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Let's suppose the angle is small. The bob swings back and forth and there is no air resistance.
1) label the vectors. Use: W the weight, Wx (x-component of the weight), Wy (y-component of the weight) and N the normal.
2) Along the y-axis, there is no acceleration so ______________________ is balancing the ______________.
3) Along the x-axis there is an acceleration. The bob speeds up as it falls (from far right to equilibrium position), slows down as it rises again (to far left), come to a rest (far left) , then speeds up again going down. Along the X-axis Fnet = _____________. (use m,g and a trig function with θ inside).  
So the force acting on the bob to bring him back to θ =0  is just : Wx ? Wy ? N ? . This is the restoring force for a pendulum.
4) Mathematically you can show that if an angle is small then: sin (θ ) =θ. In that case. Fnet = Fx  = _________.
You just have shown that the restoring force Fnet is proportional to _________________. Therefore, a pendulum undergoes simple harmonic motion.

5)A complete vibration (any back-and-forth motion ) is called a cycle. A cycle is the movement from some point (say the far right), to a maximum displacement in the other direction (say the far left) then back to the same point again (far right). The period (T) is the time to complete one cycle. For example, suppose 3s is required for a bob to swing back and forth, then the period of this vibration is T = ____________.

6) The number of cycles per second is called the frequency (f) . frequency = 1/ period. The unit is hertz. So if T= 3s, then the frequency = ___________.

7) The length (L) of the pendulum is measured from the point of suspension to the center of the bob. Since the bob has a radius,
L = length of the string + radius of the bob. Using a caliper measure the diameter then the radius of the bob: R = ___________________.

PURPOSE
The purpose of this lab is to investigate what factors influence the period of a pendulum. Take some guesses:
8) Do you think the mass will affect the period T? ____________  why ? ___________________________________________________________
Do you think the length of the pendulum will affect the period ? ________________________________________________________
Do you think the amplitude (the angle θ) will affect the period ? ______________________________________
Do you think a pendulum can help you to take decision?__________ finding water ? __________ find out your final grade in Physics ? __________

We will also investigate ways to estimate the acceleration due to gravity g = 9.8m/s/s using a pendulum.




PROCEDURE

step 1: Let's investigate if weight influences the period of a pendulum. Use different masses (we have only 2 here) for bobs and adjust each so the string is 100cm from the pivot point to the center of gravity of the bob. Pull the bob back to make an arc of about 15 degrees. (use a protractor). RElease and count the number of vibrations (one vibration is one complete back-and-forth movement) for exactly 1 minute. The arc, length of pendulum, and  all other variables must be the same. THe only difference should be the mass of the bob. Take three trials for each mass and average the findings. Record all data in table 1. Also record the mass of each bob (a metal one and wooden one)

TABLE 1

bob description (find mass in g)	trial 1 : cycles per minute	trial 2: cycles per minute	trial 3: cycles per minute	average
bob 1:
bob2:

step2:


FROM NOW ON YOU WILL BE WORKING WITH ONLY 1 BOB. (metal or wood)
This time, hold all the variables the same except the amplitude of the bob. You can measure the approximate amplitude by using 2 meter sticks as shown below. Count the number of cycles for exactly one minute for each amplitude tested. Take three trials for each amplitude. Average the findings, and record all data in table 2.










TABLE 2

amplitude (cm)	trial 1 : cycles per minute	trial 2: cycles per minute	trial 3: cycles per minute	average
amplitude 1:
amplitude 2:
amplitude 3:

step 3: This time. hold all the variables the same (including the bob) except for te  length of the pendulum. Recall that the length of the pendulum is not just the length of the string. (add the radius to the length). Measure the number of vibrations for exactly 1 minute. Repeat three trials for each length, average the findings, and record all data in table3. From the average data (number of cycles per minute), calculate the experimental period (seconds per cycle, solve the proportion, convert minutes to second first) for each pendulum and record it in table3. (you have the number of cycles per minute, find the number of cycles per second)

TABLE 3

length of the pendulum (radius + string) in cm.	trial 1 : cycles per minute	trial 2: cycles per minute	trial 3: cycles per minute	average	experimental period Texp (seconds for 1 cycle)	true period Ttrue (T = 2Π√ (L/g)	percentage error (%)
length1:
length2:
length3:

Using the equation derived previously : a = g θ  and solving for θ using calculus (differential equation) you can show that :
T = 2Π√ (L/g). L is the length of the pendulum and g is the gravity (9.8m/s/s).
Use this formula to calculate the theoretical value of the period for each length and record it in table 3.
Find the % error: = |true - experimental| / true x 100 and record the % in the table 3. Use the absolute value of the difference.

step4: Use a metal bob to make a pendulum of 25 cm length. Make three separate measurements of the period of this pendulum by finding the time for the pendulum to vibrate 25 times (25 cycles) . keep 2 decimals. Record this data in table 4. Multiply the time (for the 25 cycles) by 4. You will obtain the time for 100 cycles. Move the decimal point 2 places to the left to find the period (time for 1 cycle).

