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Introductory Mechanics Experimental Laboratory
Pendulum
Goals: Observe periodic motion. Use an electronic measuring device to acquire data. Measure the effects of different variables on the period.
APPARATUS A pendulum can be made by supporting a mass ( m ) at the end of string. In this experi- ment the mass is held by two equal length strings, supported from a horizontal rod about 10 cm apart, as shown in Figure 1. This arrangement will let the mass swing only along a line, and will prevent the mass from striking the photogate. The length ( l ) of the pen- dulum is the distance from the point on the rod halfway between the strings to the center of the mass.
FIGURE 1. Experimental setup for the pendulum.
The electronic photogate will be used to measure the time ( t ) as the mass makes each swing at its lowest point. The photogate should always be positioned so that the mass
2 Pendulum
blocks the sensor in the photogate when it hangs straight down. The photogate is attached to the graphing calculator for readout.
THEORY A pendulum mass ( m ) at the end of string is subject to two forces: gravity ( F (^) g ) and ten-
sion in the string ( F (^) T ). Since the string doesn’t change length, Newton’s first law, the law of inertia, says that there is no net force on the string from end to end. If the string hangs vertically this means that F (^) T = F (^) g = mg. When the pendulum mass displaced from the vertical at an angle (θ) the tension needed to oppose gravity is F (^) T = mg cosθ. Part of the force of gravity is not opposed by the tension and that results in a net force, Fnet = mg sinθ. From Newton’s second law, that net force causes an acceleration toward the vertical position.
The maximum position displacement ( x ) from the vertical is related to the angle and the length ( l ) of the string in EQ 1.
(EQ 1)
For small angles the sine of the angle is approximately equal to the angle when mea- sured in radians. Radians measure the number of radii around the circumference of cir- cle, so 360º = 2π radians, or 1 radian = 57.3º.
As the pendulum passes through the vertical position, the net force acts to slow the mass down and eventually cause it to reverse direction. The force alternates back and forth, and the mass moves back and forth at a regular rate. The time it takes for the mass to go back and forth in one complete cycle is called the period ( T ). Note that the mass will travel through the verical position twice during each period. If the maximum angle of amplitude isn’t too large the period can be approximated by EQ 2.
(EQ 2)
Squaring both sides gives EQ 3.
(EQ 3)
If the period squared is plotted versus the length, the line should pass through the origin and the slope ( s ) of the line should be the quantity in parentheses in EQ 3. The accelera- tion due to gravity is then related to that slope by EQ 4.
(EQ 4)
DATA COLLECTION 1. Set the strings supporting the pendulum mass to be at least 100 cm and use a 100 g mass. Measure and record the length ( l ) and record the mass ( m ).
2. Check that the calculator is on and running the DATAGATE program. 3. Select SETUP from the main screen. Select PENDULUM from the PHOTOGATE SETUP screen. Temporarily hold the mass out of the center of the Photogate.
x = l sinθ
T 2 π l g
= ---
T^2 4 π
2 g = ---------^ l
g^4 π
2 s =^ ---------
4 Pendulum
OBSERVATIONS For each of these questions make a prediction, and support your answer by answering the question “Why?”.
How consistently can you release the mass at the correct displacement? How would you improve this step of the experiment?
Based on the graph in step 11, does the period appear to depend (appreciably) on mass? Do you feel that you have enough data to answer this question conclusively?
According to your data and graph in step 12, does the period depend (appreciably) on amplitude? Explain.
Based on the graph in step 13, does the period appear to depend (appreciably) on length? Explain.
By comparing graphs in steps 13, 14 and 15, can you conclude that a directly propor- tional relationship exists between the plotted variables? Note that a straight line plot going through the graph's origin (0,0) is necessary for a direct proportion.
How well does your value of g (step 16) agree with the accepted value, 9.8 m/s^2? Give a percent error as part of the observation.
