Rotational Energy Homework - Calculate Moment of Inertia & KE

[bkl4life]

Homework Statement

A dancer spinning at 72 rpm about an axis through her center with her arms outstretched. The distribution of mass in a human body is
Head: 7%
Arms 13%
Trunk and legs: 80%
Using your own measurements on your body calculate your
a) moment of inertia about your spin
b) rotational kinetic energy

Homework Equations

I decided to use two equations: the solid sphere and solid cylinder
I=2/5 MR^2
I=1/2 MR^2

The Attempt at a Solution

75 kg for weight, 10 cm for head, 60 cm for arms, 90 cm for trunk.

.07*75=5.25
.13*75=9.75
.8*75=60

Head: 2/5*(5.25)(5)^2=52.5
Arm: 1/2*(9.75)*(30)^2=4387.5
Trunk: 1/2(60)(45)^2=60750
I add those up and got 65190I'm not sure if I did this right.

 

[bkl4life]

Never mind, I saw what I did wrong!

 

[thomate1]

Your calculations seem to be correct. Moment of inertia is a measure of an object's resistance to rotational motion, so the higher it is, the more energy is required to rotate the object. In this case, the distribution of mass in the body plays a significant role in determining the moment of inertia. Your calculations show that the trunk has the highest moment of inertia, followed by the arms and then the head. This makes sense since the trunk and legs together make up the majority of the body's mass.

To calculate the rotational kinetic energy, we can use the equation KE = 1/2 Iω^2, where I is the moment of inertia and ω is the angular velocity (in radians per second). In this case, the angular velocity can be calculated by converting 72 rpm to radians per second (1 rpm = 2π/60 rad/s). So, ω = (72*2π/60) = 12π/5 rad/s.

Plugging in the calculated moment of inertia (65190) and the angular velocity (12π/5) into the equation, we get KE = 1/2 (65190) (12π/5)^2 = 126,229 J.

This is the rotational kinetic energy of the dancer spinning at 72 rpm. It is important to note that this is just an estimate based on your body measurements and other factors such as air resistance and friction may affect the actual value. Overall, your calculations are correct and provide a good estimate of the moment of inertia and rotational kinetic energy in this scenario.