LABORATORY 4. The Simple Pendulum

Transcription

1 ABORAORY 4 he Simple Pendulum Purpose Some of the earliest studies of motion were based on the observation of a swinin pendulum. Althouh the motion is neither linear nor uniform, for small anles of swin the motion is nearly straiht line, back-and-forth motion (see diaram below), which makes it easy to measure. Both Galileo and Newton used the pendulum to develop their ideas about motion and forces. ike the early researchers, we will investiate the effects of pendulum lenth, amplitude and weiht on the pendulum s motion. We will also use it to measure the acceleration of ravity. he Pendulum & Newton s aws A simple pendulum consists of a massive bob suspended by a strin or rod of neliible mass and swinin in a vertical plane. If we inore the mass of the strin or rod, the system then consists simply of a mass oscillatin alon an arc. he period of the pendulum is the time it takes to make one complete swin, back and forth. he amplitude of the motion is the size of the anle from the vertical it attains at each end of its swin. Curiously, the pendulum tends to maintain the same period, even as its amplitude decreases due to air resistance and it travels in smaller and smaller arcs. his was one of the observations that excited Galileo s interest. (Another was the fact that a chanin pendulum mass appeared to have no effect on the period. A heavy bob would take the same time to swin as a liht bob.) he forces on the bob are its own weiht, directed vertically, and the tension in the strin, directed toward the center of the arc (see the free-body diaram). he easiest way to analyze the motion is in terms of its anle from the vertical, θ. he pendulum travels alon an arc, s, which relates to the anle θ by s = θ, where is the lenth of the pendulum and the radius of the arc. Instead of usin x-y coordinates, we can break the weiht vector W into components alon the arc (Wsinθ) and perpendicular to the arc (Wcosθ). ikewise, we can break the acceleration vector a into one component tanential to the arc (a ) and another component directed radially (a R ). :

2 -θ θ Equilibrium position (θ=) s (arc lenth) Diaram 1: Pendulum oscillatin in x-y plane. Note that s = θ x when θ (measured in radians) is small. -Wsinθ Wcosθ W=m Free Body Diaram: he weiht W is broken up into radial and tanential components. he tension is in the neative radial direction. From the free-body diaram and Newton s nd aw, we et two equations: Radial direction: anential direction: m cos θ = m sin θ = ma mar (Usin W = m.)

3 he radial equation tells us what the tension in the strin must be as a function of the anle, θ, and the centripetal acceleration, a R. (Note: a R = v /, where v is the tanential speed and is the radius of the path). We can determine the bob s speed and position alon the arc from the tanential equation. First, we use the approximation thatsin θ θ (in radian measure) for small anles. Second, we express the acceleration in terms of the anular acceleration α = a. his ives us or m θ = mα θ = α he minus sin arises because the tanential force points in the direction of decreasin anle, towards θ =. he equation tells us three thins: (1) he acceleration of the bob is proportional to its displacement θ, but points towards the equilibrium position, where θ =. his means the bob is always driven back towards equilibrium, reardless of whether the displacement θ is positive or neative; () he acceleration is also proportional to the quantity /, so that pendula with loner lenths () will have smaller accelerations and vica versa; (3) he acceleration is not dependent on the mass of the bob, consistent with Galileo s observations. he Motion in ime Now we would like to et an expression for θ as a function of time. he anular acceleration is just the second derivative of θ with respect to time, so that we can write the tanential equation as: d θ ( t) dt = θ ( t) It is straihtforward to verify that the function (1) θ ( t) = θ cos( t )

4 is a solution for θ (t). (ake the derivative of this function twice and compare the result to θ.) From this, we can find the period of the pendulum. Since the pendulum starts at θ at t = and returns to θ at t =, we have: θ ( ) = θ cos( ) = θ his can only be true if the cosine function has the value 1 at time. his means = π or () = π his is the prediction of Newton s aw for the period of the pendulum. It is valid as lon as the approximation sinθ = θ is valid. We will test this result experimentally in three ways: 1. Varyin the lenth and measurin the period ;. estin the amplitude independence (equation does not depend on the startin anle, or the amplitude of the motion; 3. estin the mass independence of the period.

