Investigate the oscillations of a simple Pendulum

                               

-mg sin θ = ma,

 

The negative sign indicates that it the direction is towards O.

        

         P

        θ

        l        T

        B

        y        ●

        O        θ

        mg sin θ        

        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=2π√ l/g          

So, while plotting a graph:-

 

T 2 = 4π 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.

The independent variable is the length of the string from the retort stand to the bob.

The controlled variables are the density of the bob and the mass of the bob taken every time the experiment is done.

Procedure

       

1. Attach the pendulum bob to one end of the thread. Next, place the other end between the two blocks of wood. Set the arrangement as shown in the diagram on the previous page.
2. Let the length, l, of the thread be 100 cm. Allow the pendulum to oscillate through a small angle, θ, of about 5˚.
3. Start the stop watch when the bob it on the extreme right position.
4. Find the time, t1, taken to complete 20 oscillations. Then repeat the 20 oscillations to find the time, t2, taken.
5. Evaluate the mean time, t, and the periodic time T. ﴾T = t / 20)
6. Repeat the steps to get 5 more value of t for values length of l varying from 100 cm to 50 cm.
7. Tabulate the values of l, t1, t2, t, T and T 2 .
8. Plot the graph of T against l and determine the slope, m.
9. Calculate the acceleration due to gravity from the formula g = (4π 2 / m) cm s -1

Tabulation

Calculations

m = y/x

m = 0 .4  ÷  0.1

m = 4

                                                   T 2 = 4π 2 ×l                    y = m x

                                                        g

So, m = (4π 2 /g)

g= 4π 2 /4

g= π 2

g= 9.87m s-2

Precautions

• Do not take any elliptical oscillations in to consideration
• 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 %