Rotational Motion - West Linn · Linear Motion Graphs related to Rotational Motion Graphs:...

© 2010 Pearson Education, Inc. Slide 7-2

7 Rotational Motion

© 2010 Pearson Education, Inc.

Recall from Chapter 6…

Angular displacement = θ

θ= ω t

Angular Velocity = ω (Greek: Omega)

ω = 2 π f and ω = θ/ t

All points on a rotating object rotate through the same angle in the same time, and have the same frequency.

Angular velocity: all points on a rotating object have the same angular velocity, ω, but different speeds, v, and v =ωr.

v =ωr


ω is positive if object is rotating counterclockwise. (Negative if rotation is clockwise.)

Conversion: 1 revolution = 2 π rad


Checking Understanding

Two coins rotate on a turntable. Coin B is twice as

far from the axis as coin A.

A. The angular velocity of A is twice that of B.

B. The angular velocity of A equals that of B.

C. The angular velocity of A is half that of B.

Slide 7-13


Answer

All points on the turntable rotate through the

same angle in the same time. ω = θ/ tAll points have the same period, therefore, all

points have the same frequency.

ω = 2 π f

Two coins rotate on a turntable. Coin B is

twice as far from the axis as coin A.

A. The angular velocity of A is twice

that of B.

B. The angular velocity of

A equals that of B.

C. The angular velocity of A is half

that of B.

Slide 7-14




A. The speed of A is twice that of B.

B. The speed of A equals that of B.

C. The speed of A is half that of B.


Slide 7-15


Answer

Twice the radius means twice the speed, v.

Therefore, B moves with twice the speed of A.



A. The speed of A is twice that of B.

B. The speed of A equals that of B.

C. The speed, v, of A is

half that of B.

Slide 7-16

v rw=


Angular Acceleration, α (Greek: Alpha)

Angular acceleration α measures how

rapidly the angular velocity is changing:

Units for Angular Acceleration

are rad/s/s or rad/s2

If the turning rate, or angular

velocity, ω, is constant, then there

is no angular acceleration, α, but

if ω is changing, then the object is

NOT undergoing UCM and the

object has angular acceleration…


Sign of the Angular Acceleration—careful!

+ω, +α

=speeding up

-ω, +α =

slowing down +ω, -α =

slowing down

-ω , -α =

speeding up


Do NOT confuse angular acceleration, α = Δω/ Δt,

with centripetal acceleration,

ac = v 2/r = ω2r. Angular acceleration, α, indicates how rapidly

the angular velocity, ω, is changing. (No UCM)

Centripetal acceleration, ac, is a vector

quantity, directed toward the center of a particle’s circular path, and is nonzero even if the angular velocity (turning rate) is constant. (ac exists always– whether experiencing UCM or not.)


Linear and Circular Motion Compared

Slide 7-18


Linear and Circular Kinematics Compared

Slide 7-19

ωf = ωi +αt


Linear Motion Graphs related to Rotational Motion Graphs:

Displacement vs. Time: Slope = Velocity

Angular displacement, θ, vs. Time:

Slope = Angular Velocity, ω= Δθ/Δt

θ

t


Linear Motion Graphs related to Rotational Motion Graphs:

Velocity vs. Time: Slope = Acceleration

Angular velocity vs. Time,

Slope = Angular Acceleration, α =Δω/Δt

ω

t


Centripetal and Tangential Acceleration

Slide 7-22

No UCM:


Example Problem

A high-speed drill rotating CCW

takes 2.5 s to speed up to 2400 rpm.

A. What is the drill’s angular

acceleration?

B. How many revolutions does it make

as it reaches top speed?

Slide 7-21


Answer:

Given: ωi = 0, ωf = 2400 rev/min, t = 2.5 seconds

ωf = 2400 rev/min (2πrad/1 rev)(1 min/60 sec) = 251.3 rad/s

Find: α = ?, θ = # revolutions = ?

Plan: Find α using ωf = ωi +αt, then find θusing θ=ωi t + 1/2 αt2. Convert θ to revolutions.

