If the length of the pendulum increases four times the the frequency ius
halved.
The frequency of a pendulum is given by the formula:
f = 1/T
where f is the frequency and T is the period. The period of a pendulum is given by the formula:
T = 2π√(l/g)
where l is the length of the pendulum and g is the acceleration due to gravity.
If the length of the pendulum is increased four times, then the period becomes:
T' = 2π√(4l/g) = 4π√(l/g)
The frequency of the pendulum with the increased length is:
f' = 1/T' = 1/(4π√(l/g))
To find the relationship between the frequency of the original pendulum and the frequency of the pendulum with the increased length, we can take the ratio:
f'/f = (1/(4π√(l/g)))/(1/(2π√(l/g))) = 1/2
Therefore, if the length of the pendulum increases four times, the frequency is halved.
To determine the relationship between the length of a pendulum and its frequency, we can use the formula for the period of a pendulum. The period is the time it takes for the pendulum to complete one full swing back and forth.
The relationship between the period (T) of a pendulum and its length (L) can be expressed as:
T = 2π * √(L/g)
Where:
T = Period of the pendulum
L = Length of the pendulum
g = Acceleration due to gravity (approximately 9.8 m/s²)
Now, let's analyze the scenario you presented. If the length of the pendulum increases by four times, let's say the original length was L₁ and the new length is L₂. We can express this relationship as:
L₂ = 4 * L₁
To determine the relationship between the frequencies, we need to find the inverse of the periods. The frequency (f) is the reciprocal of the period, so:
f = 1 / T
By substituting the formulas for the periods in terms of the lengths, we get:
f₁ = 1 / (2π * √(L₁/g))
f₂ = 1 / (2π * √(L₂/g))
Now, plug in the relationship between the lengths:
f₂ = 1 / (2π * √((4 * L₁)/g))
f₂ = 1 / (2π * 2 * √(L₁/g))
f₂ = 1 / (4π * √(L₁/g))
f₂ = 1 / 4 * f₁
Thus, the frequency of the pendulum is reduced to one-fourth of its original frequency when the length of the pendulum is increased by four times.