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# Investigate the oscillations of a simple Pendulum

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Introduction

Experiment no. 1

Aim: -

To investigate the oscillations of a simple pendulum and find the value of ‘g’, acceleration due to gravity, in the laboratory.

Apparatus: -

Retort stand with clamp                                           Pendulum bob

Piece of thread (≈ 110 cm)                                       Stop watch

Two small wooden blocks                                       metre rule

Hypothesis and Theory:-

In this experiment the set up below is arranged. This is a simple pendulum system which is a simple harmonic motion. In the figure below consider a small mass m attached to the end of a length l of wire. The other end is attached to a fixed point P and the mass oscillates along the arc. Suppose the mass during the motion at a point B at an instant, such that OB = y (displacement) and the angle at that moment is θ to PO. The downward force by the mass is mg

Middle

mg

When θ is small then sin θ = θ in radians and θ= y/l (as the angle in a circle is equal to its radius multiplied by its arc.).

− mg θ = ma     (θ = y / l)

− mg (y/l) = ma

So, a= − (g/l) y

As the acceleration is directly proportional to the displacement so it is proven that the motion of a simple pendulum is simple harmonic.

So,

g y =  w2 y

l

Thus, w2 = g/l

T=2π/w

T=2πl/g

So, while plotting a graph:-

T 2 = 2 ×l                    y = m x

g

So, plot a graph of T 2 against l and the slope of the graph will be = (4π 2 /g)

Therefore g can be calculated.

The value of g will be approximately in this range:-

9.78 ≤  g  ≤9.88

Variables

The dependent variable is the time taken for the bob to complete one oscillation.

Conclusion

The length of the thread must be measured from the point of suspension to the centre of gravity of the bob.Ignore the first few oscillations. One time steady oscillations.The angular displacement should not exceed 5˚.Avoid doing the experiment in front of the air conditioner.

Modifications

Use varying masses or bob of different density that is of different variable. Calculating the value of  g with many variables will be more accurate than compared to using only one variable.

Evaluation

The hypothesis stated in the beginning is proven correct and the value of g, acceleration due to gravity, is obtained as 9.87 m s-2. This value is very correct as normally seen in this part of the world. The actual value of g is 9.8 m s -2, so the absolute and percentage error can be calculated.

Absolute error:

Absolute error = observed result – actual result

= 9.87 – 9.8

= 0.07

Relative error:

Relative error =

=

= 0.00714

Percentage error:

Percentage error = relative error × 100

= 0.00714 × 100

= 0.714 %

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