A pendulum with a bob of mass 150 g has the same period as a thin rod (not massless) of the same mass and length 17 cm when the rod is pivoted about its end. What is the length of the pendulum?
who cares what the mass is?
period of long rod pendulum
= 2 pi sqrt (2(0.17)/3g)
period of regular pendulum with mass at end
= 2 pi sqrt (L/g)
so
L/g = 0.17 * .667 / g
L = .113 = 11.3 cm
To find the length of the pendulum, we need to use the concept of the period of oscillation of a pendulum.
The period of oscillation (T) of a simple pendulum can be calculated using the formula:
T = 2π * √(L/g),
where L is the length of the pendulum and g is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
In this problem, we are given a situation where the pendulum and the thin rod have the same mass and the same period of oscillation.
The formula for the period of oscillation (T) of a thin rod pivoted about its end is given by:
T = 2π * √(I/mgl),
where I is the moment of inertia of the rod, m is the mass of the rod, g is the acceleration due to gravity, and l is the length of the rod.
Since the mass and the period of oscillation are the same for both the pendulum and the rod, we can set the two formulas equal to each other:
2π * √(L/g) = 2π * √(I/mgl).
We can cancel out the 2π on both sides:
√(L/g) = √(I/mgl).
Squaring both sides of the equation, we get:
L/g = I/mgl.
Simplifying, we find:
L = I/m.
Since the mass and the length of the rod are given as 150 g and 17 cm respectively, the moment of inertia (I) of the rod can be calculated using the formula:
I = (1/3) * m * l²,
where m is the mass of the rod and l is the length of the rod.
Substituting the given values into the formula, we find:
I = (1/3) * 0.150 kg * (0.17 m)².
Calculating this, we find:
I ≈ 0.00179 kg•m².
Substituting the value of the moment of inertia (I) and the mass (m) into the equation L = I/m, we find:
L = 0.00179 kg•m² / 0.150 kg.
Calculating this, we find:
L ≈ 0.011932 m.
Therefore, the length of the pendulum is approximately 0.011932 meters or 11.932 cm.