In the 1968 Olympic games, University of Oregon jumper Dick...

In the 1968 Olympic games, University of Oregon 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 currently used by nearly every world-class jumper. In this technique, the jumper goes over the bar face up while arching her back as much as possible as shown in Figure P9.76a. This action places her center of mass outside her body, below her back. As her body goes over the bar, her center of mass passes below the bar. Because a given energy input implies a certain elevation for her center of mass, the action of arching her back means that her body is higher than if her 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 90.08 at the center of the arc as shown in Figure P9.76b. In this configuration, how far outside the rod is the center of mass?

Key Concepts

Center of Mass

The center of mass is the weighted average position of the mass in a body and is a central concept in mechanics. It plays a critical role in understanding the motion of a body because it is the point where the gravitational force effectively acts. In problems like the Fosbury flop, knowing the location of the center of mass is essential since it determines the minimal energy required to achieve a certain elevation and helps explain how the jumper’s curvature allows the center of mass to be lower than the body’s highest point.

Uniform Rod Model

Modeling the jumper as a uniform rod is a common simplification that assumes the mass is evenly distributed along the length of the body. This model allows us to use geometric and calculus methods to compute the location of the center of mass. The uniform rod model simplifies the analysis by reducing complex body dynamics to a one-dimensional object with a constant linear density, making the computation of the center of mass more tractable.

Circular Arc Geometry

When a straight rod is bent into a circular arc, its shape can be described using the properties of circles, such as the radius and the subtended angle. The circular arc geometry is used to determine how the curvature affects the distribution of mass along the rod. For a given subtended angle, the arc’s geometric properties allow us to calculate the new location of the center of mass by integrating over the arc length, which is essential in understanding how bending the body improves jumping performance.

Integration of Continuous Mass Distributions

Calculating the center of mass for a continuous body such as a bent rod requires integrating over the mass distribution along its length. This process involves summing up the contributions of infinitesimal mass elements, each weighted by its position. The integration leverages the symmetry and geometric properties of the arc to determine the overall center of mass relative to the body's reference frame, highlighting the importance of calculus in solving physical problems related to energy and motion.

Transcript

00:01 So here you have a situation where you have a rod representing this athlete that's jumping over a bar and you're told that this rod makes an angle, makes some angle here that's equal to 90 .08 degrees.

00:28 Trees and this after here is jumping over the bar and they want you to find how far below the rod the center of mass lies.

00:44 So essentially we want to calculate the center mass with this bent rod system and we know that so the position of the center of mass in this case in the wide direction is just one over the total mass and then we integrate this with some y times the change in mass here t -f so this is what will give us the center mass position i'll call that y prime but the center mass position is going to be calculated from some reference here so in the end we're going to want to say that so the total height here which is again going to be just the radius of that might so that's the radius minus our y prime is going to be equal to the center mass location right below, so the distance from.

01:46 So this distance here that the center of mass is is going to be ycm there.

01:53 So now we have, now we have set up the problem.

01:58 Now the only thing that's left is to actually calculate it in terms of the value.

02:05 That we're given.

02:06 So first, let's simplify these angles for a minute.

02:09 So we know that this thing, let's just assume that it's in the middle.

02:15 And so we know that that's 180 degrees.

02:19 So if it's 90 degrees in the middle here, then you know that that's 44 .96.

02:27 And this angle here will be 135 .5.

02:35 04.

02:38 So now what you want to do is convert these angles to radiance because you want to be able to find the limits here in terms of the angles.

02:53 But in this case, so let's first now work on our integral definitions here.

03:01 So we want to find dm.

03:03 So essentially we assume that we take a small piece of this rod here and want to to calculate its mass.

03:10 But we know that the total mass of the rod is m and the rod is of length l...

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