- #1

Cepterus

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## Homework Statement

A ball with radius ##r## is inside a hollow cylinder with radius ##r+R##.

In the first part of the assignment, one has to calculate the minimum kinetic energy the ball has to have at the bottom in order to complete a full loop in the cylinder. It turns out to be ##E_{\text{kin, min}}=\frac{27}{10}mgR##.

Now we suppose the kinetic energy of the ball is 10% less than needed to do the loop. We want to compute the angle ##\theta## at which the ball will lose contact to the wall of the cylinder, using the angle ##\alpha = \pi-\theta##.

## Homework Equations

## The Attempt at a Solution

My idea was to calculate the speed ##v(\alpha)## the ball has for a given angle ##\alpha##. Then I would calculate the vertical component of the centripetal acceleration and equate it with the gravitational acceleration:

\begin{align*}

E_{\text{tot}}& = E_{\text{kin}}+E_{\text{rot}}+E_{\text{pot}} \\

& = \frac12mv(\alpha)^2+\frac12 I\omega(\alpha)^2+mgh\\

&=\frac12mv(\alpha)^2+\frac12 I\left(\frac{v(\alpha)}{r}\right)^2+mg(R+R\cos\alpha)\\

&=\frac12mv(\alpha)^2+\frac12 \cdot\frac25mr^2\left(\frac{v(\alpha)}{r}\right)^2+mgR(1+\cos\alpha)\\

&=\frac12mv(\alpha)^2\left(1+\frac25\right)+mgR(1+\cos\alpha)\\

&=\frac7{10}mv(\alpha)^2+mgR(1+\cos\alpha).

\end{align*}

Now use ##E_{\text{tot}}=0.9\cdot\frac{27}{10}mgR=\frac{243}{100}mgR##:

\begin{align*}

\frac{243}{100}mgR& = \frac7{10}mv(\alpha)^2+mgR(1+\cos\alpha)\\

\frac7{10}v(\alpha)^2&=gR(\frac{243}{100}-1-\cos\alpha)\\

v(\alpha)^2&=\frac{10}7gR(\frac{143}{100}-\cos\alpha)

\end{align*}

Now the vertical component of the centripetal acceleration would be ##\cos\alpha\cdot\frac{v(\alpha)^2}R## and has to be smaller than ##g##, which gives us

\begin{align*}

\frac{\cos\alpha}{R}\cdot\frac{10}7gR(\frac{143}{100}-\cos\alpha)&<g\\

\cos\alpha\cdot\frac{10}7(\frac{143}{100}-\cos\alpha)&<1\\

\frac{143}{100}-\cos\alpha&<\frac{7}{10\cos\alpha}

\end{align*}

so in the end we get the inequality ##\cos\alpha+\frac{7}{10\cos\alpha}>\frac{143}{100}##. If I regard this as an equality to find the minimum angle, however, the equality does not have any solutions. What am I doing wrong?