Question

Solve the simplified form, the period of vibration p for a pendulum varies directly as the square root of its length l. if p is 2.5 sec. when l is 25 in., what is the period when the length is 49 in.?group of answer choices

O 24.5 sec
O 98 sec
O 3.5 sec
O 14 sec

Answer 1

The period when the length is 49 in. is 3.5 seconds. So, the correct answer is C:

`\[ \text{Period} = 3.5 \, \text{sec} \]`

The relationship between the period
`(\(p\))` of a pendulum and its length
`(\(l\))` is given as:

`\[ p \propto √(l) \]`

This can be written as an equation with a constant of proportionality
`(\(k\))`:

`\[ p = k √(l) \]`

To find
`\(k\)`, use the initial values where
`\(p\)` is 2.5 sec. when
`\(l\)` is 25 in.:

`\[ 2.5 = k √(25) \]`

Solve for
`\(k\)`:

`\[ k = (2.5)/(√(25)) = (2.5)/(5) = 0.5 \]`

Now that you have
`\(k\)`, you can use it to find the period
`(\(p'\))` when the length
`(\(l'\))` is 49 in.:

`\[ p' = 0.5 √(49) \]`

`\[ p' = 0.5 * 7 = 3.5 \, \text{sec} \]`

Therefore, the period when the length is 49 in. is 3.5 seconds. So, the correct answer is C:

`\[ \text{Period} = 3.5 \, \text{sec} \]`

Answer 2

To find the new period when the length of the pendulum is 49 inches, we use the ratio of the given period and length and find that the period p is 3.5 seconds.The correct option is option c .

Step-by-step explanation:

The period of vibration p for a pendulum varies directly as the square root of its length l. We are given that when p is 2.5 seconds, l is 25 inches. To find out the period when the length is 49 inches, we use the direct variation relationship between p and the square root of l. The formula for the period of a simple pendulum is T = 2π√(l/g), where g is the acceleration due to gravity and doesn't change with the pendulum's length or mass. Since g remains constant and was not provided, we don't need its value to calculate the new period; we can use ratios instead:

• p₁/√l₁ = p₂/√l₂
• 2.5/√25 = p₂/√49
• p₂ = 2.5 * √49 / √25
• p₂ = 2.5 * 7 / 5
• p₂ = 3.5 seconds

Therefore, when the length of the pendulum is 49 inches, the period p is 3.5 seconds.