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### Chapter 9

Rotation of Rigid Bodies

Goals for Chapter 9

- To describe rotation in terms of:
- angular coordinates (q)
- angular velocity (w)
- angular acceleration (a)
- To analyze rotation with constant angular acceleration
- To relate rotation to the linear velocity and linear acceleration of a point on a body

Goals for Chapter 9

- To understand moment of inertia (I):
- how it depends upon rotation axes
- how it relates to rotational kinetic energy
- how it is calculated

Introduction

- A wind turbine, a CD, a ceiling fan, and a Ferris wheel all involve rotating rigid objects.
- Real-world rotations can be very complicated because of stretching and twisting of the rotating body. But for now we’ll assume that the rotating body is perfectly rigid.

Angular coordinate

- Consider a meter with a needle rotating about a fixed axis.
- Angle (in radians) that needle makes with +x-axis is a coordinate for rotation.

Angular coordinates

- Example: A car’s speedometer needle rotates about a fixed axis.
- Angle the needle makes with negativex-axis is the coordinate for rotation.
- KEY: Define your directions!

55

45

35

25

15

q

Units of angles

- Angles in radians = s/r.
- One complete revolution is 360° = 2π radians.

Angular velocity

- Angular displacement of a body is = 2 – 1 (in radians)
- Average angular velocityof a body is av = /t(in radians/second)

Angular velocity

- The rotation axis matters!
- Subscript z means that the rotation is about the z-axis.
- Instantaneous angular velocity is z = d/dt.
- Counterclockwise rotation is positive;
- Clockwise rotation is negative.

Calculating angular velocity

- Flywheel diameter 0.36 m;
- Suppose q(t) = (2.0 rad/s3) t3

Calculating angular velocity

- Flywheel diameter 0.36 m; q = (2.0 rad/s3) t3
- Find q at t1 = 2.0 s and t2 = 5.0 s
- Find distance rim moves in that interval
- Find average angular velocity in rad/sec & rev/min
- Find instantaneous angular velocities at t1 & t2

Angular velocity is a vector

- Angular velocity is defined as a vector whose direction is given by the right-hand rule.

Angular acceleration

- Average angular acceleration is avg-z = z/t.
- Instantaneous angular acceleration is z = dz/dt = d2/dt2.

Angular acceleration

- Average angular acceleration is avg-z = z/t.
- Instantaneous angular acceleration is z = dz/dt = d2/dt2.

For same flywheel with dia = 0.36 m; q(t) = (2.0 rad/s3) t3

find average angular accelerations between t1 & t2, & instantaneous accelerations at those t’s

Angular acceleration as a vector

- For a fixedrotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.

Angular acceleration as a vector

- For a fixedrotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.
- BUT THEY DON’T HAVE TO BE IN THE SAME DIRECTION!

w Speeds up!

w Slows down!

Rotation with constant angular acceleration

- Linear and Angular Motion with constant acceleration equations are very similar!

Rotation of a Blu-ray disc

- A Blu-ray disc is coming to rest after being played.
- @ t = 0, w = 27.5 rad/sec; a = -10.0 rad/s2
- What is w at t = 0.3 seconds?
- What angle does PQ make with x axis then?

Relating linear and angular kinematics

- For a point a distance r from the axis of rotation:

its linear speed is v = r (meters/sec)

Relating linear and angular kinematics

- For a point a distance r from the axis of rotation:

its linear tangential acceleration is atan = r (m/s2)

its centripetal (radial) acceleration is arad = v2/r = r

An athlete throwing a discus

- Whirl discus in circle of r = 80 cm; at some time t athlete is rotating at 10.0 rad/sec; speed increasing at 50.0 rad/sec/sec.
- Find tangential and centripetal accelerations and overall magnitude of acceleration

Designing a propeller

- Say rotation of propeller is at a constant 2400 rpm, as plane flies forward at 75.0 m/s at constant speed.
- But…tips of propellers must move slower than 270 m/s to avoid excessive noise.
- What is maximum propeller radius?
- What is acceleration of the tip?

Designing a propeller

- Tips of propellers must move slower than than 270 m/s to avoid excessive noise. What is maximum propeller radius? What is acceleration of the tip?

Moments of inertia

How much force it takes to get something rotating, and how much energy it has when rotating, depends on WHERE the mass is in relation to the rotation axis.

Moments of inertia

- Getting MORE mass, FARTHER from the axis, to rotate will take more force!
- Some rotating at the same rate with more mass farther away will have more KE!

Rotational kinetic energy

- The moment of inertia of a set of discrete particles is
- I = m1r12 + m2r22 + … = miri2
- Rotational kinetic energy of rigid body with moment of inertia IKE(rotation) = 1/2 I2 (still in Joules!)
- Since I varies by location, KE varies depending upon axis!

Rotational kinetic energy example 9.6

- What is I about A?
- What is I about B/C?
- What is KE if it rotates through A with w=4.0 rad/sec?

An unwinding cable

- Wrap a light, non-stretching cable around solid cylinder of mass 50 kg; diameter .120 m. Pull for 2.0 m with constant force of 9.0 N. What is final angular speed and final linear speed of cable?

More on an unwinding cable

Consider falling mass “m” tied to rotating wheel of mass Mand radius R

What is the resulting speed of the small mass when it reaches the bottom?

More on an unwinding cable

Consider falling mass “m” tied to rotating wheel of mass Mand radius R

What is the resulting speed of the small mass when it reaches the bottom?

Method 1: Energy!

More on an unwinding cable

mgh = ½ mv2+ ½ Iw2

I = ½ MR2 for disk

mgh = ½ (m+ ½ M) v2

V = [2gh/(1+M/2m)] ½

The parallel-axis theorem

- What happens if you rotate about an EXTERNAL axis, not internal?
- Spinning planet orbiting around the Sun
- Rotating ball bearing orbiting in the bearing
- Mass on turntable

The parallel-axis theorem

- What happens if you rotate about an EXTERNAL axis, not internal?
- Effect is a COMBINATION of TWO rotations
- Object itself spinning;
- Point mass orbiting
- Net rotational inertia combines both:
- The parallel-axis theorem is: IP = Icm + Md2

The parallel-axis theorem

- The parallel-axis theorem is: IP = Icm+Md2.
- Mass 3.6 kg, I = 0.132 kg-m2 through P. What is I about parallel axis through center of mass?

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