The period, T, of a pendulum can be approximated by the formula ≈ 2𝜋√𝐿𝑔, 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

Question

The period, T, of a pendulum can be approximated by the formula ≈ 2𝜋√𝐿𝑔, 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

Answer 1

We can rearrange the formula for the period of a pendulum to solve for the length of the pendulum, L:

T ≈ 2𝜋√𝐿𝑔

Dividing both sides of the equation by 2𝜋√𝑔 gives:
T / (2𝜋√𝑔) ≈ √𝐿

Now, we can substitute the given values for the period and gravitational constant:
2s / (2𝜋√(9.8m/s²)) ≈ √𝐿

Simplifying:
1 / (√(9.8𝜋)) ≈ √𝐿

To find the approximate length of the pendulum, we square both sides of the equation:
1 / (√(9.8𝜋))² ≈ 𝐿

Simplifying further:
1 / (9.8𝜋) ≈ 𝐿

Thus, the approximate length of the pendulum is 1 / (9.8𝜋) meters.

Answer 2

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 ≈ 2𝜋√𝐿𝑔. Rearranging this formula:

T = 2𝜋√𝐿𝑔

Now, we can substitute the value of T (2 seconds) and g (9.8 m/s^2) into the formula:

2 = 2𝜋√𝐿(9.8)

Simplifying further:

1 = 𝜋√𝐿(9.8)

To isolate L, we'll divide both sides by 𝜋√(9.8):

1 / (𝜋√(9.8)) = √𝐿

Now, square both sides to get rid of the square root:

(1 / (𝜋√(9.8)))^2 = 𝐿

Calculating this expression:

(1 / (𝜋√(9.8)))^2 ≈ 0.010148

So, the approximate length of the pendulum is 0.010148 meters or 10.148 centimeters.

Answer 3

To find the approximate length of the pendulum, we can rearrange the formula as follows:

T ≈ 2𝜋√𝐿𝑔

Squaring both sides of the equation:

T^2 ≈ (2𝜋)^2𝐿𝑔

Now, we can substitute the given values:

(2 s)^2 ≈ (2𝜋)^2𝐿(9.8 m/s^2)

4 s^2 ≈ 4𝜋^2𝐿(9.8 m/s^2)

Dividing both sides by 4𝜋^2(9.8 m/s^2):

𝐿 ≈ 4 s^2 / 4𝜋^2(9.8 m/s^2)

Simplifying:

𝐿 ≈ s^2 / 𝜋^2(9.8 m/s^2)

𝐿 ≈ (2 m)^2 / 𝜋^2(9.8 m/s^2)

𝐿 ≈ 4 m^2 / (9.8𝜋^2) m/s^2

Using a calculator to evaluate the expression:

𝐿 ≈ 0.1292 m

Therefore, the approximate length of the pendulum is 0.1292 meters.