The formula for the period of a simple pendulum is T is equals to 2 π ...
Calculation of Fractional Error in the Value of g
The period of a simple pendulum is given by the formula:
T = 2π√(l/g)
Where T is the period of the pendulum, l is the length of the pendulum and g is the acceleration due to gravity.
Fractional Error in the Measurement of the Period
If the measured value of the period is t-x, where x is the fractional error, then the actual value of the period is:
T' = T + Tx
Substituting the value of T from the above equation in the formula of the period, we get:
T' = 2π√(l/g + lx)
The fractional error in the calculated value of the period is:
(T' - T)/T = lx/√(l/g)
As x is small, the fractional error in the measurement of the period is negligible.
Fractional Error in the Measurement of the Length
If the measured value of the length is L-y, where y is the fractional error, then the actual value of the length is:
l' = l + Ly
Substituting the value of l from the above equation in the formula of the period, we get:
T' = 2π√((l + Ly)/g)
The fractional error in the calculated value of the period is:
(T' - T)/T = (Ly/2l) + (Ly/2g√((l + Ly)/g))
The maximum value of the fractional error occurs when y = -1/2. Substituting this value in the above equation, we get:
(T' - T)/T ≤ 1/4
Conclusion
Thus, the fractional error in the calculated value of g is not greater than 1/4, even if the fractional errors in the measurement of the period and the length are -x and -y respectively.