High School

If a pendulum has a period of 2.5 seconds when its length is [tex]l_1[/tex] and a period of 3.5 seconds when its length is [tex]l_2[/tex], what is the relationship between [tex]l_1[/tex] and [tex]l_2[/tex]?

A. [tex]l_1 = l_2[/tex]

B. [tex]l_1 > l_2[/tex]

C. [tex]l_1 < l_2[/tex]

D. Not enough information given to determine.

Answer :

Answer:

In this case, we can see that l1 is less than l2.

Therefore, the correct answer is: c) l1 < l2

Explanation:

The relationship between the length of a pendulum and its period can be determined by the formula for the period of a pendulum:

1. The period (T) of a pendulum is directly proportional to the square root of the length of the pendulum (l). Mathematically, this relationship is represented as T ∝ √l.

2. Given that the period of the pendulum is 2.5 seconds when the length is l1 and 3.5 seconds when the length is l2, we can set up the following proportions:

T1 = 2.5 seconds and l1

T2 = 3.5 seconds and l2

3. Using the formula T ∝ √l, we can write the proportions as:

T1 / T2 = √(l1 / l2)

4. Substitute the given values into the equation:

2.5 / 3.5 = √(l1 / l2)

5. Simplify the equation:

2.5 / 3.5 = √(l1 / l2)

0.714 ≈ √(l1 / l2)

6. Square both sides of the equation to solve for the relationship between l1 and l2:

0.714^2 = l1 / l2

0.51 ≈ l1 / l2

7. From the relationship 0.51 ≈ l1 / l2, we can see that l1 is less than l2. Therefore, the correct answer is: c) l1 < l2