HSC Physics Practical 1
Aim: To investigate the oscillations of a simple pendulum
Theory: 1. Simple Harmonic Motion is a form of periodic periodic motion of a particle, etc etc in which the acceleration is always directed towards an equilibrium point and is proportional to displacement. The abbreviation of Simple Harmonic Motion is SHM.
2. Derivation of the equation for the period of a simple pendulum with a study of the properties of force and an d acceleration in a simple pendulum by examining a free diagram of a pendulum bob.
starts
As the pendulum swings, it is accelerating b oth centripetally, towards the point of suspension and tangentially, towards its equilibrium position. Its’s linear , tangential acceleration that connects a pendulum with simple harmonic motion.
The weight component, mgsinθ is accelerating the mass towards the equilibrium along the arc of the circle. The component is known as the restoring force of the pendulum. Frestoring = matangential mg sinθ = matangential
To ensure that this restoring force is qualified as SHM, the restoring force should be directly proportional to the bob’s linear displacement from equilibrium along the length of the chord.
Geometrically, the arc length, s is directly proportional to the magnitude of the central angle, θ as referred to the formula s = rθ. In the diagram, the radius of the circle of the circle, r is equal to the L, the length of the pendulum. Thus, s = Lθ where θ must be measured in radians. Substituting this into SHM, we get:
Frestoring= - ks mg sinθ = - k(Lθ)
The solving of the “spring constant” or k for a simple pendulum brings:
mg sinθ = k(Lθ) k = mg sinθ / Lθ When an angle is represented in radians, mathematicians technically represent the angle with the variable x instead of θ. Here, it must be noted that the value of sin x approximates the value of x for small angles, that is:
Or equivalently, for x equals to small values can be seen from this power series below that the value of sin x would approach x:
Using this relationship can allow us to reduce our expression for the pendulum’s “spring constant” to k = mg / L Substituting this value of k into the SHM equation for the period of an oscillating system results in
3. p can be obtained from the graph by calculating the gradient of the graph, and to get the value of q from the graph, the equation y = mx + c can be used and when we substitute the equation into y = mx + c, q be obtained from the equation as the y-intercept. Apparatus: Split rubber stoppers, thread, pendulum bobs, stopwatch, retort stands, ruler.
Procedure:
1. Two simple pendulums side by side are set up as shown in Figure 1.1, with each string clamped between two split rubber stoppers. The length of pendulum A was set to about 0.65m. Pendulum A was set left at its set length throughout the experiment. 2. Pendulum B was adjusted so that its length l is about 0.5m. The value of l was measured and recorded. 3. Both pendulums were set into motion with small oscillations. The stop watch is Started when the two pendulums are lined up as shown in Figure 1.2 and are moving in the same direction.
4. The time t that elapses before the next occasion when the two pendulums are lined Up and moving in the same direction are determined. 5.
l and
Step 4 was changed and repeated until six sets of values for l and t are obtained. l should be from about 0.3m to about 0.6m.
6. The equation relating t and l is
where p and q are constants.
Tabulation of data
Length of the thread of the pendulum/m 0.55 0.50 0.45 0.40 0.35 0.30
Time readings(s)
19.25 11.94 7.56 6.12 4.75 3.44
19.50 12.25 8.06 6.00 4.94 3.75
18.87 11.56 8.25 5.50 4.37 3.56
Average reading 19.21 11.92 7.96 5.87 4.69 3.58
1 t 0.05 0.08 0.13 0.17 0.21 0.23
1 l
1.35 1.41 1.50 1.60 1.70 1.81
Calculation To obtain the value of p, the gradient of the graph is calculated Gradient = 0.23 0.48 = 0.4791= 0.48 To obtain the value of q, we use the y = mx + c equation and apply it with the following equation:
After substituting the values into the equation, it would be like this: 0.28 = p( ) 0.28 = 0.48(1.83) 0.28 = 0.8784- 0.48 q 0.48 = 0.8784-0.28 q 0.48 = 0.5984q q = 0.802
Conclusion P and q are constant values in this experiment. The value of p is 0.48 while the value of q is 0.8.
Discussion 1. Among the problems I encountered when conducting this experiment is the problem of obtaining an accurate result when recording the data during the experiment as our result might have been affected due to the blowing of the air-con when the pendulum was swinging. Besides that, another problem I encountered during the experiment was that during the time where we had to measure the length of the thread of the pendulum, it might not have been accurate at all times and thus, we might have had measurement errors during the experiment. 2. Among the possible sources of errors in this experiment is the air- con in the Physics Lab where we conducted the experiment as the air- con kept blowing in the lab when we conducted the experiment and may have affected the accuracy of our result. Besides that, another possible source of error in this experiment is
measurement errors as when we measure the length of thread of the pendulum, it might not have been accurate all the time. Finally, another possible source of error in this experiment includes when the two pendulums are release together, the time recorded for the time elapses before the next occasion when the two pendulums are lined up and moving in the same direction also might not have been too accurate as well.
3. Among the precautions I have taken during the experiment is that I made sure t That we take 3 readings of the time that elapses before the next occasion when The two pendulums are lined up and moving in the same direction and obtaining the average of the readings to ensure a more accurate result. Next, I also ensured that the pendulum bob was completely still and in its place before we started using the pendulum to ensure that the result would be more accurate. Finally, I also ensured that the thread was tied tightly to the retort stand so that the pendulum would not suddenly become loose and fall down to the floor as this action would definitely affect our final result.
References: 1. Catherine H. Colwell (n.d) PhysicsLAB: Derivation: Period of a simple Pendulum. Retrieved 9 February 2011, from http://dev.physicslab.org/Document.aspx? doctype=3&filename=OscillatoryMotion_PendulumSHM.xml 2. definition of simple harmonic motion by the free online dictionary (n.d) Retrieved 9 February 2011, from: http://www.thefreedictionary.com/Simple+Harmonic+Motion
