Respuesta :
Given that,
The acceleration due to gravity on the surface of the earth = 32 feet/s²
We need to calculate the time
Using formula of time period
[tex]T=2\pi\sqrt{\dfrac{l}{g}}[/tex]
On differentiating with respect to l
[tex]\dfrac{dT}{dL}=2\pi\times\dfrac{1}{2}(\dfrac{l}{g})^{-\frac{1}{2}}\times\dfrac{1}{g}[/tex]
[tex]\dfrac{dT}{dL}=\dfrac{\pi}{g}\times(\dfrac{g}{L})^{\frac{1}{2}}[/tex]
Put the value into the formula
[tex]\dfrac{dT}{dL}=\dfrac{\pi}{32}\times(\dfrac{32}{4})^{\frac{1}{2}}[/tex]
[tex]\dfrac{dT}{dL}=0.277\ sec[/tex]
If the length of the pendulum is decreased to 3.97 feet.
We need to calculate the time
Using formula of time period
[tex]\dfrac{dT'}{dL}=\dfrac{\pi}{g}\times(\dfrac{g}{L})^{\frac{1}{2}}[/tex]
Put the value into the formula
[tex]\dfrac{dT'}{dL}=\dfrac{\pi}{32}\times(\dfrac{32}{3.97})^{\frac{1}{2}}[/tex]
[tex]\dfrac{dT'}{dL}=0.278\ sec[/tex]
We need to calculate the gain time
Using formula for time
[tex]\dfrac{dT''}{dL}=\dfrac{dT'}{dL}-\dfrac{dT}{dL}[/tex]
Put the value into the formula
[tex]\dfrac{dT''}{dL}=0.278-0.277[/tex]
[tex]\dfrac{dT''}{dL}=0.001\ sec[/tex]
Hence, The clock gain the time in 24 hours is 0.001 sec.
