Rotational Motion Torque

Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html

Lecture 17

Chapter 10

Rotational MotionTorque

11.13.2013Physics I


Chapter 10

Rotational Motion Rotational kinematics Rotational dynamics Torque

Outline


Exam2 Results

295 students took the examAverage 51.8/100


Exam2 Results

295 students took the examAverage 51.8/100


Rotational MotionIn addition to translation, extended objects can rotate

Need to develop a vocabulary for

describing rotational motion

There is rotation everywhere you look in the universe, from the nuclei of atoms to spiral galaxies


Polar coordinates for circular motion

Rectangular coordinates are well suited to describing motion in a straight line. However, rectangular coordinates are not useful for describing circular motion.

Since circular motion plays a prominent role in physics, it is worth introducing a special coordinate system –POLAR coordinate system.


Angular Position in polar coordinatesConsider a pure rotational motion:

an object moves around a fixed axis.

x

arclength

R

R, θ in radians!

y

R

We define object’s position with: R,

R

So, one radian is defined as the angle subtendedby an arc whose length is equal to the radius.

Rrad

1

The radian is dimensionless, so there is no need to mention it in calculations


Angular Position in polar coordinates

x

R

rad2

radians!

y

R

length arc

ApplyR

R 2/

2

x

R R

RR2

rad2360

3.572/3601 rad

2/4R2 R


Use Radians to get an arclength

R

R 60

R3

R


The average angular velocity is defined as the total angular displacement divided by time:

Angular displacement

12 Angular displacement:

The instantaneous angular velocity:

t

dtd

tt

0

lim

is the same for all points of a rotating object.

The angular velocity of any

point on a solid object rotating about a

fixed axis is the same. Both

Bonnie and Klyde go around one

revolution (2 radians) every 2 seconds.

ConcepTest 1 Bonnie and Klyde

BonnieKlyde

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every2 seconds.Klyde’s angular velocity is:

A) same as Bonnie’sB) twice Bonnie’sC) half of Bonnie’sD) one-quarter of Bonnie’sE) four times Bonnie’s

t

sec21rev

secsec2

2 radrad is the same for both

rabbits


12

12

ttt

Angular Acceleration

Average angular acceleration:

Instantaneous angular acceleration: dtd

tt

0

lim

The angular acceleration is the rate at which the angular velocity changes with time:

121t2t

Since is the same for all points of a rotating object, angular acceleration also will be the same for all points.

Thus, and α are properties of a rotating object


Vector Nature of Angular Quantities

However, direction of rotation can be clockwise and counterclockwise. How can we show a difference between them?


Vector of Angular VelocityBoth ω and α can be treated as vectors:

(by convention)Right Hand Rule

Curl fingers on right hand to trace rotation of object

Direction of thumb is vector direction for angular velocity.

+z

x

y

)ˆ( k

)ˆ( k+z

x

y


Vector of Angular acceleration

the object will be "speeding up" if the angular acceleration is in the same direction as the angular velocity, and

the object will be "slowing down" if the angular acceleration is in the opposite direction of the angular velocity.

Just as was the case for linear motion:

i

f

f

i

0 0

0

if

if

tt

0

if

if

tt


Relation between linear and angular velocities

Each point on a rotating rigid body has the same angular displacement, velocity, and acceleration!

The corresponding linear (or tangential) variables depend on the radius and the linear velocity is greater for points farther from the axis.

vtan ddt

R

d

dltanv

RIn the 1st slide, we defined:

So we can write: Rdd vtan R d

dt R

Rv tan

Relation between linear and angular velocities ( in rad/sec)

By definition, linear velocity:

Remember it!!!!You will use it often!!!


Example: Linear/angular velocityRope wound around a circular cylinder unwraps without stretching or slipping, its speed and acceleration at any instant are equal to the speed and tangential acceleration of the point at which it is tangent to the cylinder

tanv

v

Rvv tan

But their linear speeds v will be

different because and

Bonnie is located farther out (larger

radius R) than Klyde.

Bonnie

Klyde

BonnieKlyde V21V

ConcepTest 2 Bonnie and Klyde IIA) KlydeB) BonnieC) both the sameD) linear velocity is zero

for both of them

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?

Rv tan

We already know that all points of a rotating body have the same angular velocity .


Relation between linear and angular acceleration

dtdva tan

tan

dtdRa

tan R

atotal atan

aR

RvaR

2tan

aR

tana

Rv tanBy definition of linear acceleration: Let’s use:

Ra tan

Total acceleration

Finally, recall that any object that is undergoing circular motion experiences an inwardly directed radial acceleration (centripetal one).

Rv tanR2

totala


Rotational kinematic equationsThe equations of motion for translational and rotational motion

(for constant acceleration) are identical

o ot 12t2

)(221

22

2

oo

oo

o

xxavv

attvxx

atvv

oo 222

o tv

a

x

Translational kinematic equations

Rotational kinematic equations


Rotational Dynamics

What causes rotation?


Torque is a turning force (the rotational equivalent of force).

It depends on force, lever arm, angle:

Why? What causes rotation?

TorqueWhen we apply the force, the door turns on its hinges(a turning effect is produced). In the 1st case, we are able to open the door with ease. In the 2nd case, we have to apply much more force to cause the same turning effect.

A longer lever arm is very helpful in rotating objects.

rF sinF

r

Torque due to a force F applied at a distance r from the pivot, at an angle θ to the radial line.


Axis of rotation

There are two ways of calculating Torque

rF sin

r(F sin )

F(rsin )

F

rsinF

Let’s arrange it like this:

Perpendicular component of force acting at a distance r from the axis

Axis

F

r

sinrOr Let’s arrange it like this:

Force times arm lever extending from the axis to the line of force and perpendicular to the line of force

1

2

You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in loosening the nut?

A

C D

B

Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest lever arm (case #2) will provide the largest torque.

E) all are equally effective

Follow-up: What is the difference between arrangement A and D?

ConcepTest 3 Using a Wrench


Torque causes angular acceleration:

Force causes linear acceleration: (N.2nd law):

Newton’s 2nd law of rotation

I

I is the Moment of Inertia (rotational equivalent of mass)

amF


Thank youSee you on Monday

ConcepTest 3 Closing a DoorA

B

C

D

E

The diagram shows the top view of a door, hinge to the left and door-knob to the right. The same force F is applied differently to the door. In which case is the turning ability provided by the applied force about the rotation axis greatest?

The torque is t = Fd sinq, and so the force that is at 90° to the lever arm is the one that will have the largest torque. (Clearly, to close the door, you want to push perpendicularly!!) So A or B? B has larger lever arm

A B C

D

E


Frequency and Period

We can relate the angular velocity of rotation to the frequency of rotation:

1 rev/s =2 rad/s

f 1T

2

1 hertz (Hz) = 1 rev/s

Documents

Rotational Motion Torque