the time period T for a simple pendulum of length l is given by T=2π⋅√(l/g) where g is a constant.
find the percentage change in T when l changes by 6%
taking derivatives, you have
dT = π/√(gL) dL
Now, to find the % change of T, you take
dT/T
You would expect it to be about 1/2 the change in L, since you have roughly
T^2 = kL
2T dT = k dL
To find the percentage change in T when l changes by 6%, we need to calculate the relative change in T and then express it as a percentage.
Let's start by finding the absolute change in T. We know that T is given by the formula T = 2π√(l/g). If l changes by 6%, we can express this as a decimal as 0.06.
The new value of l, l', would be (1 + 0.06)l, or 1.06l.
Substituting the new value of l into the formula for T, we get:
T' = 2π√(1.06l/g).
The absolute change in T is then given by ΔT = T' - T:
ΔT = 2π√(1.06l/g) - 2π√(l/g).
To find the relative change in T, we can divide the absolute change by the original value of T:
Percentage change = (ΔT / T) * 100.
Substituting the values we found earlier:
Percentage change = [(2π√(1.06l/g) - 2π√(l/g)) / (2π√(l/g))] * 100.
Simplifying the formula:
Percentage change = [(√(1.06l/g) - √(l/g)) / √(l/g)] * 100.
Now, let's calculate the value using a specific numerical example.
Suppose we have a pendulum of length l = 100 cm (1 meter) and g = 9.8 m/s² (acceleration due to gravity).
The original value of T can be calculated as follows:
T = 2π√(l/g) = 2π√(1/9.8) ≈ 2π√(0.102) ≈ 2π(0.319) ≈ 2.006 seconds (approximately).
Now, let's calculate the new value of T after a 6% change in l:
l' = (1 + 0.06)l = 1.06 * 100 = 106 cm (1.06 meters).
T' = 2π√(l'/g) = 2π√(1.06/9.8) ≈ 2π√(0.108) ≈ 2π(0.328) ≈ 2.063 seconds (approximately).
Using these values, we can find the percentage change:
Percentage change = [(√(1.06 * 100/9.8) - √(100/9.8)) / √(100/9.8)] * 100.
= [(√(1.06 * 10.2) - √(10.2)) / √(10.2)] * 100.
≈ [(√10.812 - √10.2) / √10.2] * 100.
≈ [(3.289 - 3.195) / 3.195] * 100.
≈ 0.094/3.195 * 100.
≈ 0.029 * 100.
≈ 2.9%.
Therefore, the percentage change in T when l changes by 6% is approximately 2.9%.
To find the percentage change in T when l changes by 6%, we need to calculate the derivative of T with respect to l and then apply the percentage change formula.
1. Start with the formula for T: T = 2π⋅√(l/g).
2. Take the derivative of T with respect to l:
dT/dl = d(2π⋅√(l/g))/dl.
Using the chain rule, we can simplify this derivative:
dT/dl = 2π⋅(1/2)⋅(l/g)^(-1/2)⋅(1/g) = π/√(gl).
3. Now we need to find the percentage change in T when l changes by 6%.
Let's denote the original value of T as T_0, and the new value as T_1.
Percent change = (T_1 - T_0) / T_0 * 100%
Let Δl represent the change in l. In this case, Δl = 0.06l.
The change in T, ΔT, can be approximated by the derivative multiplied by the change in l:
ΔT = dT/dl * Δl.
We can use this approximation to calculate the percentage change:
Percent change = (ΔT / T_0) * 100%
Substitute ΔT = dT/dl * Δl and simplify:
Percent change = (dT/dl * Δl / T_0) * 100%
Percent change = (π/√(gl) * 0.06l / T_0) * 100%
Percent change = (π * 0.06 / √(gl)) * 100%.
Therefore, the percentage change in T when l changes by 6% is (π * 0.06 / √(gl)) * 100%.