Equal and Opposite variable forces on a mass - does it accelerate?

[Erica799]

1. A mass is attached to a spring that is mounted on the end of an air track. The mass is pushed by your hand along the track, providing a force, F hand. The spring provides a force on the mass, F spring, equal to F hand but opposite in direction. Assume (frictionless?) air track is being used. The forces vary linearly (classic F vs x, work is area under the curve, net work equals zero on the spring/mass system.) How do I explain why the mass moves, if the forces on the mass are always equal and opposite? Is the mass accelerating, or moving at a constant velocity? I am confused!

Homework Equations

f(t) = m (dv/dt) ?

The Attempt at a Solution

 

[Rake-MC]

If there is a force, there is an acceleration. Think of $$F = ma$$

As for why it moves. You are correct, your hand pushing on the mass is the same force of the mass pushing back on your hand. However, this is looking at it as a closed system. More force is coming outside of your defined system that is 'biasing' the direction of the force (ie. from you).

I just realized that biasing is a really bad word to use in this context. More force is added from outside the system to overcome the force that is pushing back on your hand. The equal and opposite force for this force is also located outside the system.

Hope that makes sense.

 

[baba89]

The mass is indeed accelerating, despite the equal and opposite forces acting on it. This is because acceleration is not solely determined by the magnitude of forces, but also by the mass of the object. In this case, the mass being pushed by your hand has a greater mass than the spring providing the equal and opposite force. This means that the mass will experience a greater acceleration due to the same magnitude of force acting on a larger mass.

Furthermore, the forces vary linearly because the spring is compressed or stretched by a certain distance as the mass moves along the air track. This distance, or displacement, is directly proportional to the force applied, following Hooke's Law. As the mass moves, the spring provides a force in the opposite direction, counteracting the force applied by your hand. This results in a net force of zero, which means the mass will continue to move at a constant velocity, without any external forces acting on it.

In summary, the mass is accelerating due to the unequal masses involved, and the equal and opposite forces are cancelling each other out, resulting in a constant velocity. This is a fundamental concept in Newton's Second Law of Motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The equation f(t) = m(dv/dt) is indeed applicable in this scenario, where the net force (f) is equal to the mass (m) multiplied by the change in velocity over time (dv/dt).