Physics 220 Class Exercise October 6 2000

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                                                        hanging block

Choose a coordinate system for each diagram.

Fill out the table showing the components for the forces and of the acceleration in each diagram.
Block on table
Hanging block
Force x1-component y1-component Force x2-component y2-component
normal 0 n1 tension 0 T2
friction -f 0 weight 0 -mg
weight 0 -mg
tension T1 0
Acceleration a1 0 Acceleration 0 -a2

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)

So:                                                          g(1 - mk) = 2a.

Thus                                                                 a = g/4.