Question of Physics

Question

A block of mass m coming with an initial speed hits the spring

of a spring constant k at x=0. At this instant, it begins

to experience a friction force in addition to the spring force. The maximum

compression of the spring is D. The coeffµient of friction is non-uniform and

given by µ = β×e^(-α×x) where β and α are positive constants. Find

a) the initial speed of the block, v₀ and

b) the speed of the block, v final, when it comes back to x=0 again.

Answer 1

Let g be gravitational acceleration.

We solve this problem by conservation of energy.

At the point of maximal compression, D, the initial kinetic energy of the block, mv₀²/2, got converted to the energy of the spring, kD²/2, plus the work performed by friction.

Lets first calculate the work of friction.

The force of friction is mgμ, and when multiplied by distance gives us the work.

So work is the definite integral between 0 and D of mgμ.

Lets first calculate the indefinite integral:

∫mgβe^(-αx)dx = -mgβe^(-αx)/α

Thus the work performed by friction between 0 and D is

W = mgβ(1 - e^(-αD))/α

Now we can express the conservation of energy:

mv₀²/2 = kD²/2 + W (1)

a)

Simplifying (1)

v₀² = kD²/m + 2W/m (2)

v₀ = sqrt(kD²/m + 2gβ(1 - e^(-αD))/α)

b)

When the block comes back to x=0 its initial kinetic energy ,mv₀²/2,

got converted to its final kinetic energy, mv²/2, plus the work of friction in going back and forth -- 2W.

So by conservation of energy

mv₀²/2 = mv²/2 + 2W

v² = v₀² - 4W/m

Using (2) to replace v₀²

v² = kD²/m - 2W/m

v = sqrt(kD²/m - 2gβ(1 - e^(-αD))/α)

Please notice that the solutions for both (a) and (b) may be undefined

if the quantities under the square root turn out to be negative.

That corresponds to the situation where the coefficient of friction is so high

that the block does not make it all the way to D, or does not make it back.