Introduction.
The simple pendulum is a favorite introductory exercise because Galileo's experiments onpendulums in the early 1600s are usually regarded as the beginning of experimentalphysics. This experiment is still the same with the one we are thuoght in high school till today. Meanwhy, the mathematical analysis will be more complex. Is all about beeing familiar (again) with radian measure and trigonometric functions, take derivatives oftrigonometric functions in the prelab exercise, and error analysis must be observe as presentedin the lecture. Firststep,finding the acceleration of gravity and enphesize on the error in themeasurements. Second step, taking data for your first plot using Mathcad, and in third step, your data will explain why a small angle is used in first step.
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Theory
A simple pendulum consists of a mass m hanging at the end of a string of length L. The period of a pendulum or any oscillatory motion is the time required for one complete cycle, that is, the time to go back and forth once. If the amplitude of motion of theswinging pendulum is small, then the pendulum uatomatically behaves approximately as a simple harmonic oscillator, and the period T of the pendulum is given approximately by(1)L T 2g = π
where g is the acceleration of gravity. This expression for T becomes the same in the limit of zero amplitude motion and is less accurate as the amplitude of the motionbecomes larger.Inthisexpression, we can use measurements of T and L to compute g. Let us review simpleharmonicmotion to see where eq'n (1) comes from. Recalling that simple harmonic motion occurs whenever there is a restoring force which is proportional to the displacement from equilibrium. One of the most simplest example of simple harmonic motion is that of a mass m on a spring which obeys Hooke's Law,
(2) Frestore = Fspring = – k x
where x is the displacement from equilibrium and k is a constant called the spring constant. From Newton's
Second Law, Fnet = ma , we can write 22d x m kdt = − x , or Spring 2005 M1.1 (3)22d x kxdt m = − .
This is a second-order linear differential equation − "second-order" meaning that the equation involves a second derivative, and "linear" meaning that it contains no non-linearx-terms such as x2 or sin(x). The solution to equation (3) is (4) x t( ) = A sin(ωt + φ), where kmω = . The constant A is the amplitude of the motion; the position x oscillates between -A and+A. The constant φ is called the phase constant. A and φ depend on the initial conditions,
that is, the position and velocity at time t=0. The period T is related to ω by(5) 2Tπω = .
Now consider the simple pendulum — a mass m hanging on a string of length L. We can describe the displacement from equilibrium of the mass m either with the distance along a curved x-axis or with angle θ of the string from the vertical. The distance x is related to the angle θ by x = Lθ, where θ is in radians. There are two forces on the mass m: the force of gravity, mg, and the tension in the string, Fstring. The tension Fstring has no component along our curved x-axis, while gravity has acomponent along the x-axis equal to −mg sin θ. The minus sign indicates that the direction of the force is toward the origin, always opposite to the direction of x and θ.
If θ is very small and expressed in radians, then sin θ is about equal to θ , and we have,approximately,Spring 2005 M1.2 (6) restoremg F mg L ≈ − θ = − x , and, using F =ma,wehave(7)2 22 2d x mg d x g m x dt L dt L = − , . = − x
In this case of small x or θ , the restoring force is proportional to the displacement
x, and we have a simple harmonic oscillator whose equation of motion is just like eq'n(3) except we have replaced the constant k/m with another constant g/L. The solution to equation (7) is exactly like the solution to eq’n (3) except we replace k/m with g/L. So, by looking at eq’n (4), we have, for a simple pendulum
(8) 2 LT gπω = = . To solve T, we i'll consider eq’n (1). We analize that this expression is true in the limit ofsmall amplitude motion both small θ and small x. Note that, in eq’n (1), the period is independent of both the mass m and the amplitude A. If we do not assume that θ is small so we cannot make the simplifying assumptionsin θ ≅ θ , then the restoring force is (9) Frestore = – m g sinθ ,and the equation of motion becomes (10)2 22 2d d m L mg dt dt Lsin , sin . θ θ = − θ = − θ g
Meanwhy, we have a non-linear differential equation, which is not the same to solve. In this case, the period T still don't depend on m (since m does not appear in the equation ofmotion, it cannot appear in the solution); however, the period T depend on the amplitude of motion. We will use the symbol θo for the amplitude, or maximum value, of θ . The accurate solution for the period T can be written as a infinite power series in θo, and the first three terms of the solution are(11) 2 40 0 L111 T 2 1 g 16 3072
⎛ ⎞ = π ⎜ ⎟ + θ + θ + ⎝ ⎠ " .
(The terms above the first three are very small and can always be ignored.) In
this expression, the angle θo must be in radians, not degrees. Note that this form
reduces to eq'n (1) in the limit of small angle θo.
Spring 2005 M1.3
EXPERIMENTAL PROCEDURE.
