Why period is half ? Spring problem

[k31453]

Homework Statement

A mass m slides along a frictionless horizontal surface at speed v initial. It strikes a spring of constant k attached to a rigid wall. After a completely elastic encounter with the spring, the mass heads back in the direction it came from.

(a)In terms of k, m, and v initial, determine how long the mass is in contact with the spring.

(b)In terms of k, m, and v initial, determine the maximum compression of the spring.

Homework Equations

The one thing i don't know is why period = 1/2 ??

The Attempt at a Solution

I know horizontal force will be only the force exert by spring.
I know the algebra below:
f= 1/2π * √(k/m)

Period = P = 1/f ergo
t = P/2
= 1/2* 1/f
= 1/2 * 2π √(m/k)
= π √(m/k)

 

[gneill]

The one thing i don't know is why period = 1/2 ??

Think about the difference between this scenario and the "usual" spring oscillator where the mass is fixed to the end of the spring. What constitutes a full cycle of the oscillation?

 

[Basic_Physics]

I think you mean that the mass will be in contact with the spring for half the period of oscillation (if the mass was attached to the spring).

 

[k31453]

Yes BASIC But how ?

And I know when max displacement the T is going to be 1.

 

[cepheid]

Yes BASIC But how ?

And I know when max displacement the T is going to be 1.

You should listen to what gneill said. Suppose you started at zero compression or stretch (as is the case here) and max inward speed. If the mass had been attached, what would *happen* during a full period of oscillation?

How much of that happens here? Hint: the mass loses contact with the spring when the spring force goes to zero.

 

[Basic_Physics]

half period

Assume the mass is attached to the spring - a necessary condition for SHM to occur.
At position 1 the spring is relaxed.
At position 2 the mass is at its rightmost extreme position and the spring is at max compression.
At position 3 the spring is back to its relaxed state.
At position 4 the mass is at its leftmost extreme position and the spring is at max extension.
At position 5 the spring is again relaxed.