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The length of a simple pendulum executing simple harmonic motion is increased by 21%. The percentage increase in the time period of the pendulum of increased length is
- a)11%
- b)21%
- c)42%
- d)10%
Correct answer is option 'D'. Can you explain this answer?
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Explanation:
Let's assume the original length of the pendulum is L, and the time period is T.
According to the formula for the time period of a simple pendulum:
T = 2π√(L/g)
Where:
T = Time period
L = Length of the pendulum
g = Acceleration due to gravity
1. Increase in Length:
If the length of the pendulum is increased by 21%, the new length would be:
New Length = L + 0.21L = 1.21L
2. New Time Period:
To find the new time period, we substitute the new length into the formula:
New Time Period = 2π√(1.21L/g)
3. Percentage Increase in Time Period:
To find the percentage increase in the time period, we compare the new time period with the original time period:
Percentage Increase = (New Time Period - Original Time Period) / Original Time Period * 100
Substituting the values, we get:
Percentage Increase = (2π√(1.21L/g) - 2π√(L/g)) / 2π√(L/g) * 100
Simplifying the expression:
Percentage Increase = (√(1.21L/g) - √(L/g)) / √(L/g) * 100
Using the property of square roots (a√x - b√x = (a - b)√x):
Percentage Increase = (√(1.21L/g) - √(L/g)) / √(L/g) * 100
= (√(1.21) - √(1)) * √(L/g) / √(L/g) * 100
= (√(1.21) - √(1)) * 100
Calculating the values:
Percentage Increase = (1.1 - 1) * 100
= 0.1 * 100
= 10%
Therefore, the percentage increase in the time period of the pendulum with the increased length is 10%.
Let's assume the original length of the pendulum is L, and the time period is T.
According to the formula for the time period of a simple pendulum:
T = 2π√(L/g)
Where:
T = Time period
L = Length of the pendulum
g = Acceleration due to gravity
1. Increase in Length:
If the length of the pendulum is increased by 21%, the new length would be:
New Length = L + 0.21L = 1.21L
2. New Time Period:
To find the new time period, we substitute the new length into the formula:
New Time Period = 2π√(1.21L/g)
3. Percentage Increase in Time Period:
To find the percentage increase in the time period, we compare the new time period with the original time period:
Percentage Increase = (New Time Period - Original Time Period) / Original Time Period * 100
Substituting the values, we get:
Percentage Increase = (2π√(1.21L/g) - 2π√(L/g)) / 2π√(L/g) * 100
Simplifying the expression:
Percentage Increase = (√(1.21L/g) - √(L/g)) / √(L/g) * 100
Using the property of square roots (a√x - b√x = (a - b)√x):
Percentage Increase = (√(1.21L/g) - √(L/g)) / √(L/g) * 100
= (√(1.21) - √(1)) * √(L/g) / √(L/g) * 100
= (√(1.21) - √(1)) * 100
Calculating the values:
Percentage Increase = (1.1 - 1) * 100
= 0.1 * 100
= 10%
Therefore, the percentage increase in the time period of the pendulum with the increased length is 10%.
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The length of a simple pendulum executing simple harmonic motion is increased by 21%. The percentage increase in the time period of the pendulum of increased length isa)11%b)21%c)42%d)10%Correct answer is option 'D'. Can you explain this answer?
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