Ritostart/Timer is always use to make very precise measurements of the period
T of a simple pendulum as function of the length L and the amplitude θo. A photogate
consists of an infra-red diode, called the source diode, which emits an invisible infra-red light beam. This beam is detected by another diode, the detector diode. When the mass m passes between the source and the detector, the infra-red light beam is interrupted, indicating an electronic signal that is used to start or stop an electronic timer. In "period mode", the timer starts when the mass first passes through the gate, and does not stop until the mass passes through the gate a third time, as shown below. Thus the timer measures
one complete period of the pendulum. Our timers are quite precise and read to 10-4 s = 0.1 ms. Pressing the "reset" button the timer most be set at zero and readies the timer for anothermeasurement. Check that the timer is working properly by passing a pencil three times through the photogate. The timer should start on the first pass, neglecting the second pass, and stop on the third pass.
When making measurements, begin by positioning the photogate so that the mass is directly between the diodes when it hangs straight down. Then, after pulling the mass made by it is swinging so that it does not appear into a collision with the delicate and expensive photogate. It is suspended over a support rod that make measurements with 5 or asurem the top of the mass to the pivot ht ring and measure the length L as described is longer than 1 meter, use a 2meter stick. If you are s a good idea for easure L independently and stimate and record the uncertainty δL in the length L. h swing). Make three measurements of the period T.
Your "best" the side, release it carefully so that it swings through the photogate without a collision. Allow the mass to swing back and forth a few times before makingmeasurements to allow for any wobble to settle out. After the mass
is set swinging, several measurements of T can be quickly simply pressing the reset button while the mass is still swinging. You should not stop the mass and restart it after each measurement, since this will introduce unnecessary wobble. Always keep an eye on the pendulum while In your pendulum, the str turns freely on its axis. You will more strings of varying length from about 40 cm to 120 cm. The length L of the pendulum is the distance from the center of the mass m to the pivot point, which the center of the top axle. To me L, first measure the distance λ fro point; then measure the height h of the mass. L = λ + h/2. You should only measure λ with the mass suspended, since the weig
of the mass stretches the string somewhat.
Precision determination of g. Choose the longest st above. Since the longest string working with a partner, it' both are to be compare the answers. E Now measure the period T. Begin by positioning the photogate carefully and set the mass swinging with a very small amplitude (a few degrees, just enough to pass completely through the photogate each value for T will be the middle of your three measurements. For your uncertainty δT, use half the difference between the highest and lowest measurement. Compute g from eq’n (1) using your measured values of L and T. Using Mathcad, compute δg, the uncertainty in g, which is (12) 2 2 g L T 2 g L T
δ δ⎛ ⎞ ⎛ δ ⎞ = + ⎜ ⎟ be lectures.
⎜ ⎟ ⎝ ⎠ ⎝ ⎠ .
We don’t expect you to understand where this expression comes from just yet. It will covered in the. Report your final answer in the standard format: g + δg, with units, being careful n to put more significant figures than you are sure of. (Write the final result using a text box because Mathcad will almost surely give too many significant figures for δg.)
Calculate ∆g = g – g_known, the difference between your answer and g_known = 9.796 m/s 2, a value obtained in a more precise way than we can achieve in the lab. How does this difference ∆g compare with the error δg that you calculated? Discuss the largest of systematic error which might explain the ment should have a definition like:
contribution to the uncertainty in your result and how this might be reduced. Compare your measured g with gknown = 9.796 m/s 2. If your gmeas and gknown do not agree within the uncertainty δg, suggest possible source discrepancy.
NOTE: In Mathcad, you cannot define an expression like δg/g ; you can only define a variable like δg. So, your
Mathcaddocu 2 2 L T g g 2 L T: ⎛ ⎞ δ δ defined n certainty due to random measurement errors only. It does not take account of any possible systematic errors. ur n just cut some more string from the ball, pick a different fifth length.
e lengths. Keep the amplitude of motion very
⎛ ⎞ δ = ⋅ ⎜ ⎟ + ⎜ ⋅ ⎟ ⎝ ⎠ ⎝ ⎠ .
And remember: all the variables used in the definition (g, L, T, δL, δT) have to be previously in the Mathcad document.
Eq’n (12) is the best estimate of the u Part 2. Dependence of T on L.
There should be 5 strings with approximate lengths of 40, 60, 80, 100, and 120 cm. If your apparatus is missing some of these lengths, provided. If your apparatus is not tall enough for 120 cm Measure the period T for each of the five small, so that eq’n (1) is approximately correct. By pressing the reset button several times mass is swinging, observe the period for several trials. By looking at the or many tria stimate the average value of the period and its uncertainty. (Do not record all the trials and do not compute the average - that is too time-consuming. We ngth in accord with eq’n (1). First, using while the numbers f ls, e want you to observe and estimate.)
