In the 1968 Summer Olympic Games, University of Oregon high...

In the 1968 Summer Olympic Games, University of Oregon high jumper Dick Fosbury introduced a new technique of high jumping called the "Fosbury flop." It contributed to raising the world record by about 30 cm and is presently used by nearly every world-class jumper. In this technique, the jumper goes over the bar face up while arching his back as much as possible as shown in Figure P9.40a. This action places his center of mass outside his body, below his back. As his body goes over the bar, his center of mass passes below the bar. Because a given energy input implies a certain elevation for his center of mass, the action of arching his back means his body is higher than if his back were straight. As a model, consider the jumper as a thin, uniform rod of length $L$. When the rod is straight, its center of mass is at its center. Now bend the rod in a circular arc so that it subtends an angle of $90.0^{\circ}$ at the center of the arc as shown in Figure P9.40b. In this configuration, how far outside the rod is the center of mass?

Key Concepts

Geometric Analysis of Curved Bodies

Understanding the geometry of curved bodies entails analyzing properties like arc length, subtended angle, chord length, and the relationship between the arc and its center of curvature. This analysis is essential for determining how modifications in shape (such as bending) affect the position of the center of mass relative to the body, which in turn can influence the system’s mechanical performance.

Center of Mass

The center of mass of a body is the single point at which the entire mass of the body can be considered to be concentrated. In mechanics, analyzing the center of mass simplifies the study of motion and energy because the gravitational potential energy of a system depends only on the height of its center of mass, not on the details of its mass distribution.

Centroid of a Circular Arc

When dealing with objects that have been bent or curved, such as a uniform rod formed into a circular arc, finding the centroid (or the geometric center) is crucial. For a uniform circular arc, the centroid is not located at the midpoint of the arc, and its calculation involves integrating over the length of the arc to determine the average position of the mass distribution relative to the arc’s geometry.

Transcript

00:01 In this question, it is about modeling a high jumper as a thin rod, and the rod is banded in this manner, and such that at the center is 90 degrees, and then we are asked to find the center of mass of this system, of this rod, and how far outside is the the cm from the center.

00:35 So based on this diagram, we kind of know that the center of mass should be somewhere here.

00:42 And we are interested in finding this distance or delta y.

00:53 So first we say that the x, cm is going to be zero.

00:59 So this is our y -axis.

01:01 This is our origin and this is our x -axis.

01:05 And so the x the center of mass the x coordinates of the center of mass is zero and then for the y coordinate it will be given as one over m y integral y dm okay so um so the first task is to uh define dm okay so for the so we consider a small uh small arc length on the rod which has a mass of dm.

01:44 So the dm would be equal to lambda, which is the mass per unit length times ds.

01:55 So the ds is the small arc length.

01:58 Okay, lambda is mass over l and and s, no, the length is pi over 2 times r.

02:13 So this is r and our s, the s is r the data.

02:27 Okay, so from here we can also write our mass, it is the mass per unit length times high over 2 times r.

02:41 Okay, so this is the art length and times the mass, linear mass density, you get the total mass.

02:48 And also we want to indicate, you want to express the y coordinates of this point, and it is going to be r -signata...

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