Figuring Out Total Kinetic Energy of Rotating Object

[marcia888]

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

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

Then the book gives an alternate formula:

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

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

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

factor out the 1/2v^2

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

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

$$\frac{v^2}{2m} (1+\frac{I}{mr^2})$$

It works for the right term, but not the left. What am I doing wrong?

 

[Fermat]

You just didn't factor out completely.

You ended up with,

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

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.

 

[thepatientmental]

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.