A special spring is constructed in which the restoring force is in the opposite direction to the displacement but is proportional to the cube of the displacement; i. e., F= -kx^3 this spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released. Determine each of the following in terms of k, A, and M
a) The potential energy in the spring at the instant the mass is released
b) the maximum speed of the mass
c) The displacement of the mass at the point where the potential energy of the spring and the kinetic energy of the mass are equal

Explanation:

a ) Restoring force in special spring F = k x³

let it is stretched by length l and it is further stretched by dl .

work done on spring in stretching by dl

= F dl

= k l³ dl

work done in stretching by A

= ∫k l³ dl between limit 0 to A

= k [ l⁴ / 4 ] between limit 0 to A

= k A⁴ / 4 ;

b )

Let v be the maximum speed attained ,

kinetic energy of mass = potential energy of spring

1/2 M v² = k A⁴ / 4

v² = kA⁴ / 2

v = A² √( k/2 )

c ) Let the required displacement be d .

potential energy = k d⁴ / 4

at displacement d , potential energy and kinetic energy are equal so potential energy = 1 /2 of total potential energy = 1/2 x k A⁴ / 4

= k A⁴ / 8

So

k A⁴ / 8  = k d⁴ / 4

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explanation:

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