Solve: ωf = ωi +αt

251.3 rad/s = 0 + α (2.5 s)

α= (251.3 rad/s)/(2.5 s) = 100.5 rad/s2 = α

Go to next slide for more…

K

j

k

;

K

j

k

;


Δθ=ωi t + 1/2 αt2

Δθ= 0 + ½ (100.5 rad/s2)(2.5 s)2

Δθ= 314.16 rad

Δθ = 314.16 rad (1 rev/2πrad) = 50 rev

Δθ= 50 revolutions


Torque… rhymes with fork…

Every time you open a door, turn on a water faucet, or tighten a nut with a wrench, you exert a turning force.

This turning force produces a torque.

Torque is different from a force.

If you want to make an object move, apply a force. Forces tend to make things accelerate.

If you want to make an object turn or rotate, apply a torque. Torques produce rotation.


Interpreting Torque, ( (Greek: Tau)

Slide 7-25

Torque is due to the component of the force

perpendicular to the radial line.

t = rF̂ = rFsinf


A Second Interpretation of Torque

t = r̂ F = rFsinfSlide 7-26


Signs and Strengths of the Torque

Slide 7-27


Torque: that which causes rotational velocity to change.

Variable for torque= (tau)

Increasing F causes greater linear acceleration

Increasing causes greater rotational acceleration

Demonstration: pushing door open…When does door open best?

A) from outside edge to axis of rotation

B) pushing toward hinge

C) at an angle

D) from center of door to axis of rotation (see next slide)



The four forces shown have the same

strength. Which force would be most

effective in opening the door?

Slide 7-23

A. Force F1

B. Force F2

C. Force F3

D. Force F4

E. Either F1 or F3


Answer

The four forces shown have the same strength. Which force

would be most effective in opening the door?

Slide 7-24

A. Force F1B. Force F2

C. Force F3

D. Force F4

E. Either F1 or F3


What does torque depend on?

1. The size of the force (how hard you push)

2. The point where the force is applied relative to the rotation point.

3. The direction of the force.


= F rWhere F and r have to be to each other!

= F r or…

= F r (Units: N·m)

F = the force applied

r = the moment arm or the lever arm = distance between axis of rotation and line of action so that where the lever arm (r) and line of action meet, a 90o angle is formed.

Pivot point = axis of rotation(in this case, the hinge)

Torque is positive when the force tends to produce a counter-clockwise rotation, and negativewhen force tends to produce clockwise rotation.


Even though the magnitude of the applied force is the same in each case, why does the bottom situation produce the greatest amount of torque?


For You to Solve:

A person pushes on the edge of a 0.90m wide door with a force of 5.0N acting at an angle of 350 to the plane of the door. What is the torque of the force acting on the door?


Another problem for you to solve…

Revolutionaries attempt to pull down a

statue of the Great Leader by pulling on

a rope tied to the top of his head. The

statue is 17 m tall, and they pull with a

force of 4200 N at an angle of 65° to

the horizontal. What is the torque they

exert on the statue? If they are standing

to the right of the statue, is the torque

positive or negative?

Slide 7-28


And the answer is…

F = F (cosθ)

F = 4200N (cos 65o)

F = 1775 N

= F r

= (1775 N)(17 m)

= 30,175 Nm, clockwise

= - 30,175 Nm

θ=65o

F=?

F = 4200N

r = 17 m


Which factor does the torque on an

object not depend on?

A. The magnitude of the applied force.

B. The object’s angular velocity.

C. The angle at which the force is

applied.

D. The distance from the axis to the

point at which the force is applied.

Slide 7-7


Answer

Which factor does the torque on an object

not depend on?

A. The magnitude of the applied force.

B.The object’s angular velocity.

C. The angle at which the force is

applied.

D. The distance from the axis to the

point at which the force is applied.

Slide 7-8


A net torque applied to an object

causes…???

A. a linear acceleration of the

object.

B. the object to rotate at a constant

rate.

C. the angular velocity of the object

to change.

D. the moment of inertia of the

object to change.

Slide 7-11


Answer

A net torque applied to an object

causes…

A. a linear acceleration of the object.

B. the object to rotate at a constant

rate.

C.the angular velocity of the

object to change.

D. the moment of inertia of the object

to change.

Slide 7-12

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