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Simple pendulum:

# Objective

Our objectives are;

• To plot a L-T2 graph using a simple pendulum.
• To find the effective length of the simple pendulum for a given time period using the graph.
• To calculate the acceleration due to gravity at a place

# The Theory

### What is a Simple Pendulum?

An ideal simple pendulum consists of a heavy point mass (called bob) tied to one end of a perfectly inextensible, flexible and weightless string. In practice, we make it by tying a metallic spherical bob to a fine cotton stitching thread.

### Length of a Simple Pendulum

The distance between the point of suspension of the pendulum and its Centre of Gravity (C.G.), which is the C.G. of the bob, is called the length of the simple pendulum. It is represented using the aphabet ( l ).

### Time Period of a Simple Pendulum

Time period is the time taken by the bob of the simple pendulum to make one complete oscillation. It is represented by the letter T.

### Finding the acceleration due to gravity

The time period of a simple pendulum depends on the length of the pendulum (l) and the acceleration due to gravity (g), which is expressed by the relation,

For small amplitude of oscillations,

ie;

If we know the value of l and T, we can calculate the acceleration due to gravity, g at that place.
The L-T2 graph

We can plot a graph between l and T2 by taking l along the X axis and T2 along the Y axis. The graph is a straight line.

From the graph,

## Learning Outcomes

• Students get to know about the simple pendulum.
• Students understand the relation between the length of the pendulum, time period and the acceleration due to gravity at a place.
• They get an idea about the importance of the l-T2 grap

## Reviews

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# Materials Required

• A clamp with stand
• A split cork
• A bob
• Vernier calliper
• Stop /watch
• Metre scale.

# Real Lab Procedure

1. Find the vernier constant and zero error of the vernier calipers and record it.
2. Determine the mean diameter of the simple pendulum bob using the vernier calipers.
3. Find the mean radius of the bob and represent it using ‘r’.
4. Attach a string to the bob. The length of the pendulum, l is adjusted by measuring a length of (l-r) from the top of the bob.
5. Put ink marks M1,M2 and M3 on the thread at distance of 50cm,60cm and 70cm from the C.G of the bob .
6. Pass the thread through the splited cork with the 50 cm mark at the bottom of the cork and tighten the two cork pieces between the clamp.
7. Fix the clamp in a stand kept on the table such that the height that the bob is just 2 cm above the laboratory floor.
8. Mark a point A on the floor just below the position of the bob at rest.
9. The equilibrium position of the pendulum is indicated by drawing a vertical line with a chalk on the edge of the table, just behind the string.
10. Find the least count and the zero error of the stop watch. Bring its hands to the zero position.
11. Move bob using the hand at an angle not more than 450 and leave it. See that the bob returns over the line without spinning.
12. The stop watch is started when the pendulum crosses the equilibrium position to any one side.
13. When it passes the equilibrium position in the same direction the next time it has completed one oscillation.
14. Just when the 20th oscillation is complete, count 20 and at once stop the stop watch.
15. Note the total time taken for twenty oscillations from the position of both the hands of the watch.
16. As we need two observations for the same length, repeat steps 12 to 15 one more time.
17. Repeat the experiment for lengths 60cm, 70cm, 80cm, 90 cm, 100cm, 110 cm, 120cm and 130cm.
18. In each case    is calculated. In all cases it is found that is a constant.
19. The mean value of   is calculated and then the acceleration due to gravity is calculated using the relation (2).

## To draw the l-T2 graph

The experiment is preformed as explained above. A graph is drawn with l along X axis and T2 along Y axis. The graph is a straight line, as shown in the figure. To find the length of the second’s pendulum A second’s pendulum is one for which the period of oscillation is 2 seconds. From the graph the length l corresponding to T2=4 s2 is determined. This gives the length of the second’s pendulum. To find the length of the pendulum whose period is 1.5 seconds The length l corresponding to T2 =1.52=2.25 is determined from the graph. To find the period (T) for a length 105cm T2 corresponding to l=105 cm is determined from the graph. The square root of this gives T, the period of the pendulum for a length 105 cm. From the graph = ------cm/s2

# Procedure

1. Select the environment to perform the experiment from the 'Select Environment' drop down list.
2. Select the shape of the bob of the pendulum from the 'Select Shape' drop down list.
3. Select the material of the bob from the 'Select Material' drop down list.
4. Select the type of the wire to be used from the 'Select Wire' drop down list.
5. Use the 'Change Length' slider to change the length of the pendulum.
6. Use the 'Change Dimension' slider to change the dimension of the bob used.
7. Now release the bob.
8. Clicking on the 'Show Protractor' button helps us to ensure that the angle of swing does not exceeds 450.
9. Now click on 'Play /Pause' button to start the stopwatch. We can alternatively click on the the 'Start' or 'Stop' button on the stopwatch.
10. The time for twenty oscillations is noted.
11. Now the real lab procedure from steps 12 to 18 can be followed to complete the observations for finding the acceleration due to gravity.
12. Clicking on the 'Answer' button displays the acceleration due to gravity for the corresponding environment.

# Observations

To find the diameter of the bob 1 M S D = 1mm 10 V S D =9 M S D 1 V S D=9/10 M S D=0.9 mm Vernier Constant, V.C.= 1 M.S.D.-1 V.S.D. = (1-0.9) mm = 0.1 mm = 0.01cm. Zero error of vernier callipers(e)
1. e=..............cm
2. e=..............cm
3. e=..............cm
Mean zero error e =.......cm Mean zero correction c = -e = ......cm
 SL No Main Scale Reading MSR(cm) Vernier scale Reading VSR(dvs) (VSRxL.C) (cm) Diameter of the bob,D=MSR+(VSRx L.C)+c(zero correction) (cm) Mean Diameter,D
Mean Diameter of the Bob, D= ……………cm Mean radius of the bob, r =D/2 = .........cm Least count of stop watch =..........s Zero error of stop watch =...........s Zero correction of stop watch =...........s Table for length () and time (T)
 Sl No (l-r)cm Length of the pendulum l (cm) Time for 20 oscillations Time Period T2 (s) t1(s) t2(s) Mean t(s)

# Calculations

Mean value of .=…………..ms-2 The acceleration due to gravity, g = …………m/s2 Acceleration due to gravity from graph Value or l = AB = -----cm Value for T= BC = -----------cm AB / BC = ……….. Acceleration due to gravity, g=---------m/s2

# Result

1.  Acceleration due to gravity (g) at the place
• By calculation =………….ms-2
• From the graph =………….ms-2
• Mean g =………….ms-2
2. Length of the seconds pendulum =………….m
3. Length of the pendulum whose period is 1.5 s=……..m
4. Period of the pendulum of length 105 cm=…….s