5 Procedures 1. Measure the period as a function of lenth. Get a piece of strin about.1 m lon. Run the strin throuh the center hole of a pendulum bob. Clamp each end of the strin to a clamp on the pendulum clamp, with the bob hanin in the center of the strin. Your set up should look like this (the bob will swin into and out of the pae): Pendulum, side view Note: In your report, show that the double strin arranement does not chane the above analysis of the forces on the bob. However, this arranement helps keep precession to a minimum. (Precession is the rotation of the x-y plane of the bob s motion about the vertical axis. he result is that the bob will end up travelin in a circle instead of in a plane. ry the pendulum with one strin and see what happens when its amplitude ets small.) ape or clip a protractor to the end of the pendulum clamp so you can measure the initial anle ( θ ) of the pendulum when you release it. In this part of the lab, use 5 derees as θ. ime the period of oscillation for 8 pendulum lenths () between 1 cm and 1 cm. is measured from the bottom of the pendulum clamp (where the strin bends) to the center of the bob. o determine the period:

6 o et each pendulum oscillate times and record the time for swins. o Calculate the period, exp for each lenth by dividin the total time by.. Measure the period as a function of amplitude. Use the procedures above to measure the period of the pendulum as a function of the initial displacement of the bob, θ. Use a lenth of about 3 cm and startin anles of 5, 15, 5, 35 and 45 derees. 3. Measure the period as a function of mass. Use the procedures above to measure the period of the pendulum as a function of the mass of the bob. Replace the bob you have been usin with one of a different mass. Use the same lenth as you used in one of the experiments above. Repeat this, usin a third mass. Measure and record the masses of all three bobs. Analysis: Part 1 For each lenth, square exp and record the value of in your data/analysis table. Graph as a function of the lenth of the pendulum,. Do this by hand on raph paper. Find the slope of your raph and write down the empirical equation of your raph: = slope x. (We assume the raph runs throuh, and so has y- intercept = zero. If your raph shows a non-zero intercept, explain why.) Compare your empirical equation with = 4π. In other words, compare exp with theory. hink of a quantitative way to compare the two equations. Use the accepted value for (9.8 m/s/s). Now use your data to find a measured value for. Comparin the empirical and theoretical expressions for we see that: 4π = m where m is the slope of the raph. Compare your value with the accepted value of 9.8 m/sec.

7 Find the uncertainty in, Δ, from the uncertainty in the slope of the raph, Δ m. (Refer to the lab manual introduction on calculatin uncertainties.) Use the standard deviation in the slope as m. his is calculated from the square of the differences between the best fit values for and the actual values. Use the curve fittin option on DataStudio, or any curve fittin proram, or use your hand-plotted raph. For your reference, the standard deviation in the slope, s m is iven by s m n Δyi n n xi ( xi ) = where the x i are the abscissas ( in our case), y i are the ordinates ( in our case) and y i are the differences between the fitted values for the ordinates and the measured values: y i = y i Y i. Report your value for in the form: = exp erimental ± Δ. Part Plot the period as a function of initial amplitude (the beinnin anle). his can be an informal plot, or use a raphin proram. ry to find a fit for the raph. What function fits it best? (his is not a question with a riht or wron answer. Use your judment.) Part 3 Assinment Compare the periods for the different masses by takin percent differences between the periods, and the percent differences between the masses. What conclusion is indicated by your data? Draw a free body diaram for the simple pendulum, usin an x-y coordinate system. What would be the two equations for the motion in this case? Show usin a free-body diaram that the double-strin system does not chane the direction of the net force on the bob. In this case you will have to diaram the y-z plane.