A block (6 kg) initially compresses spring #1 (k = 2000 N/m) by 60 cm from its equilibrium point. When the block is released, it slides a total horizontal distance of 1 m on a rough surface with kinetic coefficient of friction μk = 0.5. The block then compresses a stiffer spring #2 and momentarily stops. Find the following numerical energy values of the block: Initial spring potential energy at Spring #1: U1S =

In this problem we have that the elastic energy of the spring becomes part kinetic energy and the part in work against the force of friction, so, to use the law of conservation of energy, the decrease in energy is the rubbing force work

[tex]W_{fr}[/tex]= Ef - E₀

Let's look for the energies

Initial

E₀ = Ke = ½ k₁ x₁²

Final, this is just before starting to compress the spring

Ef = Ke = ½ m v²

The work of the rubbing force is

[tex]W_{fr}[/tex]= -fr x

Let's write Newton's second law the y axis

N-W = 0

N = W

fr = μ N

fr = μ mg

Let's replace

-μ mg x = ½ m v² - ½ k₁ x₁²

v² = 2/m (½ k₁ x1₁² -μ mg x)

v² = 2/6 (½ 2000 0.6²2 - 0.5 6 9.8 1) = 1/3 (360 - 29.4)

v = 3.13 m / s

With this value we calculate the energy of the block

K = ½ m v²

K = ½ 6 3.13²

K = 29.39 J

Calculate eenrgy of the spring ke 1

Ke = ½ k₁ x₁²

Ke = ½ 2000 0.60²

Ke = 360 J