Question

which dynamics equation represents the behavior of mass and spring systems?
○ $f = -k (xf - x0)$
○ $vf = a t + v0$
○ $f = m (vf - vo) / t$
○ $f = \
ho v g$
○ $f = m a$
○ $x = 1/2 a t^2 + v0 t - x0$

Explanation:

Step1: Recall Hooke's Law

Hooke's Law for a spring states that the force exerted by a spring is \( F = -k(x_f - x_0) \), where \( k \) is the spring constant, \( x_f \) is the final displacement, and \( x_0 \) is the initial (equilibrium) displacement. The negative sign indicates the force is restoring (opposes displacement).

Step2: Analyze other options

• \( v_f = at + v_0 \): Kinematic equation for velocity (not spring - mass force).
• \( F = m\frac{v_f - v_0}{t} \): Impulse - momentum relation (not spring - mass specific).
• \( F=

ho Vg \): Buoyant force (Archimedes' principle, not spring - mass).

• \( F = ma \): Newton's second law (general, not specific to spring - mass dynamics).
• \( x=\frac{1}{2}at^{2}+v_0t - x_0 \): Kinematic equation for position (not spring - mass force).

Answer:

\( F=-k(Xf - X0) \) (the first option)