Question

Which of the following combinations of variables results in the greatest period for a pendulum? length = L, mass = M, and maximum angular displacement = degrees length = L, mass = M, and maximum angular displacement = 3 degrees length = 2L, mass = M/2, and maximum angular displacement = 1 degree length = 1.5L, mass = 2M, and maximum angular displacement = 2 degrees length = L, mass = 4M, and maximum angular displacement = 4 degrees

Answer 1

Answer:

length = 2L, mass = M/2, and maximum angular displacement = 1 degree

Explanation:

We consider only small amplitude oscillations (like in this case), so that the angle θ is always small enough. Under these conditions recall that the equation of motion of the pendulum is:

$\ddot{\theta}=\frac{g}{l}\theta$

And its solution is:

$\theta=Asin(\omega t + \phi)$

Where $\omega=\sqrt\frac{g}{l}$ are the angular frequency of the oscillations, from which we determine their period:

$T=\frac{2\pi}{\omega}\\T=2\pi\sqrt\frac{l}{g}$

Therefore the period of a pendulum will only depend on its length, not on its mass or angle, for angles small enough. So, the answer is the one with the greater length.