Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block a(t) as a function of time. Express your answer in terms of k, m, and the coordinate of the block x(t). Combine the expressions for Hooke's law F=−kx and Newton's 2nd law F=ma. Note that, since the initial coordinate is zero, the deformation of the spring at any time equals the coordinate of the block x(t).

where [tex]k[/tex] is the spring constant and [tex]x(t)[/tex] is the displacement function measured from the origin. The negative sign indicates the force acts in opposite direction to the displacement. In fact, it is a restoring force; it acts to return the spring to its original undisturbed position.

Since both forces are the same,

[tex]F = ma= - kx(t)[/tex]

[tex]a=-\dfrac{k}{m}x(t)[/tex]

The implication of this is that the acceleration is proportional to the displacement but opposite to it. That last statement is the definition of a simple harmonic motion which this is.

The ratio [tex]\dfrac{k}{m}[/tex] is a constant except in situations where the mass is varying (say, the mass on the spring is a decaying material).

throwdolbeau throwdolbeau · Answer 2 · 2021-12-28T13:33:35+01:00

On combining the expressions for Hooke's law and Newton's 2nd law we will get:

[tex]a=-\frac{k}{m}x(t)[/tex]

From Newton's second law:

It states that acceleration (gaining speed) happens when a force acts on a mass (object). It is given by:

[tex]\text{F}=m*a[/tex]

where F is the force, m is the mass and a is the acceleration.

From Hooke's law:

It states that the displacement or size of the deformation is directly proportional to the deforming force or load. It is given by:

[tex]\text{F}=-kx(t)[/tex]

where k is the spring constant and x is the displacement function measured from the origin. The negative sign indicates the force acts in opposite direction to the displacement.

In fact, it is a restoring force; it acts to return the spring to its original undisturbed position.

Since, both forces are the same. Thus,

[tex]\text{F}=m*a=-kx(t)\\\\a=-\frac{k}{m} x(t)[/tex]

This implies that the acceleration is proportional to the displacement but opposite to it. This defines simple harmonic motion which is a special type of periodic motion where the restoring force on the moving object is directly proportional to the magnitude of the object's displacement.

The ratio [tex]\frac{k}{m}[/tex] is a constant except in situations where the mass is varying (say, the mass on the spring is a decaying material).

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