Rotational Motion

Rotational Motion

Chapter 7

Angles• Been working with degrees for our angles• 90 degrees, 180, 56.4, etc.

• There is another way to measure an angle, which is called radians

Radians

• Radians are found by the following:Θ=(s/r)

• s is the arc length of the circle• r is the radius of the circle• Radians are usually some multiple of pi.

Unit circle

Radians vs. degrees

• 360 degrees is the same as 2π radians-Degree to radian: radian = (π/180) * degree-Radian to degree: degree = (180/π) * radian

One revolution = 2π radians = 360 degrees

Convert:35 degrees to radians5.6π radians to degrees

Angular displacement

• Angular displacement is how much an object rotates around a fixed axis

• Such examples would be a tire rotating, or a Ferris wheel car.

Angular displacement

• Finding angular displacement is simply a matter of finding the angle in radians:

Δθ=(Δs/r)

• So the change in angular displacement is equal to the change in arc length over the radius.

Sample Problem

• A Ferris wheel car travels an arc length of 30 meters. If the wheel has a diameter of 45 meters, what is the car’s displacement?

Angular speed

• Angular speed is how long it takes to travel a certain angular distance.

• Similar to linear speed, angular is found by:

ωavg= Δθ/Δt

and its units are rad/s, though rev/s are often used as well

Sample Problem

• An RC car makes a turn of 1.68 radians in 3.4 seconds. What is its angular speed?

Angular acceleration

• Lastly, angular acceleration is how much angular speed changes in that time interval.

αavg=(ω2-ω1)/Δt

The units are rad/s2 or rev/s2, depending on angular velocity

Sample problem

• The tire on a ‘76 Thunderbird accelerates from 34.5 rad/s to 43 rad/s in 4.2 seconds. What is the angular acceleration?

Episode V: Kinematics Strike Back

• Displacement, speed, acceleration…should all sound familiar

• Recall the linear kinematics we discussed earlier.

Linear vs. Angular

• Linear and angular kinematics, at least in form, are very similar.

NOTE

• These kinematic equations only apply if ACCELERATION IS CONSTANT.

• Additionally, angular kinematics only for objects going around a FIXED AXIS.

Sample problem• The wheel on a bicycle rotates with a constant

angular acceleration of 3.5 rad/s2. If the initial angular speed of the wheel is 2 rad/s, what’s the angular displacement of the wheel in 2 seconds?

Tangential & Centripetal Motion

• Almost all motion is a mixture of linear and angular kinematics.

• Reflect on when we talked about golf swings in terms of momentum and impulse.

Tangents

• A tangent line is a straight line that just barely touches the circle at a given point.

Tangential Motion

• Similarly, for an instantaneous moment in circular motion, objects have a tangential speed.

• So for an infinitesimally small time, an object is moving straight along a circular path.

Tangential speed

• Tangential speed depends on how far away the object is from the fixed axis.

Tangential speed

• The further from the axis you are, the slower you will go.

• The closer to the axis you are, the faster you will go.

Tangential speed

• So, during a particular (infinitesimally small) time on the circular path, the object is moving tangent to the path.

• No circular path, no tangential speed

Tangential speed

• The tangential speed of an object is given as:

vt=rω

where r is the distance from the axis, or the radius of a circle.

Remember, the units for linear speed is m/s.

Sample problem

If the radius of a CD in a computer is .06 m and the disc turns at an angular speed of 31.4 rad/s, what’s the tangential speed at a given point on the rim?

Tangential acceleration

Of course, where there is speed, there probably is also acceleration

But keep in mind: THIS IS NOT AN AVERAGE ACCELERATION.

INSTANTANEOUS Tangential Acceleration

• Tangential acceleration also points tangent to the circular path, found by:

at=rα

Sample Problem

• What is the tangential acceleration of a child on a merry-go-round who sits 5 meters from the center with an angular acceleration of 0.46 rad/s2?

Centripetal Acceleration

• You can make a turn at a constant speed and still have a changing acceleration. Why?

Centripetal Acceleration

• Remember, acceleration is a VECTOR, just like velocity.

• So when you’re pointing in a different direction along a circular path, acceleration is changing, even though velocity is constant.

• This is known as centripetal acceleration.

Centripetal Acceleration

• Centripetal acceleration points TOWARDS the center of the circular path.

Centripetal acceleration

• There are two ways to determine this acceleration:

ac=vt2/r

ORac=rω2

Sample problem

A race car has a constant linear speed of 20 m/s around the track. If the distance from the car to the center of the track is 50 m, what’s the centripetal acceleration of the car?

Acceleration

• Centripetal and tangential acceleration are NOT IDENTICAL.

• Tangential changes with the velocity’s magnitude.

• Centripetal changes with the velocity’s direction.

Total Acceleration

• Finding the total acceleration of an object requires a little geometry.

Causes of circular motion

Circular Motion

• If you’ve ever gone round a sharp turn really fast, you probably feel yourself being tilted to one side.

• This is due to Newton’s Laws

Back to THOSE…

• Objects resist changes in motion.• When you go round a curve, your body wants

to keep going in a linear path but the car does not.

Once more…

• So for a linear path, if F=ma, then for a circular path, Fc=mac

• This is known as centripetal force.

Centripetal Force

• There are two other ways to find this force.

Fc=(mvt2)/r

ORFc=mrω2

Sample problem

A 70.5 kg pilot is flying a small plane at 30 m/s in a circular path with a radius of 100 m. Find the centripetal force that maintains the circular motion of the pilot.

Conundrum

• Centripetal force points towards the center of the axis.

• BUT in a car, you feel like you’re being flung AWAY from the center of axis.

• So, what gives?

When in doubt, Newton

• Your body’s inertia wants to keep going in a linear direction. Which is why you tend to tilt away from the center of axis on a curve.

• This is often labeled as centrifugal force, but it is NOT a proper force.