Answer to Question #100188 in Mechanics | Relativity for tine

Question

A student was asked to stack up uniform blocks each
measuring 25.0 cm long and 3.25 cm thick so that
alternately 2.5 cm and 5.0 cm extended beyond the block
beneath. How many blocks can be stacked before the stack
falls over? How high is the center of gravity of the last stack
that will not fall over?

Answer 1

When a student stack up one block on another with a displacement the weight force is balanced by support reaction. Not only force is balanced bat also a moment of force. This will be true until the center of gravity of the upper blocks system does not extend beyond the edge of the some down block. This is always true for the lowest block, since its fulcrum point A' coincides vertically with the point A of fulcrum point of the upper blocks. Therefore, the lower block #0 remains stationary and cannot tip over. Each upper block also cannot fall from the previous block, since its center of gravity is far from the edge of the lower block. Thus, the blocks can fall only together at the moment when the center of gravity of their system goes beyond the edge of any supporting block. Most likely it will be the edge of the lowest block #0. The center of gravity of the block system should be determined by the formula: 

(1) $X_c=\frac{\sum_i mx_i}{\sum_i m}$

As all blocks have the same mass we can rewrite (1)

(2) $X_{cN}=\frac{\sum^{N}_{i=1} x_i}{N}$

For the student's task we have

$x_1=2.5; x_2=7.5;x_3=10;x_4=15;x_5=17.5;x_6=22.5;$

$X_{c6}=\frac{2.5+7.5+10+15+17.5+22.5}{6}=12.5$

The 6th block is the last since futher the center of gravity blocks system will pass over the edge of the ground block #0. We can check the system not fall from first block #1. For this task $x_1=5; x_2=7.5;x_3=12.5;x_4=15;x_5=20;$ and

$X_{c5}=\frac{5+7.5+12.5+15+20}{5}=12$ blocks will not fall down from block #1.

The center of gravity of the last block #6 will be at height 3.25*6+3.25/2=21.125 cm from the suport.

Answer:

Seven blocks can be stacked on the support before the stack of six blocks falls over from basis block.

The height of the center of gravity of the last block is 21.125 cm.