the centers of mass and gravity are concentrated at the bob.
The experimental errors can be divided into two errors, systematic errors and random errors. The zero errors of the stopwatch and meter rule and effect of air resistance on the bob are belonged to the systematic errors. The other errors mentioned above are the random errors. An experiment with small systematic and random errors is more precise.
Conclusion
By both mathematical and graphical methods, we can find out the gravitational acceleration due to the earth in the simple pendulum set-up.
For mathematical method, the gravitational acceleration is (9.40 ± 0.42) ms-2. For graphical method, the gravitational acceleration is (10.56 ± 1.44) ms-2. Either the gravitational acceleration calculated from the mathematical method or the one from the graphical method are not equal to the value actual value (9.8 ms-2). It may result from the experimental errors mentioned above. The result will be more accurate if we follow the improvements mentioned. However, the experimental result still is close to the actual value. So it can be said that the experiment result is precise.
In the next time, we can vary the gravitational acceleration by changing the places for experiment on different floors, we can investigate the effect of g on the period.
References
Wikipedia (pendulum)
http://en.wikipedia.org/wiki/pendulum
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