# A subtle question on restoring force in SHM

1. Nov 23, 2013

### medwatt

Hello,
A was just helping my younger brother review some mechanics when I tried to explain how friction will cause the oscillations to eventually die out as the frequency keeps decreasing. So while analysing the force diagram and assuming that the frictional force is proportional to negative of the first degree of velocity, it turned out that if the mass spring system is moving from left to right, after it crosses the equilibrium position the spring force and frictional force will both be pointing to the left.
F= -kx -Friction
What I'm trying to say is since this is boosting the restoring force, it will be just like a stiffer spring, hence faster oscillations.
The same explanation occurs for the other direction of motion. Of course when particle has already reached either ends and returning to the equilibrium point then then F= -kx + Friction (increasing the period as one would expect). The thing is how then does the system come to rest if it is slowed at one end and sped up at the other.
Obviously I know that the my thinking is flawed but I cannot find where !!

2. Nov 23, 2013

### Staff: Mentor

The damping of a harmonic oscillator doesn't happen because the frequency of the oscillations decreases, it happens because the amplitude of the oscillations decreases and eventually reaches zero.

The frictional force is always acting in a way that reduces the amplitude of the next oscillation.

3. Nov 23, 2013

### medwatt

Can you explain the effect of the force variations (as described in my first post) on the amplitude because it leads to the same conundrum.

4. Nov 23, 2013

### Staff: Mentor

When the mass is moving towards the equilibrium position, the restoring force is acting to accelerate the mass, increasing its kinetic energy. The frictional force is opposing that acceleration, so some of the potential energy in the spring is lost to heat instead of being transferred to the mass as kinetic energy. Therefore, as the mass passes the equilibrium point, it will be moving more slowly, will have less kinetic energy, and won't be able to stretch the spring as far as it continues past the equilibrium point and starts moving away from it; that's a reduction in amplitude.

When the mass is moving away from the equilibrium position, the restoring force is acting to decelerate the mass, reducing its kinetic energy and increasing the potential energy in the spring. The frictional force is acting in the same direction, so the movement of the mass is opposed by both the spring and by friction. Thus, some of the kinetic energy of the mass is lost to heat instead of being converted to potential energy in the spring. Therefore, the spring will have to stretch less before the mass reaches the point of zero speed and kinetic energy and turns around. Again, less stretch of the spring means a reduction in amplitude relative to what we'd get without friction.

Thus, each oscillation is smaller than the one before, even if the frequency remains the same.

5. Nov 23, 2013

### medwatt

Thanks but what is bothering me is at certain points the frictional force and the spring force are pointing in the same direction. Although this doesn't make sense, the frictional force should contribute to increase the kinetic energy !!!
Answer this: do you agree that the spring force and frictional force both point in the same direction at some time ?

6. Nov 23, 2013

### Staff: Mentor

Yes, they both point in the same direction when the mass is moving away from the equilibrium point, so friction and the spring are both slowing the mass. However, those are the conditions under which the mass is slowing, reducing its kinetic energy by transferring some of it to the spring's potential energy and some of it to frictional heating. Thus, we're taking kinetic energy from the mass, but not giving it all to the spring as potential energy - friction is stealing some of it.

Maybe an easier way of thinking about it, suggested by the bolded text above: because the frictional force always acts against the speed of the mass, it always has to be reducing the kinetic energy of the mass.

7. Nov 23, 2013

### AlephZero

It is only boosting the restoring force while the mass is moving away from the center. The result is that the maximum distance the mass moves is smaller because of the friction force.

When the mass is moving towards the center, the friction force is opposing the restoring force, so the mass has a smaller velocity when it gets back to the center.

FWIW the friction force will cause a small reduction in the oscillation frequency compared with no friction, but that is not what causes the motion to decay, and the reduced frequency is constant, not continuously decreasing.