Show that a single mass that is connected to both the top...

Show that a single mass that is connected to both the top and bottom supports by two springs of stiffnesses $c_1, c_2$ will vibrate in the same manner as if it were connected to only one support by a spring with the combined stiffness $c=c_1+c_2$.

Key Concepts

Simple Harmonic Motion

This concept addresses systems in which the restoring force is directly proportional to the displacement from equilibrium, leading to oscillatory motion. In such systems, the dynamics can be modeled by a second-order differential equation where the solution represents periodic motion. This is a fundamental framework for understanding how masses oscillate when attached to springs or other elastic elements.

Springs in Parallel

When multiple springs are connected to a mass and arranged in parallel, any displacement of the mass causes each spring to exert a restoring force. The combined effect is equivalent to a single spring whose stiffness is the sum of the individual stiffnesses. This principle of superposition simplifies the analysis of complex spring systems by allowing them to be treated as a single effective spring.

Hooke's Law

Hooke's Law is the principle that the force exerted by a spring is proportional to its displacement from the equilibrium position, mathematically expressed as F = -kx, where k is the stiffness or spring constant. This law underpins the analysis of vibratory systems and is essential for determining the restoring force in any spring-mass setup.

Transcript

00:01 In this problem we have a, we have two springs and we're going to arrange them in either series or parallel and they want us to calculate the spring constant whether they're in series or parallel.

00:24 Let's see here.

00:26 Determine the equivalent stiffness with the same oscillations and the period of us.

00:30 Oh i guess i didn't calculate the period of isolation, but i'll do that in real time here in a second.

00:38 So if we have in series, that's actually a little harder to calculate because we can break all these things up and we have free biodeagram of the top spring, free biodeagram of the lower spring, free biodeagram of the mass.

00:56 I'm assuming that gravity, we're not, gravity isn't acting or this is in a horizontal plane.

01:05 Just assume that this is a vibration about the equilibrium position if it's sagged under gravity so there's some equilibrium position that causes these springs to stretch but if we just look at motion about that equilibrium then gravity will cancel out and we saw that early in the chapter so what we have here is what we know is we have the force from spring one on the mass and the force from mass on spring one and then that has to be equal opposite here and then this has to be equal and opposite to this because they're attached here and then this has to be equal opposite to this because the spring has no mass so we have f in this case here is k2y prime okay so that's this that's this displacement here y prime which we don't know and we also know f from here is k1 y minus y prime so the change and length of this bring is this value.

02:14 Now from here we know f equals m y double dot.

02:18 We can use those two things to find y prime.

02:24 And so y prime is k1 divided by k1 plus k2 times y.

02:32 And plug that into here and get f equals k1 plus k1 all over.

02:40 Let's see here again let me fix this up a little bit...

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