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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.

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(a) $L / 2$(b) 2$L / 3$

Physics 101 Mechanics

Chapter 8

Momentum, Impulse, and Collisions

Moment, Impulse, and Collisions

University of Michigan - Ann Arbor

University of Washington

University of Sheffield

University of Winnipeg

Lectures

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

03:30

In physics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Given a force, F, applied for a time, t, the resulting change in momentum, p, is equal to the impulse, I. Impulse applied to a mass, m, is also equal to the change in the object's kinetic energy, T, as a result of the force acting on it.

18:15

In Section 8.5 we calculat…

20:55

0:00

04:53

Consider the objects on th…

03:19

Consider a rod of radius R…

18:27

A slender rod of length $8…

04:39

The Parallel Axis Theorem …

26:33

Find the center of mass of…

04:36

Center of Mass Suppose we …

05:06

Mass from density A thin 1…

05:47

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