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| Simple Pendulum |
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| A simple pendulum consists of a heavy or point mass suspended by an inextensible or non-elastic thread from a fixed point. |
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| The length of the pendulum is the distance from the point of suspension to the center of gravity of the bob. The resting position of a simple pendulum is known as the mean position. |
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| One complete to and fro movement of a pendulum about its mean position is known as an oscillation or vibration. |
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| During the oscillation, the maximum displacement from its mean position is called amplitude. |
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| PA = PC =
A (Amplitude) |
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| The time taken for one oscillation is known as the time period (T). |
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| The number of oscillations made by the pendulum in one second is called its frequency (symbol n or f). Frequency is measured in hertz (Hz). |
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 The period of a simple pendulum of constant length is independent of its mass, size, shape or material. |
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The period of a simple pendulum is independent of the amplitude of oscillation, provided it is small. |
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The period of a simple pendulum is directly proportional to the square root of length of the pendulum. |
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The period of a simple pendulum is inversely proportional to the square root of the acceleration due to gravity. |
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| From these laws, we can arrive at the formula to determine the periodic time of a simple pendulum. |
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| where l = length of the pendulum |
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| g = gravitational constant. |
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| A seconds pendulum is a pendulum which takes 2 seconds for one oscillation. |
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| Simple Pendulum Experiment |
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To prove that T  l. |
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| Tie the hook of the bob on one end of a thread (more than 1
meter). Clamp the other end firmly between the gap of a split cork which is fixed to the clamp of the retort stand as shown in the diagram. |
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| Measure the length 'l' from the middle of the bob to the lower edge of the split cork. |
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| Pull the bob to one side (making an angle of 10o with the vertical line) and allow it to oscillate in one plane. Using a stopwatch record the time (t) taken for 20 complete oscillations. Repeat the experiment for different lengths (l) and record the corresponding time (t) in the tabular form as shown below: |
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| (i) Draw a graph of l against T2 |
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| (ii) Draw a graph of l against T |
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| Figure (a) |
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| Presentation of data in tabular and graphical form |
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| Results and observations of scientific
experiments should be properly recorded under headed columns and numbered
rows. The headings must include the units in which each quantity is
measured. |
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| Graphs are very helpful in comparing
measurements. The general rule in plotting the graph of a given data is to
plot the independent variable on the horizontal axis (X-axis). The data,
which changes, is plotted on the vertical axis (Y-axis). It is possible to
use the graph to predict the reading of measurements that lie between those
actually made. This is known as interpolation. If the pattern of the graph
is extended beyond the observed data in either direction, a prediction may
also be made of readings lying outside the observed data. Predicting
readings by this method is called extrapolation. |
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| Every graph should have the following: |
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a title (e.g., Load against extension) |
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both axes
labeled with units |
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scales being used on each axis |
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points plotted with crosses or with dot and circle. |
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| The plotted points must be joined with a single straight line or a continuous curve. The graph should cover as much of the area of the graph paper as possible. This requires a sensible choice of scale for each axis. It is unlikely that points plotted from real experimental results will all lie on a straight line or smooth curve due to errors. Hence try to produce a straight line or smooth curve which passes through as many of the plotted points as possible or which leaves an equal distribution of points on either side. Refer the graph shown in figure (a) above which shows a best fitting line. |
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| The diagram shown is a section of Vernier
Calliper. Find the least count of the instrument and also the final reading,
which is the thickness of a metal sheet. |
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| Least count of the Vernier |
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| = 0.01 cm |
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| Reading of the instrument = Main scale reading |
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| + (coinciding v.s. div x least count) |
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| = 4.3 + (8 x 0.01) |
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| = 4.3 + 0.08 |
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| = 4.38 cm |
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| In a Vernier calliper 1 cm of the main scale is divided into 20 equal parts. 19 divisions of the main scale coincide with 20 divisions on the vernier scale. Find the least count of the instrument. |
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The value of one main scale division  |
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| Number of divisions on vernier scale = 20 |
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| Least count of the vernier scale |
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| = 0.025 cm |
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| The circular scale (head scale) of a screw gauge is divided into 100 equal parts and it moves 0.5 mm ahead in one revolution. Find the pitch and the least count. |
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| Pitch = distance moved in one revolution |
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| = 0.5 mm |
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Least count  |
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| = 0.005 mm |
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| The accompanying diagram represents a screw gauge. The circular scale is divided in to 50 divisions and the linear scale is divided into
millimeters. If the screw advances by 1 mm when the circular scale makes 2 complete revolutions, find the least count of the instrument and the reading of the instrument in figure below. |
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Pitch of the screw  |
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| = 0.5 mm |
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Least count  |
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| = 0.01 mm |
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| Reading = L.S. reading + (coinciding circular scale x least count) |
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| = 3.5 mm + (32 x 0.01) |
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| = 3.5 + 0.32 |
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| = 3.82 mm |
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| Two simple pendulums are of lengths 40 cm and 1.6 m respectively. What will be the ratio of their time periods? |
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Since  therefore, |
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| or T1 : T2 = 1 : 2 |
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| The diagram below shows part of the main
scale and vernier of a calliper, which is used to measure the diameter of a
metal ball. Find the radius of the ball (sphere). |
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Least count  |
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| = 0.01 cm |
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| Reading (diameter) = M.S. reading + (coinciding V.S. reading x Least count) |
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| = 4.3 cm + (7 x 0.01) |
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| = 4.3 + 0.07 |
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| = 4.37 cm |
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| Diameter = 4.37 cm |
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 Radius  |
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| = 2.185 cm |
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| = 2.18 cm |
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| To the required number of significant figures. |
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| The diagram shown below shows a part of the linear scale and head scale (circular scale) of a micrometer screw which is used to measure the thickness of a glass plate. Calculate the thickness (pitch=0.5 mm), Total number of divisions on head scale = 50. |
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| Pitch = 0.5 mm |
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 Least count  |
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| = 0.01 mm |
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| Thickness = Linear S.D + (Circular S.D. x Least count) |
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| = 3.5 mm + (11 x 0.01) mm |
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| = 3.61 mm. |
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