Kevin S. Huang

2016: Problem 4

Topic: Dynamics
Concepts: Circular motion, Inclined plane, Limiting cases

Solution:

Looking top-down at the bead sliding down the helix, we see that it has a tangential acceleration $a_t$ determined by the pitch of the helix and a radial acceleration $a_r=v^2/r$ . The total acceleration is

a=\sqrt{a_t^2+a_r^2}

Since $v=a_tt$ , we have

a(t)=\sqrt{a_t^2+\left(\frac{a_t^2t^2}{r}\right)^2}=a_t\sqrt{1+\frac{a_t^2t^4}{r^2}}

To determine the graph of $a(t)$ , we look at small and larges times. For $t \rightarrow 0$ ,

a \rightarrow a_t=\text{const.}

For $t \rightarrow \infty$ ,

a \rightarrow \frac{a_t^2t^2}{r} \propto t^2

Thus, the answer is D.

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Kevin S. Huang

2016: Problem 4

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2016: Problem 4

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