Two syringes of different cross-sections (without needles) filled with water are connected with a tightly fitted rubber tube filled with water. Diameters of the smaller piston and larger piston are $1.0 \; cm$ and $3.0 \; cm$ respectively.
$(a)$ Find the force exerted on the larger piston when a force of $10 \; N$ is applied to the smaller piston.
$(b)$ If the smaller piston is pushed in through $6.0 \; cm$,how much does the larger piston move out?

(A) According to Pascal's Law,pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.
$(a)$ Let $F_1 = 10 \; N$ be the force on the smaller piston with diameter $d_1 = 1.0 \; cm$ and $F_2$ be the force on the larger piston with diameter $d_2 = 3.0 \; cm$.
The areas are $A_1 = \pi (d_1/2)^2$ and $A_2 = \pi (d_2/2)^2$.
Since pressure $P = F_1/A_1 = F_2/A_2$,we have $F_2 = F_1 \times (A_2/A_1) = F_1 \times (d_2/d_1)^2$.
$F_2 = 10 \; N \times (3.0 \; cm / 1.0 \; cm)^2 = 10 \times 9 = 90 \; N$.
$(b)$ Since water is incompressible,the volume of water displaced by the smaller piston must equal the volume displaced by the larger piston.
$V = A_1 L_1 = A_2 L_2$,where $L_1 = 6.0 \; cm$ is the displacement of the smaller piston.
$L_2 = L_1 \times (A_1/A_2) = L_1 \times (d_1/d_2)^2$.
$L_2 = 6.0 \; cm \times (1.0 \; cm / 3.0 \; cm)^2 = 6.0 \times (1/9) = 0.67 \; cm$.