Problem 1.3: The length of a pendulum is (100 ± 0.1) cm. If the acceleration of the free fall is (9.8 ± 0.1) m/s², calculate the percentage uncertainty in the time period of the pendulum.

Solution

Given: Length of the pendulum = (100 ± 0.1) cm

Acceleration due to gravity = (8.9 ± 0.1) cm

Required: T o find the Percentage uncertainty in the time period of the pendulum.

Strategy

• Time period of a pendulum is, T = 2π(l/g)^1/2   .
• Quotient rule for percentage uncertainty is Δx = ±(Δy + Δz) .
• Rule for percentage uncertainty in power is “multiply the percentage uncertainty with the power in the final result”.

Now the percentage uncertainty in length ‘l’ is

Percentage uncertainty in ‘g’ is

Use quotient rule to find percentage uncertainty in l/g.

Percentage uncertainty in l/g = ±(0.1 + 1.02) = ±1.12

Use the formula for time period ‘T’ and power rule to find the percentage uncertainty in the time period.

Percentage uncertainty in time period