Domino Fall

Physics Root

One domino falls into its neighbor pushing it over.

The Forces and torques in the falling domino problem.

Lean one domino into its neighbor and ask if the first domino will push over the second.

The first domino has height H1, width W1, and thickness T1.

The second domino H2,W2, and T2.

The forces on the first domino include the force of gravity, Fg1 at its center of mass.
The Normal force from the floor, FN1.
The friction force with the floor, Ff1.
And, the contact force from the second domino, FC21.
There may also be a friction force where the first domino touches the second. This force is not shown.

If we choose the pivot point for torque calculations to be the point where the first domino touches the floor then we find that FC21 *H1 cosTh = Fg1*H1/2 sin Th

Where Th is the angle between the dominos.

The distance between dominos is the width of the domino W1.

So Sin Th = W1/H1 and Cos Th = HC/H1

Substituting these for the sin and cos we get

FC21 *H1 *HC/H1 = Fg1*H1/2 W1/H1 or

FC21 = (Fg1/2) W1/HC

If the first domino is at rest then the torque produced by the force of gravity and the normal force from the floor, must equal the opposite torque from the friction force from the floor and the contact force of the second domino.

The contact force of the first domino on the second, FC12, is equal and opposite to the contact force of the second on the first FC21 by Newton's law that for every action there is an equal and opposite reaction.

This force FC12 provides the torque which topples the second domino.
The second domino pivots about the corner above the caret ^ . So the torque from the first domino is FC12 times the height at which the first domino hits the second hC. The domino topples if this torque is greater than the restoring torque from gravity Fg2 times the distance of the center of mass from the pivot point, half the thickness of the domino T2/2.

So the domino topples if FC12 * hC > Fg2 * T2/2

plugging in for FC12 from above

(Fg1/2) W1/HC > Fg2 * T2/2

or HC > Fg1W1/Fg2T2

Now:

Fg2 = 3.375 Fg1

W1 = 6

T2 = 1.5

so

If HC > 6/(3.375*1.5)

If HC > 1.185