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**6.6.6 Linear and angular momentum equations in terms of accelerations**

The linear and angular momentum conservation equations can also be expressed in terms of accelerations, angular accelerations, and angular velocities. The results are

For 2D planar motion we can use the simplified formulas

**6.6.7 Special equations for analyzing bodies that rotate about a stationary point**

We often want to predict the motion of a system that rotates about a fixed pivot â€“ a pendulum is a simple example. These problems can be solved using the equations in 6.6.5 and 6.6.6, but can also be solved using a useful short-cut.

- For an object that rotates about a fixed pivot at the origin:
- The total angular momentum (about the origin) is h =IOÏ‰
- The total kinetic energy is T = 1/2 Ï‰Â·I
_{0}Ï‰ - The equation of rotational motion is

Here I_{O} is the mass moment of inertia about O (calculated, eg, using the parallel axis theorem)

**For 2D rotation about a fixed point at th**e origin we can simplify these to

- The total angular momentum (about the origin) is h = I
_{Ozz}Ï‰_{z}k - The total kinetic energy is T=1/2 IO
_{zz}Ï‰_{z}^{2} - The equation of rotational motion is

**Proof:** It is straightforward to show these formulas. Letâ€™s show the two dimensional version of the kinetic energy formulas as an example. For fixed axis rotation, we can use the rigid body formulas to calculate the velocity of the center of mass (O is stationary and at the origin)

v_{G} = Ï‰ Ã—r_{G}=Ï‰_{z}k Ã— r_{G}

The general formula for kinetic energy can therefore be re-written as

The other formulas can be proved with the same method â€“ we simply express the velocity or acceleration of the COM in the general formulas in terms of angular velocity and acceleration, and notice that we can rearrange the result in terms of the mass moment of inertia about O.

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