Elastic Constant vs. Length

[Oblivionator]

When oscillating a ruler with an extended mass on the end, what is the direct relation of the

length against the spring constant of the ruler? I heard the spring constant had some

relation with the curve of the ruler as it oscillates

 

[timmay]

My take on this:

Deflection at the end of a cantilever beam of length L, elastic modulus E, second moment about neutral axis I, load at end (mass x g) F:

x = FL³ / 3EI

Using Hooke's Law: F = kx

Therefore k = 3EI / L³

 

[kdm06d]

don't you also have to take into account torque?

T = R x F

 

[timmay]

You're thinking of bending moment (analogous to torque, but preferred when you're talking about beams). The maximum bending moment for the beam that we're considering is the load at the end multiplied by the length: M = FL

However, the OP is talking about oscillating the system. I'm guessing they're wondering about changing the natural frequency of the ruler by changing the length that is free (how far over the edge of thet able it is before you 'twang' it). The change in natural frequency results in a change in the pitch of the sound you hear.