Percentage error =
The y-coordinate of the centroid of the three points =
Percentage error =
The slope of the best fit =
Percentage error = 2.86 + 2.84% = 5.70%
Absolute error =
The slope of the best fit = .
The theoretical slope = .
Hence, the experiment is done quite accurately.
SETUP 2:
We have then modified the simple pendulum a little by connecting it with a ticker-timer system to record the motion of the simple pendulum as shown below:
Procedures:
With the attachment of a ticker-tape, we set the mass at rest in the vertical position and mark down the equilibrium position on the tape. We then pull the mass to the right. After that, we turn on the power supply and allow the ticker-timer to take position-time record on ticker tape. We then immediately release the mass and allow it to swing to left. When the mass was completely swung to the left, we turned off the power supply to prevent any unwanted tick on the ticker tape.
Observations:
The frequency of the ticker-timer is 50Hz.
T =
The time interval between two successive dots =
= 0.02s
It is obvious to see that the separation between two successive dots before reaching the equilibrium position increases but decreases after reaching the equilibrium position.
Take the direction to the right is positive.
At t = 0, the displacement is maximum, the velocity is zero and the acceleration is negatively maximum. The displacement gradually becomes zero and reaches zero when it is at equilibrium position at t = . The separation between two successive dots increases from t = 0 to. The rate of the increasing separation, however, decreases. That means that the velocity increases and reaches maximum at t =. Acceleration also increases (as the direction to right is positive but the mass is moving to left) and gradually reaches zero at t =. Therefore, we can conclude that the displacement is zero, the velocity is maximum and the acceleration is zero at t =.
The displacement then becomes negative at t = to and reaches minimum at t =. The separation between two successive dots decreases from t = to. The rate of the decreasing separation also decreases. That means that the velocity decreases and reaches zero at t =. But the acceleration increases and gradually reaches positively maximum at t =. Therefore, we can conclude that the displacement is maximum, the velocity is zero and the acceleration is positively maximum at t =.
Measurements:
Taking the direction to the right as positive and the displacement at equilibrium position is zero, a displacement-time relations can be found by plotting a graph within a time interval.
The absolute error of each x is 0.001 m due to the ruler.
Data analysis:
Referring to the graph attached on the next page (page 7), at each point, a normal is drawn by the use of mirror. The data are as follows:
mnormal = mtangent =
Percentage error of all mtangent
=
= 11.0%
Percentage error of all mtangent is therefore 11.00%.
Using the slopes of tangent, another graph against t can be obtained.
As , the slope of tangent of each point on the displacement-time graph is the instantaneous velocity and hence another graph obtained is a velocity-time graph. The velocity-time graph is on page 9.
From the velocity-time graph, we can see that the mass is moving fastest (if we consider the magnitude only) when t =. At this instant, it is easily been seen that the mass is in its equilibrium position.
As , we then can find out the instantaneous accelerations by finding the slope of the velocity-time graph at each instantaneous time.
Note: The sign of slope of tangent depends on the increase / decrease of velocity.
Then, we can obtain an acceleration-displacement (a-x) graph by plotting the points of the instantaneous accelerations found (as y-axis) and the corresponding displacement of instantaneous times (as x-axis).
The x-coordinate of the centroid
=
= 0.00522
The y-coordinate of the centroid
=
= -0.165
The a-x graph is attached on the next page (page 12).
However, the centroid now located on the graph is on the origin. This error is due to the hand-drawing of the velocity-time graph and somehow the error performed in displace-time graph due to the scale.
The slope found in acceleration-displacement graph =
The length of the string is 1.631 m.
Therefore, the measured g is 5.881.631 = 9.59 ms-2.
Compared with theoretical g = 9.83 ms-2,
the percentage error in g =
Conclusion
Through the two experiments, we observed that the period is directly proportional to the square root of the length of the string. With a shorter string, the period is shorter. With a longer string, the period is longer. It is also proved that the slope of the graph is .
In the second experiment, we can find out the experimental gravitational acceleration. The experimental result (a = 9.59ms-2) is similar to the theoretical value (a = 9.83 ms-2). Therefore, we can prove that the theories are correct.
Because the theories are for simple harmonic motion, so we can prove that the motion is a simple harmonic motion.
Discussion:
Possible errors
Measurement errors
When measuring the length of the string, there is error made by the scale of the metre rule. In addition, the length of the string is not measured from the centre of gravity of the mass. With smaller length, by the equation , the measured g will also be smaller.
When constructing displacement-time graph, by measuring the separation between dots on the ticker tape, error may be produced because of the scale of the ruler.
Another error concerning about the ticker tape is the equilibrium position. The equilibrium may not be accurately marked on the ticker-tape. Hence, the period may not be accurately marked.
When constructing velocity-time graph and acceleration-displacement graph, the graph may not be accurately constructed due to error produced when finding the slope of displacement-time graph and velocity-time graph and the inaccurate hand-drawing.
Experimental errors
In performing the experiment, there may be friction between the string and the clamp. During the experiment, the stand oscillates a little. Some energy may be lost to the stand and clamp and hence the speed would be slower than expected. Friction between the ticker timer and the ticker tape may affect the period of the motion. The separation between two dots may not be exactly 2 seconds and hence the whole motion observed from the ticker tape may not be accurate.
Moreover, we observed that the mass would rotate a little itself. This error is very difficult to prevent from unless an extremely care is paid when performing the experiment. Again, some energy may be converted from translational energy to rotational energy. Hence, the speed would be slower.
One more error is the air resistance. There may be damping forced added onto the system. This may reduce the speed of the mass.
Precaution
The clamp should not oscillate despite the fact that the arm of the stand is not strictly held on the base.
The whole stand and clamp should be held firmly by G-clamp.
The angle of oscillation should be small. Otherwise, the assumption does not hold.
The equilibrium position should be marked accurately so that the period is correct.
The mass should be released skillfully so that the mass does not move in an ellipse form and does not move circularly itself.