How Can I Determine the Stopping Time of a Mass on a Spring?

[chandran]

I would like to clarify my basics

there is a spring fixed at one end and have a mass at the other end. Call this as system one

I have a mass kept alone. Call this as system two

If i apply a force to system two for one second the mass will accelerate in one second to some velocity depending upon the force and the mass

If i apply a force to system one for the same one second the mass will accelerate to some velocity in one second. Say it accelerates at 1m/s per second. So during this one second displacement occurs at the end of the spring(mass attached end) to a value of 1m. So the spring now exerts a reaction force proportional to this distance of 1m. If the reaction force is not equal to the applied force then the net force will continue to accelerate the mass. Now after two seconds if this reaction force equals the net force the mass comes to rest. This stopping of the mass can occur after two seconds or even after hundred seconds. How can i find this time?

Can i find the border line difference between
Statics and dynamics from the above discussion?

 

[mukundpa]

When the force applied to the mass is equal to the tension in spring, till then the mass is accelerating, gained velocity and after that the tension is greater then the applied force and the mass will keep on moving anddecelerating. The net force in either case is varying and the acceleration is varying. Using calculus we can find the relation between the velocity and time and then calculate the time when the velocity becomes zero. Definitely when the body stops the force will not be zero.

 

[Astronuc]

Both systems described are dynamic. A static system infers that forces are balanced - there is no motion and no acceleration.

Take system 2. A force (and let us say the only force) is applied to a mass for some period (1 sec) during which the mass will accelerated to some velocity over some distance. After the period, the force is removed, and the mass, unless it is subjected to some other force would continue without acceleration at constant velocity.

Now in system 1, a force is applied to mass attached to a spring. As the spring changes length, compressed or extended, it will impart a force on the mass in addition to the applied force. The acceleration of the mass will depend on the resultant force.

Let's say the applied force extends the spring. As soon as the applied force is removed, the mass will decelerate due to the spring force, until it stops (V = 0), and then the spring force will cause the mass to accelerate in the opposite direction. Then the problem becomes one of simple harmonic motion, with or without damping due to any dissipative forces.

In the spring/mass system, the spring force is a product of the spring constant (stiffness) and the displacement from some equilibrium position. Again, as mukundpa indicated, using calculus and the appropriate mathematical relationships, one can desribe the motion of the mass attached to the spring.