The Center of Gravity — Question 8

Question 8

Tilting a block raises its center until the tipping threshold

A uniform rectangular block has base width $b=.60\,\mathrm m$ and height $h=1.20\,\mathrm m$. It is slowly tilted clockwise about its lower-right corner, without sliding.

1. Find the tilt angle at which gravity has zero moment about that corner.

2. Explain which side of that angle is restoring and which side promotes overturning.

Question 8 - Solution

Tilting a block raises its center until the tipping threshold

(a) Derivation and calculation

Relative to the lower-right pivot, clockwise rotation through $\theta$ gives $$x_G=-\frac b2\cos\theta+\frac h2\sin\theta.$$ The weight passes through the pivot when $x_G=0$: $$\tan\theta_c=\frac bh=\frac{.60}{1.20},\qquad \boxed{\theta_c=26.6^\circ}.$$

(b) Result and interpretation

The torque and potential are $$\tau_z=-Mgx_G,\qquad U=Mg\left(\frac b2\sin\theta+\frac h2\cos\theta\right).$$ Below $\theta_c$, $x_G<0$ and the torque restores the block; above it, $x_G>0$ and tipping continues. At the threshold, $$U'(\theta_c)=0,\qquad U''(\theta_c)=-\frac{Mg}{2}\sqrt{b^2+h^2}<0.$$ The threshold is an unstable energy maximum.

Physical check

The critical angle decreases for a taller block with the same base width.