1. 10. Rotational Motion Angular Velocity & Acceleration Torque
Rotational Inertia & the Analog of Newton’s Law
Rotational Energy
Rolling Motion

2. Ans. blade mass toward axis
Examples of rotating objects:
Planet Earth.
Wheels of your bike.
DVD disc in the player.
Circular saw.
Pirouetting dancer.
Spinning satellite.
How should you engineer the blades so it’s easiest for the wind to get the turbine rotating?
Ans. blade mass toward axis

3. 10.1. Angular Velocity & Acceleration
Polar coord ( r,  )
Average angular velocity
 = angular displacement
( positive if CCW )
in radians
1 rad = 360 / 2  = 57.3
// rotational axis
(Instantaneous) angular velocity
Angular speed:
 in radians 
Circular motion:
Linear speed:

4. Example Wind Turbine
A wind turbine’s blades are 28 m long & rotate at 21 rpm.
Find the angular speed of the blades in rad / s,
& determine the linear speed at the tip of a blade.

5. Angular Acceleration (Instantaneous) angular acceleration
We shall restrict ourselves to rotations about a fixed axis.
(Instantaneous) angular acceleration at
Trajectory of point on rotating rigid body is a circle, i.e. r = const.
Its velocity v is always tangential: v a ar
Its acceleration is in the plane of rotation (   ) :
Tangential component:
Radial component:

6. Angular vs Linear

7. Example Spin Down
When wind dies, the wind turbine of Example 10.1 spins down with
constant acceleration of magnitude 0.12 rad / s2.
How many revolutions does the turbine make before coming to a stop?
# of rev.

8. 10.2. Torque Rotational analog of force Torque :  plane of r & F
[  ] = N-m ( not J )

9. Example 10.3. Changing a Tire
You’re tightening the wheel nuts after changing a flat tire of your car.
The manual specify a tightening torque of 95 N-m.
If your 45-cm-long wrench makes a 67 angle with the horizontal,
with what force must you pull horizontally to do the job?
Note:

10. 10.3. Rotational Inertia & the Analog of Newton’s Law
Linear acceleration:
Rotating baton
(massless rod of length R + ball of mass m at 1 end):
Tangential force on ball:
= moment of inertia
= rotational inertia
of the baton

11. Calculating the Rotational Inertia
Rotational inertia of discrete masses
ri = perpendicular distance of mass i to the rotational axis.
Rotational inertia of continuous matter
r = perpendicular distance of point r to the rotational axis.
( r) = density at point r.

12. Example Dumbbell
A dumbbell consists of 2 equal masses m = 0.64 kg on the ends of a massless rod of length L = 85 cm.
Calculate its rotational inertia about an axis ¼ of the way from one end & perpendicular to it.
GOT IT? 10.2
Would I
increase
decrease
stay the same
if the rotational axis were
at the center of the rod
at one end?
(b)
(a)

13. Example Rod
Find the rotational inertia of a uniform, narrow rod of mass M and length L about an axis through its center & perpendicular to it.

14. Example Ring
Find the rotational inertia of a thin ring of radius R and mass M about the ring’s axis.
Pipe of radius R & length L :
I = MR2 for any thin ring / pipe

15. Example Disk
Find the rotational inertia of a uniform disk of radius R & mass M about an axis through its center & perpendicular to it.

16. Table 10.2. Rotational Inertia

17. Parallel - Axis Theorem
Ex. Prove the theorem for a set of particles.

18. GOT IT? 10.3.
Explain why the rotational inertia for a solid sphere is less than that of a spherical shell of the same M & R.
Mass of shell is further away from the axis.

19. Example 10.8. De-Spinning a Satellite
A cylindrical satellite is 1.4 m in diameter, with its 940-kg mass distributed uniformly.
The satellite is spinning at 10 rpm but must be stopped for repair.
Two small gas jets, each with 20-N thrust, are mounted on opposite sides of it & fire tangent to its rim.
How long must the jets be fired in order to stop the satellite’s rotation?
To stop the spin:
Time required for a const ang accel 

20. Example 10.9. Into the Well  
A solid cylinder of mass M & radius R is mounted on a frictionless horizontal axle over a well.
A rope of negligible mass is wrapped around the cylinder & supports a bucket of mass m.
Find the bucket’s acceleration as it falls into the well.
Let downward direction be positive.
Bucket:
Cylinder:  

21. GOT IT? 10.4.
Two masses m is connected by a string that passes over a frictionless pulley of mass M.
One mass rests on a frictionless table; the other vertically.
Is the magnitude of the tension force in the vertical section of the string
(a) greater than, (b) equal to, or (c) less than
in the horizontal? Explain.
(a):
There must be a net torque to increase the pulley’s clockwise angular velocity.

22. 10.4. Rotational Energy
Rotational kinetic energy = sum of kinetic energies of all mass elements,
taken w.r.t the rotational axis.
Set of particles:

23. Example 10.10. Flywheel Storage
A flywheel has a 135-kg solid cylindrical rotor with radius 30 cm and spins at 31,000 rpm.
How much energy does it store?
Flywheel for hybrid bus (30% fuel saving).
~ energy in 1 liter of gasoline
Modern flywheels 10s of kW of power for up to a min.
Carbon composite to withstand strain of 30,000 rpm.
Magnetic bearings to reduce friction.
supercondutor to reduce electrical losses.

24. Energy & Work in Rotational Motion
Work-energy theorem for rotations:

25. Example 10.11. Balancing a Tire
An automobile wheel with tire has rotational inertia 2.7 kg m2.
What constant torque does a tire-balancing machine need to apply in order to spin this tire up from rest to 700 rpm in 25 revolutions?

26. 10.5. Rolling Motion V = velocity of CM. Composite object:
ui = velocity relative to CM.
Composite object:
Moving wheel:
 is w.r.t. axis thru cm

27. Moving wheel: Rolling wheel: V = velocity of CM
 is w.r.t. axis thru CM
Moving wheel:
Rolling wheel:

28. Example 10.12. Rolling Downhill
A solid ball of mass M and radius R starts from rest & rolls down a hill.
Its center of mass drops a total distance h.
Find the ball’s speed at the bottom of the hill.
Initially:
Finally: 
sliding ball
Note: v is independent of M & R

29. GOT IT? 10.5.
A solid ball & a hollow ball roll without slipping down a ramp.
Which reaches the bottom first? Explain.
Solid ball.
Smaller I  smaller Krot  larger v.