If the length of pendulum is 0.8+/-0.01m and time peroid is 2.5+/-0.012 sec. Then percentage error in acceleration due to gravity will be

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To find the percentage error in acceleration due to gravity, we need to first calculate the value of acceleration due to gravity using the given values.

The formula to calculate the period of a pendulum is:

T = 2π√(L/g)

Where:
T = Time period of the pendulum
L = Length of the pendulum
g = Acceleration due to gravity

Rearranging the formula to solve for g, we have:

g = 4π²L / T²

Using the given values, we have:

L = 0.8 m
ΔL = 0.01 m
T = 2.5 sec
ΔT = 0.012 sec

Calculating the value of g using the given length and time period:

g = 4π²(0.8) / (2.5)²
= 4π²(0.8) / 6.25
= (39.478) / 6.25
≈ 6.3168 m/s²

Now, let's calculate the maximum and minimum values for g using the maximum and minimum values for L and T.

Maximum value of g:
L_max = L + ΔL
T_max = T + ΔT

g_max = 4π²(L_max) / (T_max)²
= 4π²(0.8 + 0.01) / (2.5 + 0.012)²
≈ 6.4837 m/s²

Minimum value of g:
L_min = L - ΔL
T_min = T - ΔT

g_min = 4π²(L_min) / (T_min)²
= 4π²(0.8 - 0.01) / (2.5 - 0.012)²
≈ 6.1498 m/s²

Now, let's calculate the percentage error in acceleration due to gravity:

Percentage Error = ((|g_max - g_min|) / g) × 100

= ((|6.4837 - 6.1498|) / 6.3168) × 100

= (0.3339 / 6.3168) × 100

≈ 5.29%

Therefore, the percentage error in acceleration due to gravity is approximately 5.29%.

To calculate the percentage error in acceleration due to gravity, we first need to find the individual errors in the length of the pendulum and the time period.

1. For the length of the pendulum:
The given length is 0.8 ± 0.01 m. The error here is ± 0.01 m.

2. For the time period:
The given time period is 2.5 ± 0.012 sec. The error here is ± 0.012 sec.

Now, let's calculate the percentage error in acceleration due to gravity.

The formula for the period of a pendulum (T) is:
T = 2π √(L/g)

Where:
T = time period
L = length of pendulum
g = acceleration due to gravity.

Taking the natural logarithm of both sides, we get:
ln(T) = ln(2π) + ln(√(L/g))

Differentiating both sides:
(1/T) × dT = (1/√(L/g)) × (0.5) × (1/g) × dg

Rearranging the equation, we have:
dg/g = (dT/T) × (√(L/g)/2)

To find the approximate relative error in g (which is percentage error), we can use the following equation:
% error in g = (dg/g) × 100

Substituting the given values, we have:
% error in g = [(dT/T) × (√L/g)/2] × 100

Now, let's substitute the values and calculate the percentage error in acceleration due to gravity:
% error in g = [(0.012/2.5) × (√(0.8)/(9.81))/2] × 100

Simplifying this expression will give us the percentage error in acceleration due to gravity.

T=2PI(sqrt(l/g)

t^2=(2PI)^2 l/g

g= (2PI)^2*length/time^2

Now error in g is
9.8* (2PI)^2 * g*sqrt {[.01/.8]^2 + [.012/2.5]^2 )

http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm