Consider a cylinder of length L and radius R with density given by p(r) = po"/R: a) What is the mass of the cylinder? b) What is the rotational inertia about the axis of the cylinder? c) Suppose you lay the cylinder down on its side on a pool table. Where do you strike the cylinder with the cue such that it begins rolling smoothly and doesn’t spin?

College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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**Cylinder with Variable Density**

Consider a cylinder of length \( L \) and radius \( R \) with density given by the function \( \rho(r) = \rho_0 \frac{r}{R} \).

### Questions:

a) **What is the mass of the cylinder?**

To find the mass of the cylinder, integrate the density function across the volume of the cylinder.

b) **What is the rotational inertia about the axis of the cylinder?**

Calculate the moment of inertia for the cylinder using the density function and integrate over its volume about the axis.

c) **Suppose you lay the cylinder down on its side on a pool table. Where do you strike the cylinder with the cue such that it begins rolling smoothly and doesn’t spin?**

Determine the optimal point to apply a force for initiating smooth rolling motion without spinning, often related to the center of percussion.
Transcribed Image Text:**Cylinder with Variable Density** Consider a cylinder of length \( L \) and radius \( R \) with density given by the function \( \rho(r) = \rho_0 \frac{r}{R} \). ### Questions: a) **What is the mass of the cylinder?** To find the mass of the cylinder, integrate the density function across the volume of the cylinder. b) **What is the rotational inertia about the axis of the cylinder?** Calculate the moment of inertia for the cylinder using the density function and integrate over its volume about the axis. c) **Suppose you lay the cylinder down on its side on a pool table. Where do you strike the cylinder with the cue such that it begins rolling smoothly and doesn’t spin?** Determine the optimal point to apply a force for initiating smooth rolling motion without spinning, often related to the center of percussion.
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