🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning California State Polytechnic University, Pomona ### Problem 113 Easy Difficulty # In Section 8.5 we calculated the center of mass by considering objects composed of a finite number of point masses or objects that, by symmetry, could be represented by a finite number of point masses. For a solid object whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums of Eqs.$(8.28)$must be generalized to integrals$$x_{\mathrm{em}}=\frac{1}{M} \int x d m \quad y_{\mathrm{em}}=\frac{1}{M} \int y d m$$where$x$and$y$are the coordinates of the small piece of the object that has mass$d m$. The integration is over the whole of the object. Consider a thin rod of length$L,$mass$M,$and cross-sectional area A. Let the origin of the coordinates be at the left end of the rod and the positive$x$-axis lie along the rod. (a) If the density$\rho=M / V$of show that the$x$-coordinate of the center of mass of the rod is at its geometrical center. (b) If the density of the object varies linearly with$x-$that is,$\rho=\alpha x$, where$\alpha$is a positive constant - calculate the$x$-coordinate of the rod's center of mass. ### Answer ## (a)$L / 2$(b) 2$L / 3\$

#### Topics

Moment, Impulse, and Collisions

### Discussion

You must be signed in to discuss.
##### Christina K.

Rutgers, The State University of New Jersey

##### Andy C.

University of Michigan - Ann Arbor

##### Jared E.

University of Winnipeg

Lectures

Join Bootcamp

### Video Transcript

California State Polytechnic University, Pomona

#### Topics

Moment, Impulse, and Collisions

##### Christina K.

Rutgers, The State University of New Jersey

##### Andy C.

University of Michigan - Ann Arbor

##### Jared E.

University of Winnipeg

Lectures

Join Bootcamp