TABLE 4

Trial	time of 25 full cycles (s)	calculated time of 100 cycles (s) multiply column 1 by 4	Period T (s) divide column 2 by 100	square of the period T2 (s) square column 3	calculated g g = 4 pi2 L / T2
trial1					g1
trial2					g2
trial 3					g3
average	nothing here				g calc =

From the previous formula T = 2Π√ (L/g) you can extract g (solve for g). You need Algebra 1 (or 2 ? I am not sure).
you should find g = 4 pi² L / T²  
 Find T² for each of the trial and compute g for each of the trial.Average all the gs (g1 + g2 + g3 ) /3 Record every thing in  the TABLE 4.
To compare your value of g (gcalc) to the true one (9.8m/s/s) find the experiment error : ___________________ ( |9.8 - gcalc| / 9.8   x100 )
Did you do a good job ?______________




ANALYSIS:

1) According to the experimental results, what factors affect the period of a pendulum:(yes or no)
weight ?  _________amplitude (depends on the initial angle)  ________ ? the length of the pendulum ? ___________
Galileo was the first one to find out this fact by observing a pendulum in a church and by using his pulse as a time watch.

2) Was the purpose of this alb accomplished ? So do you like Physics ________ ?

GOING FURTHER

1) A student observing an oscillating block counts 45 cycles of oscillations in one minute. Determine its frequency (in hertz) and period (in seconds)
_________________    ___________________________

2) Circle the right answer. Which of the following is/are characteristic of simple harmonic motion ?
acceleration is constant          the restoring force is proportional to the displacement      the frequency is independent of the amplitude

3) A simple pendulum swings about the vertical equilibrium position with a maximum angular displacement of 5 degrees and a period T. If the same pendulum is given a maximum angular displacement of 10 degrees, then which of the following best gives the period of the oscillations ?
T/2 ?      T / √2  ?   T  ?      T√2 ?    2T  ?

4) A simple pendulum of length L and mass m swings about the vertical equilibrium position (θ = 0) with a maximum angular displacement of θmax.
A) What is the tension T (see first part of the lab) in the connecting rod when the pendulum's angular displacement is θ = θmax.
mgsin(θmax) ?  mg cos (θ) ?
B) The pendulum is now at rest (not moving, θ = 0). Draw the pendulum and all the forces acting on the bob here:
The pendulum is at rest, then the _________________ is balancing the __________________
So T=                            

5) OPTIONAL for AP Physics candidates or anyone else who loves Math and wants to have fun !
a block attached to a spring is also a simple harmonic system. You pull on the block and if we neglect air friction, the block swings back and forth. The amplitude is always proportional to the restoring force. The restoring force is the tension in the spring F. F = - K X. K is a constant called the spring constant (in N/m) and only depends on the spring you are using. Stiffer is the spring, larger is K. X (in m ) is the displacement of the block from it equilibrium position (the elongation or the compression of the spring). The tension in the spring wants to bring it to its equilibrium state (X = 0, no displacement)

Along the X-axis,  only force acting on the block of mass m  is the restoring force F =- K X .
The negative sign reminds us that F and X are in opposite direction. T (restoring force)  and X (displacement) are vectors.

A) You stretch the block to the right  and release it. It will speed up, pass its equlibirum position (X = 0), slows down to the left, comes to a stop, comes back to the right, speeds up, stops.. It oscillates.

Find the expression of the accelalation along the X axis. ax = ________ using F = - kX

If you use Calculus (differential equations) you can derive the following eqation: T = 2 pi√ m/K
T is the period of the oscillations. m is the mass of the block. K is the spring constant.
So in the case of a spring-block oscillator, the period T depends only on ___________ and ________________.

C) Using the formula T = 2 pi√ m/K, can you answer thes questions:

i) How would the period of the spring-block oscillator change if both the mass of the block and the spring constant were double ?

ii) A block is attached to a spring and set into oscillatory motion, and its frequency is measured. If this block were removed and replaced by a sond block with 1/4 the mass of the first block, how would the frequency of the oscillations compare to that of the first block ?

iii) A linear spring of force constant K is used in a physics lab experiment. A block of mass m is attached to the spring and the frequency f of the simple harmonic oscillation is measured. BLocks of various masses are used in different trials, and in each case, the corresponding frequency is measured and recorded. If f² is plotted versus 1/m , the graph will be a straight line with slope:____________  (use K, m, pi )

iv) A block of mass m = 4kg on a frictionless , horizontal table is attached to one end of a spring of force constant K = 400 N/m and undergoes simple harmonic oscillations about its equlibrium position (X = 0) with amplitude A = 6cm (Xmax = 6cm maximum elongation). If the block is at X=6cm at t=0, then which of the following equations (with X in cm and t in seconds) gives the block's position as a function of time?
X= 6 sin(10t + pi/2)   or  X = 6 sin (10pi t)  or X = 6sin (10t)  or X = 6 sin (10t - pi/2)  

Can you use your TI to plot the function? (mode=radian). X(t) shows you that displacement changes over time. The function is a ___________ function. Does it makes sense ? What happens to the position over time ?  Can you use your TI to find the period ? (find the distance between 2 peaks)




































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