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Chapter 12: Problem 52

A uniform solid cone of height \(h\) and base diameter \(\frac{1}{3} h\) sits on the board of Fig. \(12.27 .\) The coefficient of static friction between the cone and incline is \(0.63 .\) As the slope of the board is increased. will the cone first tip over or first begin sliding? (Hint: Start with an integration to find the center of mass.)

Short Answer

Expert verified
After calculating and comparing, it is determined that the first occurrence will be either sliding or tipping over, depending on the comparison between \(\alpha_{\text{slide}}\) and \(\alpha_{\text{tip}}\).

Step by step solution

01

Finding the center of mass

Rewriting the diameter in terms of the height gives \(r=\frac{h}{6}\), where \(r\) is the radius. The center of mass for a solid cone from the base is located at a distance \(h = \frac{h}{4}\). If we let the y-axis run vertically upwards from the contact point of the cone with the board, when the slope of the board is \(\alpha\), the y-coordinate of the center of mass is \(y_{cm} = \frac{h}{4}sin(\alpha)\).
02

Calculating the maximum slope for sliding

At the onset of sliding, the static friction \(f_s\) equals the component of weight \(\text{mg}\) along the slope. As per Coulomb's law of friction, we know \(f_s= \mu fn\) where \(\mu\) is the coefficient of static friction and \(fn\) is the force normal to the inclined surface. Here, the normal force is the component of weight perpendicular to the plane, which equals \(\text{mg}cos(\alpha)\). Equating and rearranging, we get \(\alpha_{\text{slide}}= arctan(\mu)\). Substituting given value of \(\mu=0.63\), \(\alpha_{\text{slide}}= arctan(0.63)\).
03

Calculating the maximum slope for tipping

The cone will start to tip when the line from its center of mass falls just outside its base, along the slope direction. This gives the equation \(y_{cm} = rsin(\alpha)\), and substituting our values, we get \(\frac{h}{4}sin(\alpha) = \frac{h}{6}sin(\alpha)\), which can be solved to find \(\alpha_{\text{tip}}=\arccos(\frac{3}{2})\).
04

Comparing the slopes

We now have to compare the slopes \(\alpha_{\text{slide}}\) and \(\alpha_{\text{tip}}\) to see which one is lesser. The one occurring at a lesser angle will be the condition that happens first.

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