The grand jeté is a classic ballet maneuver in which a...

The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure $\mathrm{P} 7.81 \mathrm{a}$. The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter $3,$ we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won't come as a surprise to learn that it's the center of gravity. But when you watch a dancer leap through the air, you don't watch her center of gravity, you watch her head. If the translational motion of her head is horizontal-not parabolicthis creates the illusion that she is flying through the air, held up by unseen forces. Figure $\mathrm{P} 7.81 \mathrm{b}$ illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It's not flight, but it will appear that way, at least for a moment. To perform this maneuver, the dancer relies on the fact that the position of her center of gravity A. Is near the center of the torso. B. Is determined by the positions of her arms and legs. C. Moves in a horizontal path. D. Is outside of her body.

The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure $\mathrm{P} 7.81 \mathrm{a}$. The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter $3,$ we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won't come as a surprise to learn that it's the center of gravity. But when you watch a dancer leap through the air, you don't watch her center of gravity, you watch her head. If the translational motion of her head is horizontal-not parabolicthis creates the illusion that she is flying through the air, held up by unseen forces.

Figure $\mathrm{P} 7.81 \mathrm{b}$ illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It's not flight, but it will appear that way, at least for a moment.
To perform this maneuver, the dancer relies on the fact that the position of her center of gravity
A. Is near the center of the torso.
B. Is determined by the positions of her arms and legs.
C. Moves in a horizontal path.
D. Is outside of her body.

Key Concepts

Biomechanics of Human Movement

In human movement, the effective position of the center of gravity is influenced by the distribution of mass, which can change as limbs and other body parts move. This concept is a cornerstone in biomechanics, explaining how altering body posture and limb positions can affect balance, motion efficiency, and the appearance of the movement.

Illusion of Flight

The illusion of flight in dynamic movements is created by manipulating the relative motion of different body parts. While the center of gravity follows a natural parabolic path due to gravity, other parts of the body, such as the head, can be moved to follow a more linear, horizontal trajectory. This divergence in movement paths can lead to the visual effect that the entire body is floating or flying, despite the underlying physics.

Center of Gravity

The center of gravity is the point in an object or body where the entire weight can be considered to be concentrated. In dynamics, especially during free fall or motion under gravity, the center of gravity follows a predictable path, typically a parabola, regardless of how other parts of the body move. This principle is crucial for understanding motion in both mechanics and biomechanics.

Projectile Motion

Projectile motion is the motion of an object or body that is thrown or projected into the air, and is subject to only the acceleration due to gravity. This motion describes a parabolic trajectory when air resistance is negligible, illustrating how bodies move under the influence of gravitational forces.

Transcript

00:01 The problem explains how a dancer can give the illusion of flight by adjusting where her head is compared to her center of gravity.

00:13 So when somebody jumps through the air, they're actually in a free fall.

00:18 So they have, you know, their center of gravity will go through parabolic motion.

00:24 Now, by adjusting where the center of gravity is, you can make.

00:30 Your head appear that it actually just staying at the same height.

00:36 So when you spread your arms and legs out, the center of gravity moves up, which means for your whole body, which means that it's closer to your head.

00:48 So your head moves down relative to the center of gravity, which is moving in a parabolic path.

00:55 So if you do it correctly, and you move your center of gravity in kind of the opposite parabolic like path, you will look like your head is staying at the same height.

01:08 And so that gives kind of the illusion that somebody is flying because their head is staying at their same height.

01:17 And that's usually what you focus on.

01:20 And so it looks like, you know, they're just kind of jumping but not actually lifting their head off the air of the you know and their head doesn't appear to follow a parabolic path like you would expect so it makes it look like they're just uh kind of gliding at this you know along which is kind of looks like somebody is flying so i'm going to answer all these questions in the in this problem here because they all relate to this this um phenomenon.

01:59 So to perform this maneuver, the dancer relies on the fact that the position of her center of gravity.

02:05 Well, as i said in this before just now, it's um, um, moves, uh, is determined by the positions of her arms and legs.

02:14 So that's answer b.

02:16 And again, why that is is because her center, the center of gravity, um, moves up with respect to, um, for her whole body moves up with respect to, um, for her whole body moves up with respect to her torso as she moves her arms and legs as she brings them from her from being down to out vertical so her center of gravity moves up and so she can do this it's hard to it's hard to make it move down you can't really make it move down if you just stand with your arms and legs your arms hanging in your legs straight down your center of gravity will be at the lower point that you can pretty much possibly make it if you're, you know, standing at least.

03:07 Now, if you spread your, if you jump and you spread your arms and legs out, then it will rise up.

03:18 So again, and it will rise up closer to your head, meaning that, again, that causes this motion to look like her head is staying, remaining at the same height.

03:31 The second question says, suppose you wish to make a vertical leap with the goal of getting your head as high as possible above the ground.

03:39 At the top of the leap, your arm should be what? held at your sides raised above your head are outstretched away from your body.

03:47 Well, if you want to get your head as high as possible, that means you should have the center of gravity as far away from your head as possible, which means that you want it to be as low as possible, which means that you want your arms at your sides.

04:06 So in the end, at least at the top of the jump, you want your arms to be at your sides.

04:11 Now, if you start the jump and your arms are above your head, now your center of gravity is higher than it normally would be.

04:25 And then as you bring your arms down, your center of gravity would move down, and that would actually make your head seem like it, is moving higher than it normally would if you were just jumping without changing your body position.

04:44 So i suppose if you want to make your head go as high as possible, you jump with your arms up and then at the top of the leap you put your arms down.

04:54 And sometimes this actually makes it look like you're kind of swimming.

04:58 You know, you're like pushing yourself, like you're flapping your wings and pushing your pushing air down to move your head up.

05:05 But it's really just you're changing your center of mass.

05:07 It's not that you're kind of flying by flapping your arms.

05:15 So in the third problem says when the dancer is in the air, is there a gravitational torque on her? and take the dancer's rotation axis to be through her center of gravity...

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