Mathy AP Physics C

Rotational Motion Part I

AP1 Mathy

Translational vs Rotational Motion

Pure translation Pure translation

Pure rotation

 Draw a line on an object  If it does not turn, it exhibits translational motion  If the line is not parallel in the motion, it is rotating

Fixed axis of rotation w/Rigid Body y

x

A rigid body has all parts locked in relationship to one another. A fixed axis of rotation means the objects axis is set and will not fluctuate as the object rotates ( fling a water balloon and watch it’s axis move around!)

AOR

  This object rotates on its axis, the other points revolve around the AOR.

 These 2 locations( ) on this object are at different displacements (r) from the axis of rotation.

 These 2 locations will move through the same angular displacement (Δθ) per time interval

 Therefore, these 2 locations have the same angular velocities (ω) . Sometimes referred to as revolutions per minute (RPM)

 Yet, these points have different linear (tangential) velocities. How so ?

The radian (read only) We know know there are 2 types of pure unmixed motion:   Translational - linear motion (entire first semester !)   Rotational - motion involving a rotation or revolution around a

fixed chosen axis( an axis which does not move).

We need a system that defines BOTH types of motion working together on a system. Rotational quantities are usually defined with units involving a radian measure.

If we take the radius of a circle and LAY IT DOWN on the circumference, it will create an angle whose arc length is equal to R.

In other words, one radian angle subtends an arc length Δs equal to the radius of the circle (R)!

Angular Displacement

Half a radian would subtend an arc length equal to half the radius

A general Radian Angle (Δθ) subtends an arc length (s) equal to r. The theta in this case represents ANGULAR DISPLACEMENT.

2 radians would subtend an arc length equal to two times the radius.

s

s

r Zero angular position

  Angular displacement θ is measured from the zero angular position on the x axis to any other location. These displacements are (+) .

 Any displacements that are “clockwise” are (-). Hint: your alarm clock is negative

  Do not reset θ to zero with each rotation, positions accumulate just as in linear motion.

Angular Velocity (average) Since velocity is defined as the rate of change of displacement. ANGULAR VELOCITY is defined as the rate of change of ANGULAR DISPLACEMENT.

Units : rad/s

Angular Velocity (instantaneous) Since velocity is defined as the rate of change of displacement, as the time interval approaches zero, an instantaneous rate of change can be known.

Units : rad/s

Angular Acceleration Once again, following the same lines of logic. Since acceleration is defined as the rate of change of velocity. We can say the ANGULAR ACCELERATION is defined as the rate of change of the angular velocity.

Also, we can say that the ANGULAR ACCELERATION is the TIME DERIVATIVE OF THE ANGULAR VELOCITY. Units “ rad/s2

Referencing Translation vs. Rotation Translational motion tells you THREE THINGS 1. magnitude of the motion and the units 2. Axis the motion occurs on 3. Direction on the given axis

Example: v =3i This tells us that the magnitude is 3 m/s, the axis is the "x" axis and the direction is in the "positive sense".

Rotational motion tells you THREE THINGS: 1. magnitude of the motion and the units 2. the PLANE in which the object rotates in 3. the directional sense ( counterclockwise or clockwise) Counterclockwise rotations are defined as having a direction of POSITVE K motion on the "z" axis

Rotational motion is not a vector

Rotation Example: Unscrewing a screw or bolt

= 5 rad/sec k

Clockwise rotations are defined as having a direction of NEGATIVE K motion on the "z" axis

Example: Tightening a screw or bolt

= -5 rad/sec k

Use “Right Hand Rule” to find the direction of motion!

Combining Linear & Angular Variables

In the end, substituting the appropriate symbols we have a formula that relates Translational velocity to Rotational velocity.

We already know the r influences the tangential velocity somehow. Lets investigate a time rate change.

* Linear velocity is directly proportional to the r, while ω is a constant!

Tangential acceleration and rotational kinematics Using the same kind of mathematical reasoning we can also define Linear tangential acceleration.

Inspecting each equation we discover that there is a DIRECT relationship between the Translational quantities and the Rotational quantities.

We can therefore RE-WRITE each translational kinematic equation and turn it into a rotational kinematic equation.

Linear Big 5 Rotational Big 5

Example A turntable capable of

angularly accelerating at 12 rad/s2 needs to be given an initial angular velocity if it is to rotate through a net 400 radians in 6 seconds. What must its initial angular velocity be?

