Conserving total energy

In summary, the conversation discusses the correct expression for the kinetic energy of a ball rolling in a semicircular track, and raises questions about the redundancy of certain terms and the impact of different radii on the expression. It is concluded that the translational kinetic energy and rotational energy about the center of the track are the same, and that one of the terms may be redundant.
  • #1
If a ball rolls in a semicircular track starting from one end of track,( the track is kept vertical) and if radius of ball is r and radius of track is R
is this expression correct? (When ball reaches lowest point)
##mg(R-r)=\frac{1}{2}I_0\omega^2 + \frac{1} {2}I\omega_1^2+\frac{1}{2}mv^2##
where ##I_0## is the moment of inertia about centre of track since the ball moves in a circle and moment of inertia is mr^2
The second is for rotation of ball
the third is translational kinetic energy

what changes should I make if R is very very larger than r?
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  • #2
You are double counting the motion of the ball. The translational kinetic energy of the ball is the same thing as the rotational energy about the center of the track, so the I0 term is redundant.
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  • #3
Your first and third term (on the right hand side) represent the same energy. Once is enough.

(Khashishi beat me to it!)
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1. What is total energy conservation?

Total energy conservation is the principle that energy cannot be created or destroyed, only transformed from one form to another. This means that the total amount of energy in a closed system remains constant.

2. Why is it important to conserve total energy?

Conserving total energy is important because it helps to reduce our overall energy consumption and promotes sustainability. It also helps to mitigate the negative effects of energy production on the environment.

3. How can I conserve total energy in my daily life?

There are many ways to conserve total energy in your daily life, such as turning off lights and electronics when not in use, using energy-efficient appliances, carpooling or using public transportation, and choosing renewable energy sources when possible.

4. What are the benefits of conserving total energy?

The benefits of conserving total energy include reducing energy costs, reducing greenhouse gas emissions and other pollutants, preserving natural resources, and promoting a more sustainable future for generations to come.

5. Can conserving total energy really make a difference?

Yes, conserving total energy can make a significant difference in reducing our overall energy consumption and promoting sustainability. Every small action adds up and together we can make a big impact on conserving total energy for a better future.

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