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(II) The human leg can be compared to a physical pendulum, with a "natural" swinging period at whichwalking is easiest. Consider the leg as two rods joined rigidly together at the knee; the axis for the leg is the hip joint. The length of each rod is about the same, 55 $\mathrm{cm} .$ The upper rod has a mass of 7.0 kg and the lower rod has a mass of $4.0 \mathrm{kg},(a)$ Calculate the natural swinging period of the system. $(b)$ Check your answer by standing on a chair and measuring the time for one or more complete back-and- forth swings. The effect of a shorter leg is a shorter swinging period, enabling a faster "natural" stride.

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A. $$1.64 \mathrm{s}$$B. It took 7.2 seconds for 5 swings, which gives a period of 1.4 seconds. That is reasonable qualitative agreement.

Physics 101 Mechanics

Chapter 14

Oscillators

Motion Along a Straight Line

Motion in 2d or 3d

Periodic Motion

Rutgers, The State University of New Jersey

Simon Fraser University

McMaster University

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

03:51

When walking; the leg not …

01:12

While seated on a tall ben…

03:13

Go Two physical pendulums …

01:50

A pendulum is made from a …

02:10

Two physical pendulums (no…

04:03

(II) A physical pendulum c…

03:37

The two pendulums shown in…

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02:17

10:30

A system consists of a spr…

so here for part A, we want to find the period, so we first must find the moment of inertia, the moment of inertia. Upper plus moment of inertia lower. So this will be equal to 1/3. Um, l squared Plus won over 12 lower case and l squared plus m times three over to El Quantity squared. And so we can say that I The moment of inertia total will be equal to capital out em over three plus seven lower case M divided by three times elsewhere. And so h would be equal to the X coordinate of the center of mass. And this would be equal to mass times. Uh, well, over two plus lower case and times three AL over, too. And then this would be divided by, um, plus lower case M. And so the X coordinate for the center of mass equaling H would be equal to al the length times 1/2 and plus three on over to and then this will be divided by the total mass. And so the period we know to be of a physical pendulum to be two pi times the square root of the moment of inertia divided by the total mass divided by the gravitational acceleration due to gravity times H and so we can actually substitute and say that t is gonna the period's gonna be equal to two pi times. What's capital am? Divide by three plus seven lower case M about by three times l squared and then this will be divided by the total mass times the acceleration due to gravity times I'm over two plus three, Um, over too divided by the total mass times l. And at this point, we can actually solve rather, let's ah, we can actually simplify a little bit more and said This is gonna be to pie times. Uh oh. I'm over three plus seven em over three. Sorry, that's gonna be capital. And this will be divided by g times. Um, over too. Plus three em over two. And we can solve So t equals two pi times the length of 0.55 meters times 1/3 of seven kilograms, plus 7/3 of four kilograms. And then this is divided by, uh, 9.8 meters per second squared times. Um, 1/2 times seven kilograms, plus three over two times four kilograms. Okay. And at that point, we can then evaluate and the period is found to be 1.65 seconds. So this would be your answer for party and then for part B. Ah, they're asking us, um, is the ah is what she observes a reasonable agreement. So the person observes five swings, and for five swings, it takes them 7.2 seconds. It takes the pendulum of 7.2 seconds. Therefore, the tea observed would be equal to 7.2 seconds divided by five swings. And this is equally 1.4 seconds. So 1.4 seconds, um, as opposed to 1.65 seconds. Ah, this is in Ah, reasonable agreement. I wouldn't say particularly good, but reasonable. It's on the same order of magnitude. Um, and it's on lee 0.2 seconds off. Ah, however, this is reasonable agreement, given that there's many other factors that are affecting the pendulum besides gravity and again 1.4 seconds. That is the end of the solution. Thank you for watching

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