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The formula for the period of a simple pendulum is T is equals to 2 π √l/g such a pendulum is used to determine the fractional error in the measurement of the period t is -x and that in the measurement of the length L is -y the fractional error in the calculated value of g is not the greater than ?
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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.
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The formula for the period of a simple pendulum is T is equals to 2 π √l/g such a pendulum is used to determine the fractional error in the measurement of the period t is -x and that in the measurement of the length L is -y the fractional error in the calculated value of g is not the greater than ?
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