Question

When a pendulum with a period of 2.00000 s is moved to a new location from one where the acceleration due to gravity was 9.80 m/s2, its new period becomes 1.99824 s. By how much does the acceleration due to gravity differ at its new location? How is the period of the pendulum related to the length of the pendulum and the acceleration due to gravity at the location of the pendulum? What happens to the length of the pendulum as it is moved to the new location?

Answer 1

1. By 0.02 m/s^2

The period of a pendulum is given by:

$T=2 \pi \sqrt{\frac{L}{g}}$

where

L is the length of the pendulum

g is the gravitational acceleration

Initially, we know:

T = 2.00000 s is the period of the pendulum

g = 9.80 m/s^2 is the acceleration due to gravity at the original location

We can solve the equation for L in order to find the length of the pendulum:

$L=\frac{T^2}{(2 \pi)^2}g=\frac{(2.0000 s)^2}{(2 \pi)^2}(9.80 m/s^2)=0.99396 m$

The length of the pendulum does not change when it is moved to the new location, so we can use the same equation with $T=1.99824 s$ (the new period) and solving it for g to find the acceleration due to gravity at the new location:

$g=L\frac{(2 \pi)^2}{T^2}=(0.99396 m)\frac{(2 \pi)^2}{(1.99824 s)^2})=9.82 m/s^2$

So, the change in gravitational acceleration is

$\Delta g = g_2 - g_1 = 9.82 m/s^2-9.80 m/s^2 = 0.02 m/s^2$

2) the period of the pendulum is directly proportional to the square root of the length, L, and inversely proportional to the square root of the gravitational acceleration, g.

The period of a pendulum is given by:

$T=2 \pi \sqrt{\frac{L}{g}}$

where

L is the length of the pendulum

g is the gravitational acceleration

So, we see that the period of the pendulum is directly proportional to the square root of the length, L, and inversely proportional to the square root of the gravitational acceleration, g.

3) The length of the pendulum does not change

The length of the pendulum does not depend on the location: in fact, only the value of the gravitational acceleration, g, depends on the location, therefore the length of the pendulum, L, does not change.