The cork is attached by a light spring to the bottom of the water vessel. Both the cork and the spring are completely immersed in the water. Will the length of the spring increase or decrease if the vessel moves vertically up (down) with constant acceleration?
Decision:
Let us write down the equation of motion of the cork in the form
$ma=F_{A}-mg-kx$,
where $F_{A}$ is the hydrostatic pressure force; $a$ is the acceleration of the system; $m$ is the mass of the plug; $k$ is the coefficient of spring stiffness; $x$ is the amount of spring extension. To determine the dependence of $x$ on $a$, it is necessary to determine $F_{A}$. Consider a vessel filled with water and moving vertically upward (along the x-axis) with constant acceleration $a$. Let us mentally isolate a volume inside the liquid and consider the forces acting on it. Since this volume also moves upward with acceleration $a$, the equidistance of all forces applied to it is directed upward and is equal to
$F=F_{A}-Mg=M$,
where $M$ is the mass of the separated volume. From this equality we find
$F_{A}=M(g+a)$,
and from the equation of motion of the plug it follows that
$ kx= (M - m)(a+g)$.
According to the problem condition, the cork is lighter than water, hence, $x > 0$ at $a > 0$.
Decision:
Let us write down the equation of motion of the cork in the form
$ma=F_{A}-mg-kx$,
where $F_{A}$ is the hydrostatic pressure force; $a$ is the acceleration of the system; $m$ is the mass of the plug; $k$ is the coefficient of spring stiffness; $x$ is the amount of spring extension. To determine the dependence of $x$ on $a$, it is necessary to determine $F_{A}$. Consider a vessel filled with water and moving vertically upward (along the x-axis) with constant acceleration $a$. Let us mentally isolate a volume inside the liquid and consider the forces acting on it. Since this volume also moves upward with acceleration $a$, the equidistance of all forces applied to it is directed upward and is equal to
$F=F_{A}-Mg=M$,
where $M$ is the mass of the separated volume. From this equality we find
$F_{A}=M(g+a)$,
and from the equation of motion of the plug it follows that
$ kx= (M - m)(a+g)$.
According to the problem condition, the cork is lighter than water, hence, $x > 0$ at $a > 0$.