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A simple pendulum is executing simple harmonic motion with a time period T. If the length of the pendulum is increased by 21% then the increase in the time period of the pendulum due to increased length is
- a)50%
- b)30%
- c)21%
- d)10%
Correct answer is option 'D'. Can you explain this answer?
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Explanation:
The time period (T) of a simple pendulum is given by the formula:
T = 2π√(L/g)
Where:
T = Time period of the pendulum
L = Length of the pendulum
g = Acceleration due to gravity
Step 1: Determining the increase in length
Let's assume the original length of the pendulum is L.
The increase in length of the pendulum is 21% of L, which can be written as 0.21L.
Step 2: Finding the new length of the pendulum
The new length of the pendulum is the sum of the original length and the increase in length.
So, the new length (L') of the pendulum is:
L' = L + 0.21L = 1.21L
Step 3: Calculating the new time period
Substituting the new length (L') in the formula for the time period:
T' = 2π√(1.21L/g)
Simplifying the equation:
T' = 2π√(1.21L/g)
T' = 2π(√1.21√L/√g)
T' = 2π(1.1√L/√g)
T' = 2π(1.1)√(L/g)
T' = 2.2π√(L/g)
Step 4: Calculating the increase in time period
The increase in time period (∆T) is given by the difference between the new time period (T') and the original time period (T).
∆T = T' - T
∆T = 2.2π√(L/g) - 2π√(L/g)
∆T = 2π√(L/g)(1.1 - 1)
∆T = 0.1(2π√(L/g))
Step 5: Expressing the increase in time period as a percentage
To express the increase in time period as a percentage, we can use the following formula:
Percentage increase = (∆T / T) * 100
Substituting the value of ∆T:
Percentage increase = (0.1(2π√(L/g)) / T) * 100
Percentage increase = (0.1(2π√(L/g)) / (2π√(L/g))) * 100
Percentage increase = (0.1 * 100)
Percentage increase = 10%
Therefore, the increase in the time period of the pendulum due to the increased length is 10%. Hence, the correct answer is option D) 10%.
The time period (T) of a simple pendulum is given by the formula:
T = 2π√(L/g)
Where:
T = Time period of the pendulum
L = Length of the pendulum
g = Acceleration due to gravity
Step 1: Determining the increase in length
Let's assume the original length of the pendulum is L.
The increase in length of the pendulum is 21% of L, which can be written as 0.21L.
Step 2: Finding the new length of the pendulum
The new length of the pendulum is the sum of the original length and the increase in length.
So, the new length (L') of the pendulum is:
L' = L + 0.21L = 1.21L
Step 3: Calculating the new time period
Substituting the new length (L') in the formula for the time period:
T' = 2π√(1.21L/g)
Simplifying the equation:
T' = 2π√(1.21L/g)
T' = 2π(√1.21√L/√g)
T' = 2π(1.1√L/√g)
T' = 2π(1.1)√(L/g)
T' = 2.2π√(L/g)
Step 4: Calculating the increase in time period
The increase in time period (∆T) is given by the difference between the new time period (T') and the original time period (T).
∆T = T' - T
∆T = 2.2π√(L/g) - 2π√(L/g)
∆T = 2π√(L/g)(1.1 - 1)
∆T = 0.1(2π√(L/g))
Step 5: Expressing the increase in time period as a percentage
To express the increase in time period as a percentage, we can use the following formula:
Percentage increase = (∆T / T) * 100
Substituting the value of ∆T:
Percentage increase = (0.1(2π√(L/g)) / T) * 100
Percentage increase = (0.1(2π√(L/g)) / (2π√(L/g))) * 100
Percentage increase = (0.1 * 100)
Percentage increase = 10%
Therefore, the increase in the time period of the pendulum due to the increased length is 10%. Hence, the correct answer is option D) 10%.
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A simple pendulum is executing simple harmonic motion with a time period T. If the length of the pendulum is increased by 21% then the increase in the time period of the pendulum due to increased length isa)50%b)30%c)21%d)10%Correct answer is option 'D'. Can you explain this answer?
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