Work and Mechanical Energy & Moment of Inertia Derivation?

In summary, the conversation discusses using work and mechanical energy to derive the expression for experimentally determined moment of inertia. The equations used include work of friction, potential energy, kinetic energy, torque, and average velocity. The final equation for the experimentally determined moment of inertia is (1/8)(a^2*t^2)((m + mf) + (I/r^2)).
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Homework Statement



We did something very similar to this in lab

http://webenhanced.lbcc.edu/physte/phys2ate/2A LAB HANDOUTS/Moment of Inertia.pdf

Use Work and Mechanical Energy to derive the expression for the experimentally determined moment of inertia.

Homework Equations


Wf= work of friction = Delta E = Ef - Ei

Wf= work of friction = Uf + Kf - Ui - Ki

U= Potential Energy

K = Kinetic Energy

Kf = (1/2)(m + mf)(Vf)^2 + (1/2)(I)(omegaf)^2

I = Moment of Inertia

Omegaf = angular acceleration

Average Velocity = v = (Vf + Vi)/(2)

If Neccessary

s=(1/2)*a*t^2

Torque= F*r= m*r*a

T= (mf + m)(g - a) = tension

Ui= mgh

Uf= mfgh

K(rotate) = (1/2)(I)(omegaf)^2

I = Moment of Inertia

K(linear) = (1/2)(m + mf)(Vf)^2

Experimentally Moment of Inertia

I=r^2(m((gt^2/2s)-t) - mf)

Trying to get to this ^

mf= mass effective not much meaning just mass in kg
If it confusing the gt^2 is divided by 2s then it is subtracted by t and multiplied by r^2 and then minus mf


The Attempt at a Solution



Wf = work of friction = Uf + Kf - Ui - Ki

The final potential energy and initial kinetic energy are both zero so this only leaves

Wf = Kf - Ui

Wf = ((1/2)(m + mf)(Vf)^2) + (1/2)(I)(omegaf)^2 - mgh

Wf = (1/2)(m + mf)(s/t)^2 + (1/2)(I)((s/t)*(1/r))^2 - mgh

Wf = (1/2)(m + mf)(s^2/t^2) + (1/2)(I)((.5*a*t^2)/(t) * (1/r))^2

Wf = ((1/2)(m + mf)((1/2)*(a*t^4)*(t^2)) + ((1/2)(I)(a*t) * (1/r))^2

Wf = ((1/8)(m + mf)(a^2 * t^2) + (1/8)(I)((a^2 * t^2)/(r^2))

Wf = (1/8)(a^2*t^2)((m + mf) + (I/r^2))
 
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  • #2
So...what is your question?
 
  • #3



I can see that your attempt at deriving the expression for the experimentally determined moment of inertia using work and mechanical energy is on the right track. However, there are a few things that need to be clarified and corrected.

Firstly, in the equation Wf = Kf - Ui, the final potential energy should not be zero. It should be the final potential energy due to gravity, which is mgh. Additionally, the initial kinetic energy should be the initial kinetic energy due to rotation, which is (1/2)(I)(omega)^2. So the equation should be Wf = Kf - Ui = (1/2)(m + mf)(Vf)^2 + (1/2)(I)(omegaf)^2 - mgh.

Secondly, in the equation for average velocity, v = (Vf + Vi)/(2), the final and initial velocities should be the linear velocities, not angular velocities. So the equation should be v = (Vf + Vi)/2.

Thirdly, in the equation for torque, T = F*r = m*r*a, the mass, m, should be the effective mass, which is the sum of the actual mass, m, and the mass of the rotating object, mf. So the equation should be T = (mf + m)*r*a.

Finally, for the experimentally determined moment of inertia, the equation should be I = (r^2(m((gt^2/2s)-t) - mf), not I = (r^2(m((gt^2/2s)-t) - mf).

With these corrections, your derivation should lead to the desired expression for the experimentally determined moment of inertia. Keep up the good work in your scientific studies!
 

1. What is the relationship between work and mechanical energy?

Work and mechanical energy are closely related. Work is defined as the product of the force applied to an object and the distance the object moves in the direction of the force. Mechanical energy is the sum of kinetic energy (energy of motion) and potential energy (energy due to position). According to the work-energy theorem, the work done on an object is equal to the change in its mechanical energy.

2. How is work calculated in relation to mechanical energy?

Work is calculated by multiplying the magnitude of the force applied to an object by the distance the object moves in the direction of the force. Mathematically, it is expressed as W = Fd, where W is work, F is force, and d is distance. The unit of work is joule (J) in the SI system.

3. What is moment of inertia and how is it derived?

Moment of inertia is a measure of an object's resistance to rotational motion. It is derived by integrating the mass of each infinitesimal element of an object multiplied by the square of its distance from the axis of rotation. The formula for moment of inertia is I = ∫r²dm, where I is moment of inertia, r is the distance from the axis of rotation, and dm is the mass of the infinitesimal element.

4. What factors affect an object's moment of inertia?

The moment of inertia of an object depends on its mass, shape, and distribution of mass around the axis of rotation. Objects with larger masses and farther distances from the axis of rotation have higher moments of inertia. Objects with more spread out mass distributions also have larger moments of inertia compared to objects with more concentrated masses.

5. How is the moment of inertia used in real-life applications?

The concept of moment of inertia is used in various real-life applications, such as designing machines and structures, understanding rotational motion of objects, and calculating the stability of objects. It is also used in engineering disciplines such as mechanical, aerospace, and civil engineering to design efficient and safe structures and machines.

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