Two objects of equal mass m are connected by a massless rope that
does not stretch. One object lies on a flat table, while the other hangs
over the edge, as shown in the diagram. The coefficient of kinetic friction
between block and table is 0.5.
Draw a free body diagram for each object showing all the forces acting on each.
Block on table
Choose a coordinate system for each diagram.
Fill out the table showing the components for the forces
and of the acceleration in each diagram.
Now apply N2 to each coordinate direction in each diagram.
n1 - mg = 0; T1 - f = ma1; T2-mg = -ma2
How many equations do you have? 3
How many unknowns? n1, f, a1, a2, T1, T2: 6 total.
Use the connection between the two objects to write 2 more equations relating the accelerations of the two particles (1 equation) and the tension forces acting on each block (1 equation).
T1 = T2 (Notice these are force magnitudes.)
a1 = a2 (Similarly these are acceleration magnitudes.)
Finally use the relation between normal force and kinetic friction force to write another equation. Solve to find the acceleration of the hanging block.
f = mkn1
Now we have 6 equations in 6 unknowns: Bingo! Let's write T1 = T2 = T and a1 = a2 = a. Then:
Then: n1 = mg; f = mkmg; T = m(g-a) = m(a + mkg)
g(1 - mk) = 2a.
a = g/4.