Now we shall see whether your data vary with le Mathcad make a plot of T vs. L (using small circles or squares for your data) and note that T does not vary linearly with L. [Side comment: a graph of "A vs. B" means A on the yaxis
and B on the x-axis: Y vs. X, always.]
Squaring both sides of eq’n (1) yields Spring 2005 M1.6
(13)2 2 4 T L π = . g So, if this equation is correct, then a graph of T2 vs. L should be a straight line with d slope 4π2/g.
Using Mathcad, make a plot of (T2)i vs. Li that shows your data as points NOT connecte by a line. On the plot of T2 vs. L, also plot the straight line y(L) = m·L, where the slope m = 4π2/g. Use the known value of g = 9.796 m/s2. The theory line should be plotted without points to avoid confusion of these points with your data points. [It is NOT necessary to repeat the error analysis done in part 1 above for each length of Dependence of T on amplitude. Notice that the Mathcad Primer shows, step by step, all the analysis for this section. string.] Part 3. Using the string which is nearest to 1 meter in length, measure the period T as a functions of the maximum angle θ on ached to the pendulum rig. Measure T for several angles, from about 5o up to about 70o or so, attervals of about 5o. (Avoid angles larger than 70o, simply forsafety’s sake.) Be extremely careful to avoid collisions with the photogate. You need not take multiple trials
at each angle; one good measurement at each angle is enough. Convert your angles from degrees to radians and then make a graph of T(θo) vs. θo (in radians). Don't forget to let o = 1,2,...N where you define N. On the same graph, plot the eq’n (11)o. Measure the angle θ o using the protractor which is att in 2 4 0 0 L 1 11 T 2 1 g 16 3072 ⎡ ⎤ = π + θ + θ ⎢ ⎥ ⎣ ⎦ , using your measured value of L and the known value of g. Theory and experiment should agree pretty closely. Comment on any discrepancy. For example, is the error random(some points too high and some too low) or systematic (all points high or low).
0.4cm, T=2.1090±0.0009 sec Spring 2005 M1.9
SIMPLE PENDULUM EXPERIMENT
Inference : When the length of a simple pendulum increases, the period of oscillation also increases. // The period of pendulum is affected by the length of the thread. Hypothesis : The longer the length of a simple pendulum, the longer will be the period of oscillation
- Aim : To find the relationship between the length of a simple pendulum and the period
of oscillation. - Variable : a) Manipulated variable : Length, l
b) Responding variable : Period, T.
c) Fixed variable : Mass of pendulum bob. - Materials / Apparatus : Retort stand, pendulum bob, thrread, metre rule, stoo watch.
FUNCTIONAL DIAGRAM
Procedure :
a) Set up the apparatus as shown in Figure above.// A small brass or bob was attached to the thread. The thread was held by a clamp of the retort stand.
b) The length of the thread , l was measured by a metre rule, starting with 90.0 cm. The bob of the pendulum was displaced and released.
c) The time for 20 complete oscillations, t was taken using the stop watch. Calculate the period of oscillation by using, T = t / 20
d) The experiment was repeated using different lengths such as 80.0 cm. 70.0 cm, 60.0 cm, 50.0 cm and 40.0 cm.
Length of string, l / cm Time taken for 10 oscillation, t (s) Period of oscillation T T2 (s2 ) t 1 t 2 Average
Notes :
a) Plotting the graph
· The graph should be labeled by a heading
· All axes should be labeled with quantities and their respective units.
· The manipulated variable (l) should be plotted on the x-axis while the responding variable (T2 ) should be plotted on the y-axis
· Odd scales such as 1:3, 1:7 , 1:9 0r 1 :11 should be avoided in plotting graph.
· Make sure that the transference of data from the table to the graph is accurate.
· Draw the best straight line.
- the line that passes through most of the points plotted such that is balanced by the number of points above and below the straight line.
· make sure that the size of the graph is large enough, which is, not less than half the size of the graph paper or.( > 8 cm x 10 cm )
Discussion / Precaution of the experiment / to improve the accuracy
a) The bob of the pendulum was displaced with a small angle
b) The amplitude of the oscillation of a simple pendulum is small.
c) The simple pendulum oscillate in a vertical plane only.
d) Switch off the fan to reduce the air resistanceConclusion
The length of simple pendulum is directly proportional to the square of the period of oscillation. //T2 is directly proportional to l (the straight line graph passing through the origin)
Reffrence:https://www.youtube.com/results?search_query=image+of+a+simple+pendulum
https://nano-optics.colorado.edu/fileadmin/Teaching/phys1140/lab_manuals/LabManualM1.pdf
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