## APPARATUS REQUIRED

i) Simple pendulum

ii) Stop watch

iii) Meter scale

iv) Clamp and thread

## THEORY

### The principle of Least Squares:

least-square is one of the most commonly used methods in numerical computation. Essentially it is a technique for solving a set of equations where there are more equations than unknowns ie. an overdetermined set of equations. This set of notes shows the origins of a particular form of the algorithm, batch linear least-squares.

The most important application is in data fitting. The best fit in the least squares sense minimizes the sum of squared residuals, a residual being the difference between an observation value and the fitted value provided by a model. When the problem has substantial uncertainties

In the independent variable, Then simple regression and least square methods leave problems: in such cases, the methodology required for fitting errors in variable models may be considered instead of that for least squares.

**Least squares depending problem fall into two categories,**

Linear or ordinary least squares and non-linear least squares depending on whether or not the residuals are linear are all unknown. This problem occurs in statistical regression analysis; it has a closed-form solution. A closed form solution is any formula that can be evaluated in a finite number of standard operations. The non-linear problem has no closed form solution and is usually solved by iterative refinement at which iteration is approximated by a linear one, and thus the core calculation is similar in both cases.

A residual is defined as the difference between the actual value of the dependent variable and the value predicted by the model.

γ_{i} = y_{i} – f(x_{i}, β)

An example of a model is that of the straight line in two dimensions. Denoting the intercept as β_{o} and the slope as β_{1} the model function is given by

f(x, β) = β_{o} + β_{1}x

let, the equation of straight line y = a+bx – – – – – – (i)

which is fitted to given points (x_{1}y_{1}) (x_{2} y_{2}) (x_{3} y_{3})- – – – – – – – (x_{n} y_{n})

let R, be the theoretical value for x_{1} than, E_{1} = y_{1}-R_{1} = y_{1}– (d+ bx_{1})

On solving equation (iii) we get the value of ‘a’ and ‘b’ and putting this value in equation (i) we get the value of equation in line of best fit.

## OBSERVATION

Diameter of ball(d) = M.SR + C.SR x L.C.

= 18+50× 0.01 =18.50m

r=d/2 = 18.50/2 = 9.25cm = 0.925m

Time for 20 Oscillation | Time forPer oscillation | Effectivelength(l) | g(ms^{2})4𝝅^{2}l/T2 | Mean(g) | T^{2} |

17 | 17/20 =0.85 | 20 | 195.11 | 0.72 | |

25 | 25/20 =1.25 | 40 | 1011.24 | 1.56 | |

31 | 31/20 = 1.55 | 60 | 985.96 | 1021.95 | 2.40 |

35 | 35/20 = 1.75 | 80 | 1031.07 | 3.06 | |

40 | 40/20 = 2 | 100 | 985.96 | 4 |

Slope of (m) =g/4𝝅^{2} = 1021.95/ 4×(3.14)^{2} = 25.92 cms^{-2}

Time for 20 Oscillation | Time for 1 Oscillation | T^{2} | X_{i}X _{1}+X_{2}+- – | x̄ | Effective length(l) | Y_{i}Y _{1}+y_{2}+- – | Ȳ |

17 | 0.05 | 0.75 | 20 | ||||

25 | 1.25 | 1.56 | 11.74 | 2.348 | 40 | ||

31 | 1.55 | 2.40 | 60 | 300 | 60 | ||

35 | 1.75 | 3.06 | 80 | ||||

40 | 2 | 4 | 100 |

which is compared with equation of straight line y= mx

If l= 20 then l= 25.92 × 0.72 = 18.7

l= 40 then l= 25.92 × 1.56 = 40.4

l= 60 then l= 25.92 × 2.4 = 62.2

l= 80 then l= 25.92 × 3.06 = 79.31

l= 100 then l= 25.92 × 4 = 103.68

## RESULT:

Thus the slope m= 25.92 and acceleration due to gravity (g) = 10.21 m/sec^{2}

## PERCENTAGE ERROR

Observed value of (o)g= 10.21 m/s²

standard value of (s)g = 9.8 m/s²

percentage errors=| 9.8-10.21/9.8 ×100%|

= 4.32

CONCLUSION:

Hence, the acceleration due to gravity can be determined by using a simple pendulum.