BIO Human Rotational Energy. A dancer is spinning at 72 rpm...

BIO Human Rotational Energy. A dancer is spinning at 72 rpm about an axis through her center with her arms outstretched, as shown in Fig. P9.92. From biomedical measurements, the typical distribution of mass in a human body is as follows: \begin{equation}\begin{array}{l}{\text { Head: } 7.0 \%} \\ {\text { Arms: } 13 \%(\text { for both })} \\ {\text { Trunk and legs: } 80.0 \%}\end{array}\end{equation} Suppose you are this dancer. Using this information plus length measurements on your own body, calculate (a) your moment of inertia about your spin axis and (b) your rotational kinetic energy. Use the figures in Table 9.2 to model reasonable approximations for the pertinent parts of your body.

BIO Human Rotational Energy. A dancer is spinning at 72 rpm about an axis through her center with her arms outstretched, as shown in Fig. P9.92. From biomedical measurements, the typical distribution of mass in a human body is as follows: \begin{equation}\begin{array}{l}{\text { Head: } 7.0 \%} \\ {\text { Arms: } 13 \%(\text { for both })} \\ {\text { Trunk and legs: } 80.0 \%}\end{array}\end{equation}
Suppose you are this dancer. Using this information plus length measurements on your own body, calculate (a) your moment of inertia about your spin axis and (b) your rotational kinetic energy. Use the figures in Table 9.2 to model reasonable approximations for the pertinent parts of your body.

Key Concepts

Moment of Inertia

This is a measure of an object's resistance to changes in its rotation about an axis. In rotational dynamics, the moment of inertia depends on the object's mass and how that mass is distributed relative to the axis of rotation. For complex systems like the human body, calculating the moment of inertia involves summing the contributions from different body segments, each approximated by simpler geometric shapes.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation and is given by the formula ½I?², where I is the moment of inertia and ? is the angular speed in radians per second. This concept is central in understanding the energy dynamics of rotating systems, such as a spinning dancer.

Angular Velocity Conversion

In problems involving rotational motion, it is often necessary to convert angular speed from revolutions per minute (rpm) to radians per second. This conversion is crucial since standard equations in rotational dynamics require angular velocity to be in radians per second.

Biomechanical Mass Distribution

When analyzing motions such as spinning in the human body, knowing the distribution of mass across body sections is essential. Different parts of the body contribute unequally to the overall moment of inertia due to variations in mass and distance from the rotation axis. Typical percentage distributions help in modeling the body accurately.

Body Segment Modeling

Modeling a human body for rotational dynamics involves representing body parts as simpler geometric shapes (such as cylinders or rods) whose moments of inertia are easier to calculate. This abstraction allows one to approximate the overall moment of inertia and rotational kinetic energy using established formulas from physics.

Transcript

00:01 The total of the moment of inertia for a typical dancer is equal to 7 % times the moment of inertia of the head, plus 13 % of the moment of inertia of the arms, and then plus the 80 % of the moment of inertia of the trunk and legs of the dancers.

00:28 So we know when the head starts spinning, it's going to be a lot.

00:33 Kind of like the sphere spinning.

00:37 So that's why i -had here can be equal to i -sphere, which is equal to 2 over 5 m r square.

00:43 And then when the r start rotating, it was rotated like a raw, rotate through one end of the raw.

00:53 Ok, so have i -r -r -on here, is equal to ir, which is equal to 1 3rd ml2.

00:59 And when the trunk and the legs start rotating, it's rotating like the rectangular plate.

01:08 Which is rotating about the center of the rectangular plates.

01:13 And that's why we have itrunk and legs here is equal to irac, which is equal 1 over 12m times a square plus b square.

01:21 So now we can do some rearrangement here for the equation.

01:27 So eventually we'll have the total moment of inertia is equal to capital m times 0 .02a r squared plus 0 .043 l square, and m plus 0 .067 times a square plus b square.

01:47 We know capital m here is the mass for a typical dancer, which is about 65 kilograms.

01:54 And r here is the radius of a human head, which is about 0 .1 meter...

Please give Ace some feedback