30.7 rad/s

Kinetic Energy of Rotation Consider the Kinetic Energy of a circular saw!

Can we simply use the equation

Is the linear velocity the same at each dot?

What is its KE at its Center of Mass ?

Problems? No, we will add up all the particles

Wait, all the vi members are going a different speed !!!

Yes, but not the ω components! And since vt=r ω

Hey, what happened to , why was it renamed I ????

?

Kevin eats ½ Industrial Waste squared

Rotational Inertia Just like massive bodies tend to resist

changes in their motion ( AKA - "Inertia") . Rotating bodies also tend to resist changes in their motion. We call this ROTATIONAL INERTIA. We can determine its expression by looking at Kinetic Energy of a uniform object.

We now have an expression for the rotation of a mass in terms of the radius of rotation. We call this quantity the MOMENT OF INERTIA (I) with units kgm2

* As r goes up, an object is harder to spin !

Rotational Inertia Thought Check

* As r goes up, an object is harder to spin !

  A Tether Ball game is set up by Dr. Evil.The mass and radial arm length is shown in the diagram. Rank these Members individually based on their rotational inertia (Most to least)

36 kg

9 kg

4 kg

1m

2 m

3 m

  Why will a pool stick rotate easier in Figure a) than figure b) ?

a)

b)

Other Moments of Inertia bicycle rim

filled can of coke

baton

baseball bat

basketball

boulder

Example: Moment of Inertia, I

m1 m2

Let's use this equation to analyze the motion of a 4-m long bar with negligible mass and two equal masses(3-kg) on the end rotating around a specified axis.

EXAMPLE #1 -The moment of Inertia when they are rotating around the center of their rod.

EXAMPLE #2-The moment if Inertia rotating at one end of the rod

m1 m2

24 kgm2

48 kgm2

Example cont’

Now let’s calculate the moment of Inertia rotating at a point 2 meters from one end of the rod.

m1 m2

2m

120 kgm2

As you can see, the FARTHER the axis of rotation is from the center of mass, the moment of inertia increases. We need an expression that will help us determine the moment of inertia when this situation arises.

Parallel Axis Theorem

  The parallel-axis theorem states

  Where ICM is the rotational inertia about an axis containing the Center of Mass

  I is the rotational inertia about an axis parallel to ICM

  M = total mass and h = distance between axes.

Example – Parallel Axis Theorem

m1 m2

d = 2m

m1 m2

4m

(not drawn to scale)

48 kgm2

120 kgm2

Choose a new axis!

Old axis is here

Answers come out the same !!!!

Parallel Axis Theorem Example

  Use the parallel-axis theorem to find the rotational inertia about the center of mass of an uniform thin rod.

©2008 by W.H. Freeman and Company

Continuous Masses The earlier equation, I =Σmr2, worked fine for what is called

POINT masses. But what about more continuous masses like disks, rods, or sphere where the mass is extended over a volume or area. In this case, calculus is needed.

This suggests that we will take small discrete amounts of mass and add them up over a set of limits. Indeed, that is what we will do. Let’s look at a few example we “MIGHT” encounter. Consider a solid rod rotating about its CM.

Will, I =Σmr2, be the equation for a rod?

The rod We begin by using the same technique used to derive the center of mass of a continuous body.

dr

The CM acts as the origin in the case of determining the limits.

Read only

Other Moments of Inertia bicycle rim

filled can of coke

baton

baseball bat

basketball

boulder

Your turn What if the rod were rotating on one of its ENDS?

dr

As you can see you get a completely different expression depending on HOW the body is rotating.

The disk

R

r

dx

dx

2πr

The bottom line..

Will you be asked to derive the moment of inertia of an object? Possibly! Fortunately, most of the time the moment of inertia is given within the free response question.

Consult the chart ( on the notes page) called Moments of Inertia to view common expressions for “I” for various shapes and rotational situations.

Newton’s 2nd Law for Rotation Example 1

  A bike is placed on a stand so it can be used as a stationary bike. The bike is pedaled so that a force of 18 N is applied to the rear sprocket at a distance of rs = 7.0 cm from the axis. Treat the wheel as a hoop (I = MR2) of radius r =35 cm and mass m = 2.4 kg. What is the angular velocity of the wheel after 5.0 s?

©2008 by W.H. Freeman and Company