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.

Question

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.

Answer 1

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

T ≈ 2𝜋√𝐿/𝑔

Let's plug in the given values and solve for L:

2 s ≈ 2𝜋√L/9.8 m/s^2

Divide both sides of the equation by 2𝜋:

2 s / 2𝜋 ≈ √L/9.8 m/s^2

Square both sides of the equation to eliminate the square root:

(2 s / 2𝜋)^2 ≈ L/9.8 m/s^2

(4s^2 / 4𝜋^2) ≈ L/9.8 m/s^2

Multiply both sides of the equation by 9.8 m/s^2:

9.8 m/s^2 * (4s^2 / 4𝜋^2) ≈ L

L ≈ 9.8 m/s^2 * (4s^2 / 4𝜋^2)

Simplify the equation:

L ≈ (9.8 * 4s^2) / (4𝜋^2)

L ≈ (9.8 * 4s^2) / (4 * 3.14^2)

L ≈ (39.2s^2) / (39.478)

L ≈ s^2

Therefore, the approximate length of the pendulum is s^2, where s is the period of 2 seconds.

Answer 2

To find the approximate length of the pendulum, we will use the given formula:

T ≈ 2𝜋√(L/g)

We are given that the period of the pendulum is 2 seconds (T=2 s) and the gravitational constant on Earth is 9.8 m/s² (g=9.8 m/s²).

Substituting these values into the formula, we have:

2 ≈ 2𝜋√(L/9.8)

Now, we can solve for the length of the pendulum, L.

Divide both sides of the equation by 2𝜋:

2/(2𝜋) ≈ √(L/9.8)

Simplify:

1/𝜋 ≈ √(L/9.8)

To isolate L, square both sides of the equation:

(1/𝜋)² ≈ (L/9.8)

1/𝜋² ≈ (L/9.8)

Now, multiply both sides of the equation by 9.8:

9.8/𝜋² ≈ L

Finally, calculate the value using an approximate value of 𝜋 (such as 3.14159):

L ≈ 9.8/3.14159²

L ≈ 9.8/9.87

L ≈ 0.993 meters

Therefore, the approximate length of the pendulum is 0.993 meters when it has a period of 2 seconds.

Answer 3

To find the approximate length of the pendulum, we can rearrange the formula and solve for L:

T ≈ 2𝜋√𝐿/𝑔

Given that T = 2 s and g = 9.8 m/s^2, we can plug in these values:

2 ≈ 2𝜋√𝐿/9.8

Next, we can isolate √𝐿 by multiplying both sides of the equation by 9.8/2𝜋:

2(9.8/2𝜋) ≈ √𝐿

9.8/𝜋 ≈ √𝐿

Next, we can square both sides of the equation to eliminate the square root:

(9.8/𝜋)^2 ≈ 𝐿

Now we can calculate:

(9.8/𝜋)^2 ≈ 𝐿

(9.8/3.14)^2 ≈ 𝐿

(3.12)^2 ≈ 𝐿

9.7344 ≈ 𝐿

Therefore, the approximate length of the pendulum is 9.7344 meters.