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# Oscillations: The spring with varying force constant

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Many real springs are more easily stretched than compressed. We can represent this by using different spring constants for and for . As an example, consider a spring that exerts the following restoring force:

A mass on a frictionless, horizontal surface is attached to this spring, displaced to by stretching the spring and released.

1) Find the period of the motion.

2) Does the period depend on ?

yes

no

3) Are the oscillations simple harmonic?

yes

no

4) What is the most negative value of that the mass reaches?

5) Is the oscillation symmetric about the point ?
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https://brainmass.com/physics/energy/oscillations-spring-varying-force-constant-74058

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Many real springs are more easily stretched than compressed. We can represent this by using different spring constants for and for . As an example, consider a spring that exerts the following restoring force:

A mass on a frictionless, horizontal surface is attached to this spring, displaced to by stretching the spring and released.

I think the force is - kx for x > 0
And - 2kx for x < 0

In this case the motion is periodic but one half cycle (x > 0) is with force constant and the other half cycle is (x < 0) with force constant 2k. These half cycles are alternating.

1) Find the period of the motion.

The time period is the sum of the half period of the two SHMs and hence

2) Does the period depend on ?

yes

no

No the period in any SHM does not depend on the amplitude and hence on A.

3) Are the oscillations simple harmonic?

yes

no

Oscillations are not simple harmonic as such but the two half cycles are simple harmonic.

4) What is the most negative value of that the mass reaches?

The value of x is most negative when m comes to rest on the negative x-axis. This can be calculated using the conservation of energy as the total mechanical energy is conserved
Elastic potential energy of the spring at one extreme = Elastic potential energy of the spring at other extreme
Or (1/2)kA2 = (1/2) (2k)(-x)2
Or (only negative sign is applicable for the other extreme position.)

5) Is the oscillation symmetric about the point ?

yes

no

No, as the amplitudes of the two SHMs are not same and force constants are different, the oscillation is not symmetric about the mid point.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!