Figuring Out Total Kinetic Energy of Rotating Object

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In summary, the conversation is about understanding how to get from one formula to another for calculating the total kinetic energy of a rotating object. The first formula involves adding the translational and rotational kinetic energy, while the alternate formula involves factoring out the velocity squared and including a factor for the moment of inertia. The person asking the question is trying to figure out how to get from one formula to the other and seeks clarification on the process. The conversation ends with the suggestion to either factor out completely or multiply out the alternate formula to get back to the original one.
  • #1
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This isn't a homework question, but I'm just wondering how the text in the book gets from one formula to the next.

To figure out total kinetic energy of a rotating object:

K translational + K rotational

[tex]\frac{1}{2}mv^2+\frac{1}{2}I (\frac{v}{r})^2[/tex]

Then the book gives an alternate formula:

[tex]\frac{1}{2}mv^2(1+\frac{I}{mr^2})[/tex]

So I wanted to see how they get from one to the other. So I tried

[tex]\frac{1}{2}mv^2+\frac{1}{2}I (\frac{v}{r})^2[/tex]

factor out the 1/2v^2

[tex]\frac{1}{2}v^2 (m+\frac{I}{r^2})[/tex]

almost there. But how do I get a 1 in place of the m? I could divide by m, but then I get

[tex]\frac{v^2}{2m} (1+\frac{I}{mr^2})[/tex]

It works for the right term, but not the left. What am I doing wrong?
 
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  • #2
You just didn't factor out completely.

You ended up with,

[tex]\frac{1}{2}v^2 (m+\frac{I}{r^2})[/tex]

now take out m. i.e. divide both terms inside the brackets by m so that you end up with m outside the barckets.

Alternatively, you could multiply out the alternate formula, and you will end up with the original one.
 
  • #3


The alternate formula for calculating total kinetic energy of a rotating object is derived from the original formula by substituting the moment of inertia (I) for the rotational kinetic energy term (\frac{1}{2}I (\frac{v}{r})^2). This can be seen by rearranging the original formula as follows:

K translational + K rotational = \frac{1}{2}mv^2+\frac{1}{2}I (\frac{v}{r})^2

= \frac{1}{2}mv^2+\frac{1}{2}(\frac{I}{r^2})v^2

= (\frac{1}{2}m+\frac{1}{2}(\frac{I}{r^2}))v^2

= (\frac{1}{2}m+\frac{I}{2r^2})v^2

Since the moment of inertia (I) is equal to mr^2 for a point mass rotating around a fixed axis, we can substitute this in the formula:

= (\frac{1}{2}m+\frac{mr^2}{2r^2})v^2

= (\frac{1}{2}m+ \frac{m}{2})v^2

= (\frac{m}{2}+\frac{m}{2})v^2

= mv^2

Thus, the alternate formula can be written as:

K total = \frac{1}{2}mv^2(1+\frac{I}{mr^2})

which simplifies to:

K total = \frac{1}{2}mv^2(\frac{m+mr^2}{m})

= \frac{1}{2}mv^2(1+\frac{I}{mr^2})

This alternate formula is useful when calculating the total kinetic energy of a rotating object, as it takes into account both the translational and rotational kinetic energies.
 

Q: What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion.

Q: How is kinetic energy calculated?

Kinetic energy is calculated using the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

Q: What is rotational kinetic energy?

Rotational kinetic energy is the energy an object possesses due to its rotation. It is calculated using the formula KE = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity.

Q: How do you find the total kinetic energy of a rotating object?

The total kinetic energy of a rotating object is the sum of its linear kinetic energy and its rotational kinetic energy. It can be calculated using the formula KE = 1/2 * m * v^2 + 1/2 * I * ω^2.

Q: What is the moment of inertia?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the shape and mass distribution of the object. The moment of inertia is represented by the symbol I in the formula for rotational kinetic energy.

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