cutnell & johnson, wiley publishing, physics 5 th ed. copywrited by holt, rinehart, &...

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

Copywrited by Holt, Rinehart, & Winston

Torque ( )

• Torque is the product of a force and the perpendicular distance from the axis of rotation to the line of action of the force (aka the lever arm).

• when F is not perpendicular to a line from the axis of rotation

or

where F can be written as Fsinθ

lF




Fr

sinFl

Torque ( )

• Units: Newton meter (Nm)• Positive: Counter Clockwise (ccw) Rotation• Max Torque when Force is 90° to the line of

action.




Torque

• If F = ma, and a = r, then we can write:

• F·r is the torque exerted on a particle at radius ‘ r’ We may take the sum of both sides over i particles to get

2rmrF

i

iii

i rm 2

Equilibrium

• A rigid body is considered in equilibrium is it has zero translational acceleration and zero angular acceleration. OR2 conditions required for equilibrium

If object has forces acting

in both the horizontal and

vertical directions then:

0F 0 Copyright ©2007 Pearson Prentice Hall, Inc.

0 xF 0 yF

Torque Summation• In summing torques since the net torque is the

same anywhere on the body, it does not matter which point of rotation we choose for our summation of torques

• Best to choose a point that is easiest to solve (The location of one of our unknown forces)

Copyright ©2007 Pearson Prentice Hall, Inc. Copyright ©2007 Pearson Prentice Hall, Inc.

DiverA diver weighing 530N is at the

end of a diving board with a length of 3.90m. The board has a negligible weight and is bolted at one end with a fulcrum supporting it at 1.40 m from the end. Find the force of the bolt and the fulcrum on the board.


Moment of Inertia

• Alpha is the same for all particles of the object and when removed, we get the MOMENT OF INERTIA. This is a property of the object just like Inertia (mass).

• Different shapes have different values of I.

i

iii

i rm 2

IorrmI neti

ii 2



Moment of Inertia

Experiments show that I is directly proportional

• to the mass.

• The distribution of mass in the body.

To illustrate this consider two wheels having equal mass but different mass distribution.

A B

If both of these wheels accelerate from rest to the same angular velocity ω in the same time t.

• The angular acceleration, α, must be the same for both wheels. Also, the total angle turned through must be the same.

• But, when moving with angular velocity ω, the particles of wheel B are moving faster than the particles of wheel A.

• Therefore, B possesses more kinetic energy than wheel A. • More work is done accelerating wheel B than wheel A.• A greater torque was needed to accelerate B than A. and so

the moment of inertia of wheel B is greater than the moment of inertia of wheel A.

A B

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Moment of Inertia

• The moment of inertia of a body is directly proportional to its mass and increases as the mass is moved further from the axis of rotation.

• The fact that I depends on mass distribution means that the same body can have different moments of inertia depending on which axis of rotation we consider.

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

Moment of Inertia Proof

• A body, of mass, m, is moving with angular speed ω

Now, v = rω

So, K.E. = ½(mr²)ω²• For a point mass,

moving in a circle of radius, r, the quantity mr² is the rotational equivalent of m.

• I = mr²• We therefore define the

moment of inertia of a point mass to be given by

K.E. = ½mv²

I = mr²

Moment of Inertia Proof

The moment of inertia of any body can be found by adding together the moments of inertia of all its component particles.

Using this idea gives the following results:

2iibody rmI


Center of Gravity

• Point at which a body’s weight can be considered to act when calculating the torque due to weight.

• Xcg=location of the center of gravity...2211 xWxWxW cgtot

...

...

21

2211

WW

xWxWxcg

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. Cut

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& J

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A synonymous concept is the center of mass and sometimes these terms are interchanged.

Center of Gravity

• If an extended object is to be balanced, it must be supported through its center of gravity.

Copyright ©2007 Pearson Prentice Hall, Inc.


Center of Gravity

• The center of gravity can be physically found by suspending an object by a point and tracing the vertical axis it hangs along. Hang the object from a different point and again tracing its vertical axis. Where these two lines cross would be the center of gravity.


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CG of an Arm

• Calculate the center of gravity of the following arm.

• W1=17 N

• W2=11 N

• W3=4.2 N


Comparing Linear and Rotational Dynamics

We can derive new equations by simply substituting rotational quantities for linear quantities.

For example:Power is the time-rate of doing

work.

• p = mv gives us

What is the last quantity derived called?

dt

d

dt

dWP

Linear quantity

Angular (rotational) quantity

d

vo1

vf2

a

F T (Torque)

m I (Moment of Inertia)

p L

L = Iω

Angular momentum

Angular Momementum (L)

• equivalent to linear momentum

• A "principle of conservation of angular momentum" also exists.

L = Iω

Conservation of Angular Momentum

• With arms and leg outstretched (A), themoment of inertia, Iinitial, is relatively large.

• Suppose the skater has a low initial angular velocity ωinitial.

• The skater then gradually decreases her/his moment of inertia by bringing arms and leg nearer to the axis of rotation (B and C).

• Her/his angular velocity is observed to increase. • This is easily explained if we consider that the person’s angular

momentum does not change.

• If Iinitial > Ifinal then ωfinal > ωinitialIinitial × initial = Ifinal × final

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• The total angular momentum of a system remains constant as long as no external torque acts on the system.

• The angular momentum, L of a point mass is given by L = Iω and the moment of inertia of a point mass is I = mr²

• so, we have• and, as v = rω we can

write

L = (mr²)ω

L = (mv)r

Angular Momentum

• linear momentum multiplied by its distance from the point considered

• Notice that we can therefore calculate the angular momentum of a body about any point even if the body is not moving in a circular path.

L = pr

A High Diver

• A diver can change his/her rate of rotation in a similar way. The diver starts out with low angular velocity with body straight and arms outstretched. The distribution of mass is then changed in order to vary the angular velocity.

Cat

• Cats use the principle of conservation of angular momentum in order to rotate themselves "in mid-air". A cat will always land on its feet, if given enough (but not too much!) height.



Comparing Linear and Rotational Dynamics

Rotational Kinetic Energy

• An Object can experience Translational (linear) motion and/or Rotational motion. Both motions have related kinetic energies.

• Translational kinetic energy is ½ mv2

• The Rotational Kinetic energy is based upon Translation KE:

22

122

122

1 )( IrmvmKE

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Riddle of the Sphinx

Beside you are two balls. One is hollow, the other is solid. Tell me which is which.

• The balls are of equal weight.– The hollow ball is weighted inside with

metal, so they are both exactly the same weight.

• A plank of wood may be used.How will you avoid being the Sphinx’s

lunch?

cutnell & johnson, wiley publishing, physics 5 th ed. copywrited by holt, rinehart, &...

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