Physics 1ALecture 10A

"The pursuit of perfection often impedes improvement.”

--George F. Will

Rotational VariablesUp until now, we have only dealt with bodies in translation, which is motion along a line or curve.

Now, we will deal with the consequences of rotational motion.We need to define useful variables for rotational motion as we did with translational motion.

Angular position is measured by θ [in radians].

Rotational VariablesAngular displacement is the change in angular position:

Δθ = θf - θi

For ease, when dealing with angular displacement we say that we are dealing with a rigid body.

This means that each point on the object undergoes the same angular displacement.

The convention for angular displacement is that a clockwise displacement is negative and a counterclockwise displacement is positive.

Also, be careful, Δθ can be greater than 2πrad.

Rotational VariablesAverage angular speed is given by:

It is angular displacement over a time interval.

The units of angular speed are [rad/s].Angular speed will be positive if θ is increasing (ccw).

Angular speed will be negative if θ is decreasing (cw).

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ω =dθdt

Rotational VariablesAverage angular acceleration is given by:

It is the change in angular speed over a time interval.

The units of angular acceleration are [rad/s2].

These variables correspond to translational variables.

θ --> x ω --> v α --> a

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α =dωdt

Rotational VariablesWe can use the constant acceleration equations with rotational variables:

Remember (just like linear motion) these equations rely on the fact that angular acceleration remains constant over the time period in question.

Rotational VariablesExampleA wheel has a constant angular acceleration of 3.0rad/s2. During a certain 4.0 second interval, you measured that the wheel had an angular displacement of 120 radians. If the wheel had started at rest, how long if had been in motion at the start of the 4 second interval?

AnswerFirst, you must define a coordinate system.Let’s say that the wheel starts moving counterclockwise in the positive direction.

Rotational VariablesAnswerLet’s list the quantities we know:ωo = 0 <-- it starts from rest (at t=0), but our interval of 4s does not start at t=0.α = +3.0rad/s2

ω <-- don’t knowΔθ <-- don’t knowt <-- finding

But we also know that in a 4sec. span it went through an angle of 120rad.

We can use the third equation to find the angular velocity at the beginning of that time span.

Rotational VariablesAnswerUse:

Thus, it started out the 4 sec. interval with an angular velocity of 24rad/s.

We have a new known: ω = 24rad/s.

Rotational VariablesAnswerSince Δθ is missing we can use:

The only difference between translational kinematic equations and rotational kinematic equations is that as the object rotates it will gain angular displacement even as it keeps repeating the same position.

Angular displacement is cumulative.

Relating VariablesSometimes you would like to relate linear variables and rotational variables (and vice versa).

Let’s say you were traveling on the edge of a merry-go-round of radius, r. If you wanted to know your distance travelled, Δs, if you have rotated through an angular displacement, Δθ, then use:

Δs = r (Δθ) [Δθ in radians]

The bigger the radius, the more distance you have covered per angular displacement.

Relating VariablesNow, let’s say you wanted to know how fast you would travel linearly is you let go of the merry-go-round if your angular velocity was ω; then use:

where vt is known as tangential speed.

Relating VariablesAcceleration and angular acceleration (α) are a little harder to relate to each other.

First, there is centripetal acceleration, ac, which is given by:

This acceleration is also known as radial acceleration, ar.

This radial acceleration component tells us that the body is traveling in a circular type motion (this means that the velocity vector is changing direction).

Relating VariablesThe second type of acceleration is tangential acceleration, at. It is given by:

at = αr

This tangential acceleration component tells us if the body is changing its angular speed (this means that ω is changing magnitude).

at

ar

at

ar

a

Rotational VectorsAngular displacement, angular velocity, and angular acceleration are all vector quantities.

You already know how to calculate the magnitude from previous equations.

Direction is defined by the right hand rule.

Grasp axis of rotation with your right hand.

Wrap your fingers in the direction of rotation.

Your thumb points in the direction of ω.

Rotational VectorsIn the picture to the right we have two rotating disks.

In (a), the disk rotates clockwise, the angular velocity vector points into the board.

In (b), the disk rotates counterclockwise, the angular velocity vector points out of the board.

Please remember to use your right hand when performing the right hand rule.

Cross Products

The direction of C is determined by the Right Hand Rule.

C is perpendicular to both A and B.

Another way to multiply vectors is the cross product:

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C = A × B

The magnitude of C is given by:where θ is the angle between vectors A and B.

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C = A B sinθ

A cross product measures how perpendicular two vectors (A and B) are.

If A and B are parallel, their cross product is zero.

For Next Time (FNT)

Study Chapter 10.