A
15-kg block is on a ramp which is inclined at 20° above the horizontal.
It is connected by a string to a 19-kg mass which hangs over the top
edge of the ramp. Assuming that frictional forces may be neglected, what
is the downward acceleration of the 19-kg block?

Given m_{1} = 15 kg, m_{2}
= 19 kg, θ
= 20°.
Find the acceleration a.

The
system moves with a constant acceleration a, that is the net force
acting on the first mass is

F_{1}
= m_{1}a
and the net force acting on the second mass if F_{2}
= m_{2}a.
The forces acting on the first mass are the weight W_{1}
= m_{1}g,
the reaction N, and the tension force T_{1}.
Note that the force N is compensated by the normal component of the
weight W_{1,n}
= m_{1}gcos
θ and the net force is
directed along the inclined plane and equals F_{1}
= m_{1}a
= T_{1 }–
m_{1}gsinθ,
where the latter is the second component of the weight W_{1}.

For
the second mass we have F_{2}
= m_{2}a
= m_{2}g– T_{2}.
Note that the string has no mass and therefore, the forces T_{1
}and T_{2
}have the same
magnitude. By adding the two above equations, we get