# Rotational Statics

## Summary

- Statics is the branch of mechanics dealing with forces acting on objects that are not accelerating.
- Equilibrium is a general concept that can refer to states of…
- static equilibrium, which includes…
- translational equilibrium (the equilibrium of forces)
- rotational equilibrium (the equilibrium of torques)

- dynamic equilibrium, which includes but is not limited to…
- thermodynamic equilibrium (the equilibrium of internal energy transfer)

- static equilibrium, which includes…
- An adjective before the noun is not needed when the type of equilibrium is known from context.
- This section of this book is about static equilibrium only.
- Dynamic equilibrium will be dealt with in other sections of this book.

- Static equilibrium (a.k.a. the equilibrium of extended bodies)
- The translational and angular acceleration of an extended body in static equilibrium are both zero.
- An object in static equilibrium will…
- remain at rest or continue moving with a constant translational velocity and
- remain at rest or continue rotating with a constant angular velocity.

- An object in static equilibrium will…
- An extended body is in static equilibrium when the net force and net torque on it are zero.
no net force ∑ *F*_{−x}= ∑*F*_{+x}⎫

⎪

⎬

⎪

⎭⇒ ∑ **F**= 0∑ *F*_{−y}= ∑*F*_{+y}∑ *F*_{−z}= ∑*F*_{+z}no net torque ∑τ _{−x}= ∑τ_{+x}⎫

⎪

⎬

⎪

⎭⇒ ∑ **τ**= 0∑τ _{−y}= ∑τ_{+y}∑τ _{−z}= ∑τ_{+z} - An extended body is in static equilibrium when the forces and torques on it are balanced.
balanced forces ∑ *F*_{−x}= ∑*F*_{+x}⎫

⎪

⎬

⎪

⎭⇒ ∑ **F**= 0∑ *F*_{−y}= ∑*F*_{+y}∑ *F*_{−z}= ∑*F*_{+z}balanced torques ∑τ _{−x}= ∑τ_{+x}⎫

⎪

⎬

⎪

⎭⇒ ∑ **τ**= 0∑τ _{−y}= ∑τ_{+y}∑τ _{−z}= ∑τ_{+z}

- The translational and angular acceleration of an extended body in static equilibrium are both zero.
- Center of mass (center of gravity)
- The center of
*mass*of an object…- is the effective location of an extended body for for the purposes of some calculations
- is the point about which an extended body will always rotate when free of external torques
- is computed from the
*mass*distribution - may lie outside of an extended body
- is represented by the symbol
**r**or_{cm}**r**_{com}

- The center of
*gravity*of an object…- is the effective location for the force of gravity acting on an extended body
- is the pivot about which an extended body will be in stable rotational equilibrium in a gravitational field
- is computed from the
*weight*distribution - is in the same location as the center of mass when the gravitational field is uniform
- is represented by the symbol
**r**or_{cg}**r**_{cog}

- Equations
- For a discrete collection of objects
**r**=_{cm}∑ *m*_{i}**r**_{i}= 1 ⎛

⎝∑ *m*_{i}x_{i}**î**+ ∑*m*_{i}y_{i}**ĵ**+ ∑*m*_{i}z_{i}**k̂**⎞

⎠∑ *m*_{i}*m***r**=_{cm}the location of the center of mass for the set of objects *m*=_{i}the mass of the ith object in the set **r**=_{i}the position vector of the ith object in the set *m*=the total mass of the set *x*,_{i}*y*,_{i}*z*=_{i}the coordinates of the ith object in the set **î**,**ĵ**,**k̂**=the unit vectors of the coordinate system used - For a single object with…
uniform density **r**=_{cm}1 ⌠⌠⌠

⌡⌡⌡**r***dV**V*nonuniform density **r**=_{cm}1 ⌠⌠⌠

⌡⌡⌡ρ **r***dV**m***r**=_{cm}the location of the center of mass of the object *m*=the total mass of the object *V*=the total volume of the object **r**=the position vector of any infinitesimal piece of the object *dV*=an infinitesimal volume element in the object ρ = a density function

- For a discrete collection of objects

- The center of
- Stability of equilibrium for extended bodies
- The equilibrium of an extened body in a gravitational field is said to be…
- stable if…
- any small disturbance results in a net torque that tries to restore its equilibrium orientation
- any small disturbance would raise its center of mass
- its center of gravity is below the pivot point

- unstable if…
- any small disturbance results in a net torque that drives it away from its equilibrium orientation
- any small disturbance would lower its center of mass
- its center of gravity is above the pivot point

- neutral if…
- any small disturbance does not result in a net force
- any small disturbance would neither raise nor lower its center of mass
- its center of gravity is at the pivot point

- stable if…

- The equilibrium of an extened body in a gravitational field is said to be…