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.
