The period, T, of a pendulum can be approximated by the formula ≈ 2𝜋√𝐿/𝑔, 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
.
To find the approximate length of the pendulum, we can rearrange the formula for the period:
T = 2𝜋√𝐿/𝑔
Given that the period is 2 seconds (T = 2), and the gravitational constant is 9.8 m/s^2 (g = 9.8), we can substitute these values into the equation:
2 = 2𝜋√𝐿/9.8
Now we can solve for the length of the pendulum (L):
2 * 9.8 = 2𝜋√𝐿
19.6 = 2𝜋√𝐿
Dividing both sides by 2𝜋:
19.6 / (2𝜋) = √𝐿
Taking the square of both sides:
(19.6 / (2𝜋))^2 = L
Calculating the value:
L ≈ 9.9134
Therefore, the approximate length of the pendulum is approximately 9.9134 meters.
To find the approximate length of the pendulum, we can rearrange the formula for the period T.
T ≈ 2𝜋√𝐿/𝑔
Given that the period T is 2 seconds and the gravitational constant g is 9.8 m/s^2, we can substitute these values into the formula:
2 ≈ 2𝜋√𝐿/9.8
Dividing both sides of the equation by 2𝜋/9.8:
2/2𝜋 ≈ √𝐿/9.8
Simplifying:
1/𝜋 ≈ √𝐿/9.8
To isolate 𝐿, we can square both sides of the equation:
(1/𝜋)^2 ≈ (√𝐿/9.8)^2
1/𝜋^2 ≈ 𝐿/9.8
Multiplying both sides of the equation by 9.8:
9.8/𝜋^2 ≈ 𝐿
So, the approximate length of the pendulum is 9.8/𝜋^2.
To find the approximate length of the pendulum, we will use the formula for the period of a pendulum: T ≈ 2𝜋√(L/g), where T is the period, L is the length of the pendulum, and g is the gravitational constant.
Given that the period of the pendulum is 2 seconds (T = 2 s) and the gravitational constant on Earth is 9.8 m/s^2 (g = 9.8 m/s^2), we can substitute these values into the formula and solve for L.
T ≈ 2𝜋√(L/g)
2 ≈ 2𝜋√(L/9.8)
Next, let's isolate L by dividing both sides of the equation by 2𝜋 and squaring both sides to get rid of the square root:
(2/(2𝜋))^2 ≈ (√(L/9.8))^2
1/(2𝜋)^2 ≈ L/9.8
Simplifying further:
1/(4𝜋^2) ≈ L/9.8
To find the length L, multiply both sides of the equation by 9.8:
9.8/(4𝜋^2) ≈ L
Using a calculator to evaluate the right side of the equation gives us approximately:
L ≈ 0.248
Therefore, the approximate length of the pendulum is 0.248 meters.