Simple Pendulum experiment to plot L-T graphs

Aim

Using a Simple Pendulum plot L-T and L-T² graphs, hence find the effective length of second's pendulum using appropriate graph.

Apparatus and material required

Clamp stand; a split cork; a heavy metallic (brass/iron) spherical bob with a hook; a long, fine, strong cotton thread/string (about 2.0 m); stop-watch; metre scale, graph paper, pencil, eraser.

Description time measuring devices in a school laboratory

The most common device used for measuring time in a school laboratory is a stop-watch or a stop-clock (analog). As the names suggest, these have the provision to start or stop their working as desired by the experimenter.

(a) Stop-Watch AnalogA stop-watch is a special kind of watch. It has a multipurpose knob or button (B) for start/stop/back to zero position [Fig. E 6.1(b)]. It has two circular dials, the bigger one for a longer second's hand and the other smaller one for a shorter minute's hand. The second's dial has 30 equal divisions, each division representing 0.1 second. Before using a stop-watch you should find its least count. In one rotation, the seconds hand covers 30 seconds (marked by black colour) then in the second rotation another 30 seconds are covered (marked by red colour), therefore, the least count is 0.1 second.

(b) Stop-ClockThe least count of a stop-watch is generally about 0.1s [Fig. E 6.1(b)] while that of a stop-clock is 1s, so for more accurate measurement of time intervals in a school laboratory, a stop-watch is preferred. Digital stop-watches are also available now. These watches may be started by pressing the button and can be stopped by pressing the same button once again. The lapsed time interval is directly displayed by the watch.

Terms and Conditions

1. Second's pendulum: It is a pendulum which takes precisely one second to move from one extreme position to other. Thus, its times period is precisely 2 seconds.
2. Simple pendulum: A point mass suspended by an inextensible, mass less string from a rigid point support. In practice a small heavy spherical bob of high density material of radius r, much smaller than the length of the suspension, is suspended by a light, flexible and strong string/thread supported at the other end firmly with a clamp stand. Fig. E 6.2 is a good approximation to an ideal simple pendulum
3. Effective length of the pendulum: The distance L between the point of suspension and the centre of spherical bob (centre of gravity), L = l + r + e, is also called the effective length where l is the length of the string from the top of the bob to the hook, e, the length of the hook and r the radius of the bob.

The simple pendulum executes Simple Harmonic Motion (SHM) as the acceleration of the pendulum bob is directly proportional to its displacement from the mean position and is always directed towards it.

The time period (T) of a simple pendulum for oscillations of small amplitude, is given by the relation

T = 2π radic; L/g (e6.1)

where L is the length of the pendulum, and g is the acceleration due to gravity at the place of experiment.

Eq. (6.1) may be rewritten as

T²=4π ²L//g (e6.2)

Procedure

1. Place the clamp stand on the table. Tie the hook, attached to the pendulum bob, to one end of the string of about 150 cm in length. Pass the other end of the string through two half-pieces of a split cork.
2. Clamp the split cork firmly in the clamp stand such that the line of separation of the two pieces of the split cork is at right angles to the line OA along which the pendulum oscillates [Fig. E 6.2(a)]. Mark, with a piece of chalk or ink, on the edge of the table a vertical line parallel to and just behind the vertical thread OA, the position of the bob at rest. Take care that the bob hangs vertically (about 2 cm above the floor) beyond the edge of the table so that it is free to oscillate.
3. Measure the effective length of simple pendulum as shown in Fig. E 6.2(b).
4. Displace the bob to one side, not more than 15 degrees angular displacement, from the vertical position OA and then release it gently. In case you find that the stand is shaky, put some heavy object on its base. Make sure that the bob starts oscillating in a vertical plane about its rest (or mean) position OA and does not (i) spin about its own axis, or (ii) move up and down while oscillating, or (iii) revolve in an elliptic path around its mean position.
5. Keep the pendulum oscillating for some time. After completion of a few oscillations, start the stop-watch/clock as the thread attached to the pendulum bob just crosses its mean position (say, from left to right). Count it as zero oscillation.
6. Keep on counting oscillations 1,2,3,..., n, everytime the bob crosses the mean position OA in the same direction (from left to right). Stop the stop-watch/clock, at the count n (say, 20 or 25) of oscillations, i.e., just when n oscillations are complete. For better results, n should be chosen such that the time taken for n oscillations is 50 s or more. Read, the total time (t) taken by the bob for n oscillations. Repeat this observation a few times by noting the time for same number (n) of oscillations. Take the mean of these readings. Compute the time for one oscillation, i.e., the time period T ( = t/n) of the pendulum.
7. Change the length of the pendulum, by about 10 cm. Repeat the step 6 again for finding the time (t) for about 20 oscillations or more for the new length and find the mean time period. Take 5 or 6 more observations for different lengths of penduLum and find mean time period in each case.
8. Record observations in the tabular form with proper units and significant figures.
9. Take effective length L along x-axis and T 2 (or T) along y-axis, using the observed values from Table E 6.1. Choose suitable scales on these axes to represent L and T 2 (or T ). Plot a graph between L and T 2 (as shown in Fig. E 6.4) and also between L and T (as shown in Fig. E 6.3). What are the shapes of L-T 2 graph and L-T graph? Identify these shapes.

Observations

Radius (r) of the pendulum bob (given) = ... cm

Length of the hook (given) (e) = ... cm

Least count of the metre scale = ... mm = ... cm

Least count of the stop-watch/clock = ... s

Table E 6.1: Measuring the time period T and effective length L of the simple pendulum

Plotting Graph

1. L vs T graphsPlot a graph between L versus T from observations recorded in Table E 6.1, taking L along x-axis and T along y-axis. You will find that this graph is a curve, which is part of a parabola as shown in Fig. E 6.3.
2. L vs T²graphPlot a graph between L versus T from observations recorded in Table E 6.1, taking L along x-axis and T along y-axis. You will find that this graph is a curve, which is part of a parabola as shown in Fig. E 6.3.
3. From the T² versus L graph locate the effective length of second's pendulum for T² = 4s².

Result

1. The graph L versus T is curved, convex upwards.
2. The graph L versus T² is a straight line.
3. The effective length of second's pendulum from L versus T² graph is ... cm.

Sources of Error

1. Friction at the pulleys may persist even after oiling.
2. Slotted weights may not be accurate.
3. Slight inaccuracy may creep in while marking the position of thread.

Note:The radius of bob may be found from its measured diameter with the help of callipers by placing the pendulum bob between the two jaws of (a) ordinary callipers, or (b) Vernier Callipers, as described in Experiment E 1.1 (a). It can also be found by placing the spherical bob between two parallel card boards and measuring the spacing (diameter) or distance between them with a metre scale.