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Rotational Motion

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Rotational Motion

- Rotational motion is the motion of a body about an internal axis. In rotational motion the axis of motion is part of the moving object.
- All of the properties of linear motion which we have discussed so far this year have corresponding rotational (angular) properties.

- linear propertyangular property
- distance(d) = angular displacement (θ)
- velocity(v)= angular velocity (ω)
- acceleration(a)=ang. accel. (α)
- inertia (m) =rotational inertia (I)
- force (F)=torque (τ)

- The motion of an object which moves in a straight line can only be described in terms of linear properties. The motion of an object which rotates can be described in terms of linear or rotational properties.

- Since all rotational quantities have linear equivalents, we can convert between them.
- Angular displacement is the rotational equivalent of distance. To find the distance a point on a rotating object has traveled (its arc length) we need to multiply the angular displacement by the radius.

- where angular displacement is measured in “radians”:
- angular velocity= (angular displacement /time)
- ω = (Δθ)/t
- where angular velocity can be measured in radians/sec, or revolutions/sec.

- Angular speed is the rotational equivalent of linear speed. To find the linear speed of a rotating object (its tangential speed) we need to multiply the angular speed by the radius.

- Angular acceleration is also similar to linear acceleration.
- Angular acceleration=angular speed/ time
- α = ω / t

- Angular acceleration is the rotational equivalent of linear acceleration. To find the linear acceleration of a rotating object (its tangential acceleration) we need to multiply the angular acceleration by the radius.

- While an object rotates, every point will have different velocities, but they will all have identical angular velocities.
- All of the equations of linear motion which we have discussed so far this year have corresponding rotational (angular) equations.

- linear equation angular equation
- v = ∆x/∆t => ω =∆ θ /∆t
- a = ∆v/∆t =>α=∆ ω /∆t
- vf = vi + a∆t=> ωf = ωi +α∆t
- ∆d = vi∆t + 1/2a(∆t)2=>∆ θ = ωi∆t + 1/2α(∆t) 2
- vf = √(vi2 + 2a∆x)=> ωf = √(ωi2 + 2a∆d)
- F=ma=> τ=I α

- Practice problems p. 145-147

- An object moving at constant speed in a circular path will have a zero change in angular speed, and therefore a zero angular acceleration.
- That object is changing its direction, however, and therefore has a changing linear velocity and a non-zero linear acceleration. It has an acceleration directed toward the center of the circle causing it not to move in a straight line.
- This is a centripetal (center seeking) acceleration.

- This acceleration is perpendicular to the tangential (linear) acceleration.
- All accelerations are caused by forces and centripetal acceleration is caused by centripetal force. A force directed towards the center of a circle which causes an object to move in a circular path.

- A centripetal force pulls an object towards the center of a circle while its inertia (not a force) tries to maintain straight line motion.
- This interaction is felt by a rotating object to be a force pulling it outward. This "centrifugal" force does not exist as there is nothing to provide it. It is merely a sensation felt by the inertia of a rotating object

- Practice problems p. 149-150

- Force causes acceleration.
- Inertia resists acceleration.
- Torque causes rotational acceleration.
- Rotational inertia resists rotational acceleration.
- τ=I α = r F
- I = m r2

- Inertia is measured in terms of mass. Rotational inertia is measured in terms of mass and how far that mass is located from the axis.
- The greater the mass or the greater the distance of that mass from the axis, the greater the rotational inertia, and therefore the greater the resistance to rotational acceleration.