Question

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).

Answer 1

Answer:

$a=-\dfrac{k}{m}x(t)$

Explanation:

From Newton's second law,

$F = ma$

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

From Hooke's law,

$F = -kx(t)$

where $k$ is the spring constant and $x(t)$ 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,

$F = ma= - kx(t)$

$a=-\dfrac{k}{m}x(t)$

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 $\dfrac{k}{m}$ is a constant except in situations where the mass is varying (say, the mass on the spring is a decaying material).