A light rod of length L and a little weight of mass m make up the pendulum.

Question

A light rod of length L and a little weight of mass m make up the pendulum. In a vertical plane, the rod can complete a full rotation. The weight is initially in the bottom equilibrium position. A horizontally directed velocity v is applied to the weight. What values of v will cause the pendulum to oscillate, and what values will cause it to rotate in the vertical plane? At what starting velocity v will the rod be crushed at the top point, and at what initial velocity v will it be stretched?

Answer 1

The question uses the words "rod" and "crushed".

The latter sounds rather drastic for a solid object like a rod, and analyzing

its crushing would require information about the material.

Therefore I will answer the question after replacing the last sentence with:

"What starting velocity v will cause the rod to experience inward stress at the top point,

and what starting velocity v will cause outward stress?"

I believe the word "light" when applied to rod is supposed to suggest the assumption

that the rod has no mass.

Let

w be the velocity at the top, in those cases when the mass does make it there,

g = 9.81 m/s² be gravitational acceleration.

The kinetic energy of the mass at the bottom is

mv²/2

The kinetic energy of the mass at the top (if it gets there) is

mw²/2

With respect to the bottom, the potential energy of the mass at the top (if it gets there) is

2Lmg

By conservation of energy

2Lmg + mw²/2 = mv²/2

From that

w = √(v²-4Lg)

The mass will make it to the top if the expression for w is defined, that is, if

v² - 4Lg ≥ 0

v ≥ √(4Lg)

Now lets calculate the stress the rod will experience at the top.

It experiences two forces:

- downward gravitational push from the mass, which has magnitude mg,

- upward centrifugal force, which has magnitude mw²/L.

So the direction of their sum will depend on which is larger.

The rod will experience outward stress if

mw²/L > mg

w² > Lg

v²-4Lg > Lg

v > √5Lg

So these are the possible outcomes

0 ≤ v < √4Lg the mass will oscillate

v = √4Lg the mass will get stuck at the top in unstable position

√4Lg < v < √5Lg the mass with rotate and the rod will experience inward stress at the top,

v = √5Lg the mass with rotate and the rod will experience no stress at the top,

v > √5Lg the mass with rotate and the rod will experience outward stress at the top.