Concept SummaryConcept SummaryBelton High School PhysicsBelton High School Physics

Circular Motion Terms

• The point or line that is The point or line that is the the centercenter of the circle is of the circle is the the axis of rotationaxis of rotation..

• If the axis of rotation is If the axis of rotation is insideinside the object, the the object, the object is object is rotating rotating (spinning)(spinning)..

• If the axis of rotation is If the axis of rotation is outsideoutside the object, the the object, the object is object is revolvingrevolving..

Planar, rigid object rotating about origin O.

Rotational variables

Look at one point P:

rsArc length s:

Thus, the angular position is:

s

radian measurer

is measured in degrees or radians (more common)

One radian is the angle subtended by an arc length equal to the radius of the arc.

rs = r

For full circle: 22

r

r

r

s

Full circle has an angle of 2 radians,

Thus, one radian is 360°/2

Radian degrees2 360° 180° 90°

1 57.3°

2 pod racers complete a 180° turn and remain neck and neck. Which had the greater Linear Velocity?A. The one closer to the point of rotation.B. The one farther from the point of rotation.C. Both were equal.

How would you describe the time it takes for each of the 2 pod racers to complete the turn?

A. Greater for the one closer to the point of rotation.

B. Greater for the one farther from the point of rotation.

C. Both the same.

Angular Angular VelocityVelocity

• rotationalrotational or or angular angular velocityvelocity, which is , which is the the rate angular rate angular position changesposition changes..

• Rotational velocity Rotational velocity is measured in is measured in radians/sec, radians/sec, degrees/second, degrees/second, rotations/minute rotations/minute (rpm), etc.(rpm), etc.

• Common Common symbol, symbol, (Greek letter (Greek letter omegaomega))

Define quantities for circular motion

(note analogies to linear motion!!)

Angular displacement:

Average angular speed:

Instantaneous angular speed:

Average angular acceleration:

Instantaneous angular acceleration:

o

oavg

ot t t

dt

d

tt

0

lim

oavg

ot t t

dt

d

tt

0

lim

10-5 Relating the linear and angular variables

rv

Tangential speed of a point P:

ta r

Tangential acceleration of a point P:

Note, this is not the centripetal acceleration ar !!

rsArc length s:

Caution: Measure angular quantities in radians

22

rva rr

Rotational Vs. Tangential Rotational Vs. Tangential VelocityVelocity

• If an object is rotating:– All points on the object have the

same rotational (angular) velocity.

– All points on the object do not have the same linear (tangential) velocity

Rotational & Tangential Rotational & Tangential VelocityVelocity

• We now see that….– Tangential (linear) velocity of a

point depends on:•The rotational velocity of the point.

– More rotational velocity can mean more linear velocity.

– The distance from the axis of rotation.•More distance from the axis means

more linear velocity

Centripetal Acceleration and Angular Velocity

• The angular velocity and the The angular velocity and the linear velocity are related (v = linear velocity are related (v = ωr)ωr)

• The centripetal acceleration can The centripetal acceleration can also be related to the angular also be related to the angular velocityvelocity 2 2 2

2C

v r ωa r ω

r r

10-3 Are angular quantities vectors?

Right-hand rule for determining the direction of this vector.

• rotates through the same angle,

• has the same angular velocity,

• has the same angular acceleration.

Every particle (of a rigid object):

characterize rotational motion of entire object

Linear motion with constant linear acceleration, a.

ov v at

21

2o ox x v t at

1

2( )o ox x v v t

o t

21

2o ot t

1

2( )o o t

2 2 2 ( )o ov v a x x 2 2 2 ( )o o

10-4 Rotational motion with constant rotational acceleration, .

Rotational Inertia• Equates to “normal” Inertia (mass).

– An object rotating about an axis tends to keep rotating about that axis.

• Rotational Inertia: resistance to changes in rotational motion.

Rotational inertia (or Moment of Inertia) I of an object depends on:

- the axis about which the object is rotated.

- the mass of the object.

- the distance between the mass(es) and the axis of rotation.

- Note that must be in radian unit. The SI unit for I is kg.m2 and it is a scalar.

2i

ii rmI

Calculation of Rotational inertia

dVrdmrmrIi

iimi

222

0lim

Note that the moments of inertia are different for different axes of rotation (even for the same object)

2

12

1MLI

2

3

1MLI

TorqueTorque

• Every time you open a door, pull a lever, or use a wrench you exert a “turning” force.

• This turning force produces a Torque.– Forces make things accelerate.– Torques make things rotate

– Also known as a “couple” or “moment”

Producing TorquesProducing Torques

• A torque is produced by “leverage”– Greater the “lever” or

length of the lever arm, greater the torque.

– Greater the force you apply to the lever arm, greater the torque.

• AND the angle of the applied force plays a part. – Torque is the Cross

Product of Force X Lever arm.

The Cross ProductThe Cross Product

• A way of multiplying vectors to produce a different vector.

– NOT the same as the Dot product (produced a scalar).

Example: Torque = r F sin θ

sinABBA

• Torque is positive if the direction of rotation is counterclockwise.

• Torque is negative if the direction of rotation is clockwise.

• The SI unit of torque is N.m (Note that the unit of work J is also N.m . However, the name J is exclusively reserved for work/energy).

( )( sin ) trF Fr 10-8 Torque

sinr F magnitude

( sin )( )r F r F

• It is clear that torque can also be defined as

• We use the right-hand rule, sweeping the fingers of the right hand from to . The outstretched right thumb then gives the direction of .

• When several torques act on a body, the net torque is the sum of the individual torques, taking into account of positive and negative torques.

• Newton’s Second law can be applied to Torques!– An object will rotate in the direction of the net

Torque!– If the Net Torque is zero then no rotation occurs!

r F

torque defined as a vector productof r and F

r

F

Checkpoint: Assuming all of these forces are equal in magnitude, which direction does the net Torque point?A: to the rightB: to the leftC: into the boardD: out of the board

If a doorknob were placed in the If a doorknob were placed in the center of a door rather than at the center of a door rather than at the

edge, how much more force would be edge, how much more force would be needed to produce the same torque needed to produce the same torque

for opening the door?for opening the door?

a)a) Equal amountEqual amount

b)b) Twice as muchTwice as much

c)c) Four times as Four times as muchmuch

d)d) None of the None of the aboveabove

Suppose that a meterstick is Suppose that a meterstick is supported at the center, and a 20 N supported at the center, and a 20 N block is hung at the 80 cm mark. block is hung at the 80 cm mark.

Another block of unknown weight just Another block of unknown weight just balances the system when it is hung balances the system when it is hung

at the 10 cm mark. What is the at the 10 cm mark. What is the weight of the second block?weight of the second block?

a)a) 5 N5 Nb)b) 10 N10 Nc)c) 15 N15 Nd)d) 20 N20 N

Homework (by Wednesday)

• Watch Walter Lewin lecture 19: Watch Walter Lewin lecture 19: Rotating Rigid BodiesRotating Rigid Bodies

• And lecture 20: Angular And lecture 20: Angular momentum and Torquesmomentum and Torques– Available on iTunesU or youtube or Available on iTunesU or youtube or

my site.my site.