Ponderable: Activity - Spring and two masses
WID 1156913 (Problem 8.P.28 – used in M&I labs)
Encourage students to use the following process:
 Draw diagrams of initial and final state of the point particle system
o Include all forces acting on the system
o Every force should act on the center of mass
 Write Energy Principle for the point particle system
 Draw diagrams of initial and final state of the real system
o Include all forces acting on the system
 Calculate the work from each force
 Write Energy Principle for the real system
 Use ∆Ktrans from point particle system in real system
You hold up an object that consists of two blocks at rest, each of mass M = 5 kg,
connected by a low-mass spring. Then you suddenly start applying a larger upward force,
of constant magnitude F = 157 N (which is greater than 2Mg). The diagram shows the
situation some time later, when the blocks have moved upward, and the spring stretch has
increased.
The heights of the centers of the two blocks are as follows:
Initial and final positions of block 1: y1i = 0.3 m, y1f = 0.4 m
Initial and final positions of block 2: y2i = 0.7 m, y2f = 1.0 m
It helps to show these heights on a diagram. Note that the initial center of mass of the two
blocks is (y1i + y2i)/2, and the final center of mass of the two blocks is (y1f + y2f)/2.
(a) Consider the point particle system corresponding to the two blocks and the spring.
Calculate the increase in the total translational kinetic energy of the two blocks. It is
important to draw a diagram showing all of the forces that are acting, and through what
distance each force acts.
Epoint particle system  W  Q
K translation  Fnet external rCM
 y1f  y2 f y1i  y2i 
 Fnet, y yCM, f  yCM, i  (F  Fgrav ) 

2
2 





 157 N  2 5 kg  9.8
m
s2
 0.4 2 1.0  0.3 2 0.7 
 157 N  98 N 0.7 m  0.5 m   59 N 0.2 m 
K translation  11.8 J
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Many students have difficulty with calculating center of mass because they confuse
which height is which (diagrams are helpful).
For Fgrav , check that students are using 2M (mass of both blocks) instead of M (mass of
only one block).
(b) Consider the real system corresponding to the two blocks and the spring. Calculate
the increase of (Kvib+Us), the vibrational kinetic energy of the two blocks (their kinetic
energy relative to the center of mass) plus the potential energy of the spring. It is
important to draw a diagram showing all of the forces that are acting, and through what
distance each force acts.
Ereal system  W  Q
r r r
K total   F1 r1  F2 gr2  ...
K translation  K vibration  Uspring  Fy2  Fgrav yCM




K vibration  U spring  F y2 f  y2i  2Mg yCM, f  yCM, i  K translation
 157 N 1.0 m  0.7 m   98 N 0.2 m   11.8 J
 47.1 J  19.6 J  11.8 J
K vibration  U spring  15.7 J
The biggest confusion is in the fact that in the real system, the forces move through
different displacements
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