Question

A light spring of force constant 4.45 N/m is compressed by 8.00 cm and held between a 0.250 kg block on the left and a 0.450 kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is the following. In each case, assume that the coefficient of static friction is greater than the coefficient of kinetic friction. Let the positive direction point to the right.

Answer 1

Based on the kinetic friction, one can figure out the lower limit of the static friction, which is larger than the kinetic friction. Therefore we can decide if a block is going to have a non-zero net force or not. If the force by the spring is less than the static frictional force, then the corresponding block will not have an acceleration (net force being zero).

Case 1: ?k=0, so both would experience no-zero acceleration. The forces that the spring exerts on both blocks would have the same magnitude and are in opposite directions.

The two blocks will attain maximum speed when the spring is fully relaxed, at which time the spring potential energy is fully converted into kinetic energy. In this case, we can use two physics concepts: mechanical energy conservation and momentum conservation.

Energy Conservation: Ei=Ef

0.5k?x^2=0.5m1v1^2 + 0.5m2v2^2

0.5*4.45*(0.08)^2 = 0.5*0.25*v1^2 + 0.5*0.45*v2^2

Momentum Conservation: 0=m1v1+m2v2

0.25v1 - 0.45v2 = 0

So v2 = (5/9)v1

Substituting v2 in above equation, we get

0.01424 = 0.125*v1^2 + 0.225*(25/81)*v1^2

0.01424 = 0.1944*v1^2

So v1 = 0.27 m/s

v2 = 0.15 m/s

Case 2: ?k=0.1. The force from the spring is ||F__s=k\cdot \Delta x||. Calculate the frictional forces on both blocks and you will find that the block on the right with larger mass won�t move and the block on the left with less mass will have acceleration in the negative direction. Now the question becomes one block on the left moving and accelerating to the left with a frictional force to the right and a spring force to the left.

Here the important physics is that the maximum speed that block 1 would attain happens when the net force on it is zero � i.e. when the force of the spring is equal to the friction in magnitude but in opposite direction. That is when the block stops accelerating to the left and starts to decelerating (towards the left � from then on the frictional force will be larger than the spring force and the net force will point to the right).

Now we need to set up a clear coordinate system as shown below.

x₀: The location at which the spring is fully relaxed (unstretched).

x₁: The initial location of block 1 at which the spring is compressed with certain compression displacement which equals (x₁-x₀)

x₂: The location of block 1 at which the force produced by the spring is equal to the force of friction in magnitude but in opposite direction.

Then pick x₁ as the starting point and x₂ as the end point. Apply the extended energy conservation that includes work:

U1+K1+Wf=U2+K2

which gives:

0.5*k*(x1-x0)^2 + 0 - f(x1-x2) = 0.5*k*(x2-x0)^2 + 0.5*m1*v1^2

Base on the given conditions, we also have x1?x0=0.08.

At x₂, the spring force and the friction has the same magnitude:

f=?mg=Fs=k(x2?x0),

which gives:

x2?x0=?mg/k = 0.1*0.25*9.8/4.45 = 0.055

The above two also give

x1?x2=0.08??mg/k = 0.08 - 0.055 = 0.025

So ,0.5*4.45*(0.08)^2 - (0.1*0.25*9.8)(0.08-0.055) = 0.5*4.45*(0.055)^2 + 0.5*0.25*v1^2

So v1 = 0.1052 m/s

Case 3. Frictional forces on both blocks are larger than the spring force so none of the blocks is moving.

�k	0.250 kg block	0.450 kg block
0	0.27 m/s	0.15 m/s
0.1	0.1052 m/s	0
0.490	0	0