08 No Tipping Allowed

Aim¶

A cylinder sliding to a stop tips only if its diameter-to-height ratio is less than the reciprocal of the coefficient of kinetic friction.

Subjects¶

• 1K20 (Friction)

Diagram¶

Equipment¶

• 5 pvc cylinders, $\varnothing= 80 \mathrm{~mm}$

  • $l_{1}=285 \mathrm{~mm}$

  • $l_{2}=320 \mathrm{~mm}$

  • $l_{3}=335 \mathrm{~mm}$

  • $l_{4}=350 \mathrm{~mm}$

  • $l_{5}=385 \mathrm{~mm}$

• Horizontal surface (smooth table)

Presentation¶

The lateral standing cylinder is given a push by hand. (Push the cylinder on the bottom half, a number of times from left to right and vice versa).

Cylinder $l_{1}$ never tips, $l_{5}$ always tips, $l_{2}, l_{3}$ and $l_{4}$ tip sometimes/often ( $l_{3}$ tips roughly $50 \%$ of the times).

Explanation¶

When the cylinder is about to tip over, the upward normal force acts at the leading edge of the base (Figure 1, point A).

In the decelerating reference frame, $ma$ acts on the center of mass along with the vertical gravitational force $mg$ (See Figure 3).

When the resultant of $ma$ and $mg$ is directed to point $\mathrm{A}$, the cylinder is on the verge of tipping. Figure 3 shows that in that case $\mu_{k}=d / h$.

Remarks¶

Before you can cut the cylinder to the proper heights, we need to know the value of $\mu_{k}$ . $\mu_{k}$ can easily be determined by placing it on an inclined board and finding the angle of incline for which the cylinder slides at constant speed after being given a gentle push. $\mu_{k}=\tan (\alpha)$ ( $\alpha=$ angle of incline).

Sources¶

• Ehrlich, Robert, Turning the World Inside Out and 174 Other Simple Physics Demonstrations, pag. 43

• Friedrich, Artur, Handbuch der experimentellen Schulphysik, part 2, Mechanik der festen Körper, pag. 128