Dynamics High School Question [closed]

Question

Alright so, my book says that the answer for the acceleration is $3.0 \;\mathrm{m\, s^{-2}}$ but I got $1.6 \;\mathrm{m\, s^{-2}}$

Let the directions of acceleration be the positive directions

Equation 1: \begin{align} F_{net,A} &= F_t - F_g \iff \\ m_1 a &= F_t - m_1 g \end{align}

Equation 2: \begin{equation} F_{net,B,x} = F_{g,x}+F_f - F_t, \end{equation} this is what I think is correct, but my book says this equation is actually $F_{net\,B,x} = F_{g,x} - F_f - F_t$, which I don't understand because the direction of these forces are different for block $B$? Shouldn't friction and the direction gravity in the $x$-plane be added together? Why am I subtracting when I think I should be adding? This is the only part I need help with, thanks in advance.

Answer 1

Remember frictional force goes always in the opposite direction of the movement. This is expressed by changing the sign respect the force of movement, in this case the force that pull the blocks is the gravitational force,i.e, the weight of the blocks. It seems you are working in a reference frame such that Fgx is positive for the Block B sliding up the ramp. Then the frictional force, and also the tension force, should have opposite sign, that's why they are subtracting.

Answer 2

Part of your problem seems to arise because you assume the positive direction to be the direction of acceleration and then treat down for block A as the negative direction. And assuming the directions as you note the force on block B should be \begin{equation} F_{net}B_{x} = F_{tension} - F_{friction} - F_{gx}, \end{equation} that is the net force is the force due to tension minus the force due to friction of the box on the slope and then subtract the force due to gravity "down" the box.