The period, T, of a pendulum can be approximated by the formula β 2πβπΏπ, where L is the length of the pendulum and g is the gravitational constant. What is the approximate length of the pendulum if it has a period of 2 s? Note: On Earth the gravitational constant is 9.8 m/s2
We can rearrange the formula for the period of a pendulum to solve for the length of the pendulum, L:
T β 2πβπΏπ
Dividing both sides of the equation by 2πβπ gives:
T / (2πβπ) β βπΏ
Now, we can substitute the given values for the period and gravitational constant:
2s / (2πβ(9.8m/sΒ²)) β βπΏ
Simplifying:
1 / (β(9.8π)) β βπΏ
To find the approximate length of the pendulum, we square both sides of the equation:
1 / (β(9.8π))Β² β πΏ
Simplifying further:
1 / (9.8π) β πΏ
Thus, the approximate length of the pendulum is 1 / (9.8π) meters.
To find the approximate length of the pendulum using the given formula, we need to rearrange the formula and solve for L.
The formula for the period of a pendulum is T β 2πβπΏπ. Rearranging this formula:
T = 2πβπΏπ
Now, we can substitute the value of T (2 seconds) and g (9.8 m/s^2) into the formula:
2 = 2πβπΏ(9.8)
Simplifying further:
1 = πβπΏ(9.8)
To isolate L, we'll divide both sides by πβ(9.8):
1 / (πβ(9.8)) = βπΏ
Now, square both sides to get rid of the square root:
(1 / (πβ(9.8)))^2 = πΏ
Calculating this expression:
(1 / (πβ(9.8)))^2 β 0.010148
So, the approximate length of the pendulum is 0.010148 meters or 10.148 centimeters.
To find the approximate length of the pendulum, we can rearrange the formula as follows:
T β 2πβπΏπ
Squaring both sides of the equation:
T^2 β (2π)^2πΏπ
Now, we can substitute the given values:
(2 s)^2 β (2π)^2πΏ(9.8 m/s^2)
4 s^2 β 4π^2πΏ(9.8 m/s^2)
Dividing both sides by 4π^2(9.8 m/s^2):
πΏ β 4 s^2 / 4π^2(9.8 m/s^2)
Simplifying:
πΏ β s^2 / π^2(9.8 m/s^2)
πΏ β (2 m)^2 / π^2(9.8 m/s^2)
πΏ β 4 m^2 / (9.8π^2) m/s^2
Using a calculator to evaluate the expression:
πΏ β 0.1292 m
Therefore, the approximate length of the pendulum is 0.1292 meters.