A physics question with simple harmonic motion

[Wooh]

This is a little question that I am not sure how to do because of the way gravity works with vertical springs.

A 3 lb block rests on top of a 5 lb block which is attacked to a k=1000 N/m spring force spring following hooke's law. The blocks are released and allowed to reach equilibrium (.08 m, of course). After this, the blocks are pushed downward and then allowed to oscillate. Q1: what is the maximum acceleration they can have at any point and still stay together. I said -g, which means that if they are accelerating downward at more than 10 m/s^2 (allowed to estimate the value of g). Then it asks what is the maximum distance it could have been pushed down (after the .08) and just reach this acceleration.

This is where I got tripped up, as I was unsure on how to include potential energy and whatnot. Anyone care to do it so I can learn by example?

 

[dextercioby]

There's a similar problem solved in detail on this forum exactly...It's under the thread title:"Redemption:I challange Dextercioby"...That's me...

Daniel.

 

[wais]

The maximum acceleration that the blocks can have and still stay together is indeed -g, as you correctly stated. This is because if the acceleration is greater than -g, the 3 lb block will lose contact with the 5 lb block and they will separate.

To find the maximum distance that the blocks can be pushed down after reaching equilibrium, we can use the conservation of energy principle. At equilibrium, the blocks have a total potential energy of 0.08 m * (3 lb + 5 lb) * 9.8 m/s^2 = 6.24 J. This energy is stored in the spring and will be converted into kinetic energy as the blocks are pushed down and start oscillating.

At the maximum distance, the blocks will have a maximum potential energy of (0.08 + x) m * (3 lb + 5 lb) * 9.8 m/s^2 = (0.08 + x) * 78.4 J, where x is the additional distance the blocks have been pushed down. This potential energy will be converted into kinetic energy, and at the maximum distance, the kinetic energy will be equal to the maximum potential energy. Therefore, we can equate the two expressions:

(0.08 + x) * 78.4 = 6.24

Solving for x, we get x = 0.0708 m. This means that the maximum distance the blocks can be pushed down after reaching equilibrium is 0.0708 m. Any distance greater than this will result in the blocks separating.

To better understand this concept, you can think of it in terms of the spring force. At equilibrium, the spring force is equal to the weight of the blocks (8 lb * 9.8 m/s^2 = 78.4 N). If the blocks are pushed down further, the spring force will increase. At the maximum distance, the spring force will be equal to the maximum acceleration (-g) times the total mass (8 lb), which again gives us 78.4 N. This is the point at which the blocks will lose contact and start to separate.

Hope this helps!