A pendulum of Length L cm has time period T seconds. T is directly proportional to the square root of L.
The length of the pendulum is increased by 40%.
What is the percentage increase in the time period?
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Multiplier for increasing L = 1.4
Multiplier for increasing T = ?
Thank you.
18.3
T = k√L
using L'=1.4L instead, we have
T' =k√L' = k√(1.4L) = k√L √1.4 = T√1.4
To find the percentage increase in the time period, we need to determine the multiplier for increasing the time period.
We know that the time period T is directly proportional to the square root of the length L. This can be represented as:
T = k√L
where k is the constant of proportionality.
Now, let's consider the change in length of the pendulum. The length is increased by 40%, which can be expressed as:
New Length = 1.4L
Substituting this into the equation for the time period, we get:
New Time Period = k√(1.4L)
To find the multiplier for increasing the time period, we can divide the new time period by the original time period:
Multiplier for increasing T = New Time Period / Original Time Period
Substituting the expressions for the time periods, we have:
Multiplier for increasing T = (k√(1.4L)) / (k√L)
The k cancels out, leaving:
Multiplier for increasing T = √(1.4L) / √L
Simplifying this expression, we have:
Multiplier for increasing T = √1.4
To find the percentage increase, we can subtract 1 from the multiplier and multiply by 100:
Percentage increase = (√1.4 - 1) x 100
Now, let's calculate the value of the multiplier and the percentage increase:
√1.4 ≈ 1.183
Therefore,
Multiplier for increasing T ≈ 1.183
Percentage increase ≈ (1.183 - 1) x 100
≈ 18.3%
Hence, the percentage increase in the time period is approximately 18.3%.
To find the percentage increase in the time period, we need to calculate the multiplier for increasing T.
We are given that T is directly proportional to the square root of L, which can be expressed as T ∝ √L.
Using the given information, let's express this relationship as an equation:
T = k√L
where k is a constant of proportionality.
Now, let's consider the change in T when the length of the pendulum is increased by 40%. The new length of the pendulum is L + 0.4L, which simplifies to 1.4L.
Substituting the new length into the equation, we have:
T' = k√(1.4L)
To find the multiplier for increasing T, we need to compare the new time period T' to the original time period T:
Multiplier for increasing T = T' / T = (k√(1.4L)) / (k√L)
Notice that the constant of proportionality, k, cancels out in this ratio expression. This means that the multiplier for increasing T does not depend on the value of k.
Now, let's simplify the expression by canceling out the square root terms:
Multiplier for increasing T = √(1.4L) / √L
We know that √(1.4L) / √L is equal to √(1.4), so the multiplier for increasing T is √(1.4).
To find the percentage increase, we can subtract 1 from the multiplier and then multiply by 100:
Percentage increase = (√(1.4) - 1) * 100
Now, calculate the value of √(1.4) and subtract 1:
√(1.4) ≈ 1.183
Percentage increase ≈ (1.183 - 1) * 100 ≈ 18.3%
Therefore, the percentage increase in the time period when the length of the pendulum is increased by 40% is approximately 18.3%.