08 No Tipping Allowed

Aim¶

To show that a cylinder that slides 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¶

On the verge of tipping, 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¶

When constructing the demonstration, you need to know the value of $\mu_{k}$ before you can cut the cylinder to the proper heights. $\mu_{k}$ can easily be determined by placing a short cylinder on an inclined board and finding the angle of incline for which the cylinder slides at constant speed after being given an initial 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