A 7 kg mass hangs on a 10 meter long weightless cord. To the nearest cm, what should the new length of the pendulum be in order that the new frequency be 1.8 times its current value?
The correct answer is 309 but I got 250.
Heres what I did..
(1/2pi)(sqrt g/L2) = (2/2pi)(sqrt g/L1)
1/L2 = 4/L1
4L2 = L1
L2 = L1/4
So 10/4 times 100cm/1m = 225...
Sorry, I meant 250 not 225. Either way its still not the correct answer..
ω₁= √(g/L₁)
ω₂=√(g/L₂)
1.8 ω₁=ω₂
1.8√(g/L₁)=√(g/L₂)
1.8²g/L₁=g/L₂
L₂=L₁/3.24=10/3.24 =
=3.086 ≈3.19 m = 309 cm
To find the correct answer, you need to use the formula for the frequency of a pendulum, which is given by:
f = 1 / (2 * π) * √(g / L)
Where:
f is the frequency
π is a mathematical constant approximately equal to 3.14159
g is the acceleration due to gravity (approximately 9.8 m/s² on Earth)
L is the length of the pendulum
Let's denote the current length of the pendulum as L1, and the new length as L2. Since we know that the new frequency should be 1.8 times the current frequency, we can set up the following equation:
1 / (2 * π) * √(g / L2) = 1.8 * (1 / (2 * π) * √(g / L1))
To simplify the equation, we can cancel out the common terms:
√(g / L2) = 1.8 * √(g / L1)
Now, let's isolate L2 by squaring both sides of the equation:
(g / L2) = (1.8 * √(g / L1))²
(g / L2) = 1.8² * (g / L1)
L2 = (1.8² * L1) / g
Plugging in the given values:
L2 = (1.8² * 10) / 9.8
Now, calculating this expression:
L2 ≈ 3.672 meters
To convert this to centimeters, multiply by 100:
L2 ≈ 367.2 centimeters
Rounding this to the nearest centimeter, we get 367 centimeters.
Therefore, the correct answer is 367 centimeters, not 250 centimeters.