how do i predict the displacement of a spring needed for the spring to just hit the ceiling.?

Question

the height from the top of the table to the ceiling is 2.54 m. Show all calculations.

I would be so grateful if somebody could explain this to me. thanks.

Answer 1

let (k) be spring constant, let us choose displacement (compression =x) so that when released this spring will generate that much initial speed (u) for mass (m) attached to it - hitting the ceiling
------------------------------------- ceiling ^ x = h
..
================= Table x=0 .V

====================== x = - x (compressed

energy conservation >> before = after
PE + elastic PE+KE = PE1 +KE (just hits roof)
- mgx + 0.5 kx^2 + 0 = mg h +0
x^2 - (2mg/k) x - (2mg/k) h =0 -------- (1)
let p = 2mg/k
x^2 - p x - ph =0
x = [p +- {p^2+4ph}^1/2] / 2 --------- (2)
========== this is the prediction relationship of (x)
you have h (can vary it). can also vary m, k (different material)

Answer 2

In order to get the spring up to the ceiling you have to overcome the force of gravity. The energy required to raise something an height, h, in meters is given by m*g*h - where m is the mass of the spring in kg and g is acceleration due to gravity.
Hooke's Law states that the restoring force exerted by a spring is proportional to its displacement. F= -kx (http://en.wikipedia.org/wiki/Hooke's_law).
So we need to do enough work in displacing the spring to equal the gravitational energy required.
Work done = the integral of the force over a distance (displacement)
so we get: m*g*h = integral from 0 to a of -kx dx
=> -0.5k a^2 = m*g*h
where a is the displacement needed, and k is the spring constant which can be determined by attaching weights and measuring the extension (M*g = -kx)

Answer 3

Calculate the potential energy gained by going to the ceiling (weight X distance). That must equal the energy stored in the spring: (1/2)k(x^2). Solve for x.

Answer 4

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The displacement of a spring needed for the spring to just hit the ceiling.is the displacement at which energy stored in the spring equals the kinetic energy[KE] required to raise the spring just up to the ceiling

The kinetic energy required to raise the spring just up to the ceiling [KE]= potential energy of spring at height 'h' of ceiling[PE]

Potential energy of spring at height of ceiling[PE]=mgh

where m is mass of spring , h is height of ceiling and g is acceleration of gravity

Suppose displacement required is 'x' , force constant of spring is 'k' then

energy stored in the spring = the kinetic energy[KE] =[PE]=mgh

(1/2)kx^2=mgh

x = sq rt [2mgh / k ]

If h =2.54 m and g =9.8 m/s^2 , then

x = sq rt [49.784 m / k ]

For the spring of mass 'm' and force constant 'k, to just hit the ceiling, the displacement of a spring needed is sq rt [49.784 m / k ]
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