Pressing a mass against a spring on a table with friction

Question

Suppose I have a mass $m$ kept on a horizontal table with coefficients of friction $\mu_{s}$ (static) and $\mu_{k}$ (dynamic/kinetic). And suppose I have a spring with spring constant $k$ which is attached to a wall perpendicular to the table . Now, if I press a mass against the spring and let it go, will the mass move in the direction in which the spring moves after being released (i.e. towards its mean position)? I think it does, but I'm not able to explain this.

Now, for each displacement $x$ (after releasing the spring), the force exerted by the spring on the mass is $kx$. To get the block moving in the first place, does the force exerted on it by the spring, $kx$, need not to be greater than the max static friction $\mu_{s}mg$? But if $x$ isn't able to become sufficiently large (which because of static friction it won't), this can't happen.

And, if this is possible (which I know it is, I just can't explain it), then isn't Newton's second law violated for a little while? For very little time (till $kx > \mu_{k}mg$), my spring is accelerating the mass in the direction OPPOSITE to the kinetic friction, but $\mu_{k}mg > kx$, which means that this acceleration is in a direction opposite to the direction of net force.

I know there's a mistake in this logic, but I can't spot it.

Answer 1

Problems like these can best be approached with a Free Body Diagram (the vertical is the $y$-axis and the horizontal the $x$-axis):

We can see that (with no movement in the vertical direction):

$$F_N=mg$$

Since as the spring is compressed, the friction force $F_f$ points to the wall (assume the static case - index $s$):

$$F_f=\mu_sF_N=\mu_s mg$$

Now we can write using $\text{N2L}$:

$$m\mathbf{a}=\mathbf{F_S}-\mathbf{F_f}$$ Or with scalars: $$ma=-kx+\mu_s mg$$

The mass $m$ will not move for $a=0$, so:

$$\mu_s mg \geq kx$$

Or:

$$\boxed{\mu_s \geq \frac{kx}{mg}}$$

for no horizontal ($x$) motion.

But if that is not the case then (with $x$-axis motion the index $k$ is required for $\mu$):

$$ma=-kx+\mu_k mg$$

Or:

$$m\ddot{x}+kx=\mu_k mg$$

Which is the Newtonian Equation of Motion of a damped harmonic oscillator (assuming $\mu_k$ is a constant)

Answer 2

First of all if you compress a spring . It will exert a force on block towards it's mean position.If friction is present say $\mu_s m g$ , it will try to oppose the motion of block as suggested by you. In this case friction and spring force will be in opposite direction.

Now if $\mu_s m g \ge K(x)$ It will automatically decrease as static friction is variable. Therefore in this case $f_s= K(x)$ and yes block won't move.

If say block is moving : $K(x)>\mu_smg> \mu_kmg$. Thus acceleration is in direction of net force.

I think you are confusing actual length of spring to the x present in the spring force.

x is the compression/ extension of spring from the mean position.So if you even compress a spring the actual length of spring decreases but x increases.

Answer 3

first you press you the s pring x1 and $$kx1<\mu_s*m*g$$, nothing happens the two forces cancel , then you increase x so that $$kx> max(\mu_s*m*g)$$ in this moment the smaller kinetic friction sets in and you have $$kx<\mu_k*m*g$$ the force $$F=-(kx-\mu_k*m*g)$$ accelerates your mass in -x direction . I can't see where any law is broken.