User:Egm.s12.team13.rlh

R1.3 A free body diagram of the spring-mass-dashpot found in Sec 1 of the notes must be broken up into three different parts. The spring, the dashpot, and the mass.

The positive y direction equals $$ \overrightarrow{y+} \! $$

The spring's free body diagram is: $$ F_k \longleftarrow\! \land\land\land \longrightarrow\! F_k $$

The dashpot's free body diagram is: $$ F_k \longleftarrow\!\sqsubset\sqsupset\longrightarrow\! F_c $$

The free body diagram of the mass is: $$ F_c \longleftarrow\! \bigcirc \longrightarrow\! F(t) $$

Since there are only two opposing forces on the dashpot those two forces must be equal. $$ F_c = F_k = F_i $$

By definition $$ ma = my'' $$

Using Newtons second law :$$ \sum F = ma$$

we get $$ \sum F_y = F(t) - F_i = my'' $$

Rearranging yields

$$ F(t) = F_i + my''$$