How to determine the speed of the block at the equilibrium position of the spring?

Question

A 6 kilogram block free to move on a horizontal, frictionless surface is attached to a spring. The spring is compressed 0.15 meters from equilibrium and released. The speed of the block is 1.76 meters per second when it passes the equilibrium position of the spring. The same experiment is now repeated with the frictionless surface replaced by a surface with coefficient of friction 0.34. The acceleration of gravity is 9.8 meters per second squared. How can you find the velocity (speed) at the equilibrium position?

Answer 1

The difference is that now friction will do negative work against the block. The work will be equal to umgx, where u is the coefficient of static friction, m is the mass of the block, g is the acceleration due to gravity, and x is the distance traveled. u, m, and g are all given, and x at equilibrium is unchanged from the 0.15 m in the first part of the problem. This work, umgx, must be subtracted from the kinetic energy the block had at equilibrium in the absence of friction. KE = 0.5*mv^2, where v is velocity and was also given. With friction, the block will instead have kinetic energy of 0.5*mv^2 - umgx, and this kinetic energy will be equal to 0.5*m(v_f)^2, where v_f is the velocity under the influence of friction. If you write 0.5*mv^2 - umgx = 0.5*m(v_f)^2, you'll see that m cancels out from every term and you can multiply every term by 2 for neatness, giving you v^2 - 2ugx = (v_f)^2, and v_f is your only unknown. You can rewrite this as v_f = sqrt(v^2 - 2ugx), and simply calculate using the given values of v (equilibrium velocity with no friction), u, g, and x.

Answer 2

for the first part of the problem you should find K "cnstant of spring"; you can do this in this way:

V^2 - Vi^2=2.a.x V is the velocity and Vi is the initial velocity that is zero! so:

V^2=2.a.x you have x=0.15 and V=1.76 so you can find "a"

after you find "a" you can find "K" , that is:
K.x=F=m.a => K.x=m.a x=0.15 ; m=6 ; and you have a from upper equation! so you can find K ....
after that just solve your 2'nd part kinematics and dont forget that summation of forces is equal to mass times accelaration! so
(force of spring) - (force of friction = f.N)+(other posetive forces)-(other negative forces) = m.a

postive works are those that help the motion ...
after finding a you can find other parameters....