Spring Collision?

Question

A 8 kg block is pushed by an external force against a spring with spring constant 173 N/m until the spring is compressed by 2.1 m from its uncompressed length (x = 0). The block rests on a horizontal plane that has a coefficient of kinetic friction of 0.56. The acceleration of gravity is 9.8 m/s^2. Remember: The block is not attached to the spring.

After all the external forces are removed (so the compressed spring releases the mass) how far D along the plane will the block move before coming to a stop? Answer in units of m.

To civil_av8r:

I did the same thing as you and got 8.688616071m and the online submission says its wrong..but it really seems right.

Answer 1

Energy problem

1st step is to figure out how much energy is in the spring

E = 1/2*k*x^2

That energy is transfered into Kinetic Energy

1/2*k*x^2 = 1/2*m*v^2

Solve for v

v = x*sqrt(k/m)

As soon as the spring is done "pushing" on the block, the only force acting on the block is friction.

Sum of the horizontal forces = m*a = -u*m*g

a = -u*g

Use kinematics to solve for the distance

vf^2 = vi^2 + 2*a*d

In our case, vf = 0

d = -vi^2/2*a
d = (x^2*k/m)/(2*u*g)

Presto!

Plug and chug

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I got to looking at the solution for d and if you transfer the m to the bottom and the 2 to the top

d = (1/2*k*x^2)/(u*m*g)

d = Energy of the spring / force of friction

I guess another way of looking at the problem is that the energy of the spring is transfered into the block as work. We know the definition of work is W = F*d so d = W/F. Intresting...

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I got 8.96, make sure you are using the correct number of units.

The answerer below says the Spring Energy is K*delta(x) which isn't, that is the force needed to compress the spring. K is in the units N/m and x is in the units of m, so K*delta(x) = N/m * m = N.

Perhaps we forgot to look at the energy friction takes up during the spring's "release" phase.

Energy absorbed by friction = friction*d = u*m*g*x

Therefore, when the spring is done "pushing" the Total Energy of the system is 1/2*k*x^2 - u*m*g*x (= 289.3 J)

Now use d = Energy/friction = (1/2*k*x^2 - u*m*g*x)/(u*m*g) = (1/2*k*x^2)/(u*m*g) - x = 6.59