   Chapter 8, Problem 85AP

Chapter
Section
Textbook Problem

Many aspects of a gymnast's motion can be modeled by representing the gymnast b) four segments consisting of arms, torso (including the head), thighs, and lower legs, as in Figure P8.85. Figure P8.85b shows arrows of lengths rcg locating the center of gravity of each segment. Use the data below and the coordinate system shown in Figure P8.85b to locate the center of gravity of the gymnast shown in Figure P8.85a. Masses for the aims, thighs, and legs include both appendages.  Figure P8.85

To determine
The centre of gravity of the gymnast for a given segment.

Explanation

Explanation

Given Info:

The mass of the arms is 6.87kg , the mass of torso is 33.57kg , the mass of thighs is 14.07kg , the mass of legs is 7.54kg

From the table the length of arms is 0.548m , the length of torso is 0.601m , the length of thighs is 0.374m , the length of legs is 0.350m

From the table the centre of gravity of arms is 0.239m along y direction, the centre of gravity of torso 0.337m along x direction, the centre of gravity of thighs is  0.151m , the centre of gravity of legs is 0.227m . The angle between thigh and horizontal is 60° .

Formula to calculate the position of centre of gravity of gymnast along x-axis is,

x=maxa(cg)+mtxt(cg)+mthxth(cg)+mlxl(cg)ma+mt+mth+ml

• x is the centre of gravity of gymnast along x axis
• ma is the mass of arms
• mt is the mass of torso
• mth is the mass of thighs
• ml is the mass of legs
• xa(cg) is the centre of mass in the x direction of the arms
• xt(cg) is the centre of mass in the x direction of the torso
• xth(cg) is the centre of mass in the x direction of the thighs
• xl(cg) is the centre of mass in the x direction of the legs

Formula to calculate the position of centre of gravity of gymnast along y-axis is,

y=maya(cg)+mtyt(cg)+mthyth(cg)+mlyl(cg)ma+mt+mth+ml

• y is the centre of gravity of gymnast along y axis.
• ya(cg) is the centre of mass in the y direction of the arms
• yt(cg) is the centre of mass in the y direction of the torso
• yth(cg) is the centre of mass in the y direction of the thighs
• yl(cg) is the centre of mass in the y direction of the legs

Formula to calculate the centre of gravity of thighs along x direction is,

xth(cg)=Lt+rcg(th)cosθ

• Lt is the length of the torso
• rcg(th) is the centre of gravity of thighs
• θ is the angle between thighs and horizontal x axis

Substitute 0.601m for Lt , 0.151m for rcg(th) , and 60° for θ to calculate xth(cg) .

xth(cg)=0.601m+(0.151m)cos60°=0.601m+(0.151m)12=0.677m

Formula to calculate the centre of gravity of thighs along y direction is,

yth(cg)=rcg(th)sinθ

Substitute 0.151m for rcg(th) and 60° for θ to calculate yth(cg)

yth(cg)=(0.151m)sin60°=(0.151m)32=0.131m

Formula to calculate the centre of gravity of legs along x direction is,

xl(cg)=Lt+Lthcosθ+rcg(l)

• Lt is the length of the torso
• rcg(l) is the centre of gravity of legs
• Lth is the length of thighs.

Substitute 0.601m for Lt , 0.227m for rcg(l) , 0.374m for Lth and 60° for θ to calculate xl(cg)

xl(cg)=[0

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