June 26, Week 4 Today: Chapter 6, Circular Motion Homework...

June 26, Week 4

Circular Motion 26th June 2014

Today: Chapter 6, Circular Motion

Homework Assignment #4 -Due Tomorrow.

Homework Assignment #5 - Due Monday, July 7 at 5:00PM.

No office hours next Friday.

Objects in Contact


When objects are in contact with each other and being pushed,they must have an equal acceleration.

Objects in Contact



AB

Objects in Contact



AB

aA = aB

Contact-Exercise I


A 5-kg mass A is placed in front of a 7-kg mass B on a frictionlesstable. If a 12N force is applied to mass A, what is the accelerationof the masses?

AB12N

Contact-Exercise I



AB12N

(a)12N

5 kg= 2.4m/s2

Contact-Exercise I



AB12N

(a)12N

5 kg= 2.4m/s2 (b)

12N

7 kg= 1.7m/s2

Contact-Exercise I



AB12N

(a)12N

5 kg= 2.4m/s2 (b)

12N

7 kg= 1.7m/s2 (c)

12N

12 kg= 1m/s2

Contact-Exercise I



AB12N

(a)12N

5 kg= 2.4m/s2 (b)

12N

7 kg= 1.7m/s2 (c)

12N

12 kg= 1m/s2

(d)24N

5 kg= 4.8m/s2

Contact-Exercise I



AB12N

(a)12N

5 kg= 2.4m/s2 (b)

12N

7 kg= 1.7m/s2 (c)

12N

12 kg= 1m/s2

(d)24N

5 kg= 4.8m/s2 (e)

24N

12 kg= 2m/s2

Contact-Exercise I



AB12N

(a)12N

5 kg= 2.4m/s2 (b)

12N

7 kg= 1.7m/s2 (c)

12N

12 kg= 1m/s2

(d)24N

5 kg= 4.8m/s2 (e)

24N

12 kg= 2m/s2

−→n

12N

−→w

Contact-Exercise II


A 5-kg mass A is placed in front of a 7-kg mass B on a frictionlesstable. If a 12N force is applied to mass A, what is the contact forceexerted by A on B?

AB12N

Contact-Exercise II



AB12N

(a) 19N

Contact-Exercise II



AB12N

(a) 19N (b) 17N

Contact-Exercise II



AB12N

(a) 19N (b) 17N (c) 12N

Contact-Exercise II



AB12N

(a) 19N (b) 17N (c) 12N (d) 7N

Contact-Exercise II



AB12N

(a) 19N (b) 17N (c) 12N (d) 7N (e) 5N

Contact-Exercise II



AB12N

−→

FB on A

−→

FA on B

(a) 19N (b) 17N (c) 12N (d) 7N (e) 5N

Contact-Exercise II



AB12N

−→

FB on A

−→

FA on B

(d) 7N

∑

Fx = max ⇒ 12N − FB on A = (5 kg) (1m/s2)

Or FA on B = (7 kg) (1m/s2)

Uniform Circular Motion


NOTE: Chapter 6 is only for the circular motion of a particle goingaround a circle. We’ll do the more realistic problem of a “large”rotating object in the next chapter.

For circular motion, the velocity is tangent to the circle⇒ 90◦ to thecircle’s radius.





b

−→v = velocity





b

−→v = velocity

As the object goes around thecircle, its direction changesso it accelerates





b −→v = velocity






b

−→v = velocity


Angular Position


It is easier to discuss a particle’s motion in terms of its angularposition.

Angular Position



Angular Position, θ - angle from the positive x-axis to a particle’sposition.

Angular Position




b

Angular Position




b

θ

Angular Position




b

r

θ

r - circle’s radius, unit: m

Angular Position




b

r

θ


Here we introduce a different “unit” tomeasure angle called radians or rad

Angular Position




b

r

θ



s

s - arclength, unit: m

Angular Position




b

r

θ



s

s - arclength, unit: m

When θ is in radians, s = rθ

Angular Position II




b

r

θ

s


Angular Position II




b

r

θ

s


The full circle (360◦) hass = 2πr ← circumference⇒ 360◦ = 2π rad

Angular Position II




b

r

θ

s



180◦ = π rad

Angular Position II




b

r

θ

s



180◦ = π rad

Example: Convert 30◦, 45◦, and 90◦ toradians.Convert 1 rad to degrees

Angular Position III




r

θ

s






r

θ

s


Units:





r

θ

s


Units: θ = sr





r

θ

s


Units: θ = sr⇒

mm

= 1← No Unit!





r

θ

s


Units: θ = sr⇒

mm

= 1← No Unit!

“rad” is a way specify an angularquantity





r

θ

s


Units: θ = sr⇒

mm

= 1← No Unit!

“rad” is a way specify an angularquantity

One other angle unit: therevolution(rev) - one completeround trip1 rev = 360◦ = 2π rad

Angular Velocity


The rate at which a particle circles is given by its angular velocity, ω.

Angular Velocity



θi

Angular Velocity



θi

θf

Angular Velocity



θi

θf ωav =θf − θit2 − t1

=∆θ

∆t

Angular Velocity



θi


=∆θ

∆t

Unit: rad/s

Angular Velocity



θi


=∆θ

∆t

Unit: rad/s

Other Popular Unit: RPM

Angular Velocity



θi


=∆θ

∆t

Unit: rad/s

Other Popular Unit: RPM = rev/min

Angular Velocity



θi


=∆θ

∆t

Unit: rad/s


For uniform circular motion: ω = ωav

Angular Velocity



θi


=∆θ

∆t

Unit: rad/s


For uniform circular motion: ω = ωav

By convention, ω is positive for counter-clockwise motion

Related Quantities


Related to angular velocity are period and frequency.

Related Quantities



Period, T - time for one revolution

Unit = second (s.)

Related Quantities



Period, T - time for one revolution

Unit = second (s.)

Frequency, f - how many revolutions per unit of time

Unit = Hertz (Hz).

Period and Frequency Exercise


A ball on a string takes 3 s to go around a circle. What is the periodand frequency for this motion?




(a) T = 3 s, f = 3Hz




(a) T = 3 s, f = 3Hz

(b) T =1

3s, f = 3Hz




(a) T = 3 s, f = 3Hz

(b) T =1

3s, f = 3Hz

(c) T =1

3s, f =

1

3Hz




(a) T = 3 s, f = 3Hz

(b) T =1

3s, f = 3Hz

(c) T =1

3s, f =

1

3Hz

(d) T = 3 s, f =1

3Hz




(a) T = 3 s, f = 3Hz

(b) T =1

3s, f = 3Hz

(c) T =1

3s, f =

1

3Hz

(d) T = 3 s, f =1

3Hz

(e) T = 3 s, but f cannot be determined




(a) T = 3 s, f = 3Hz

(b) T =1

3s, f = 3Hz

(c) T =1

3s, f =

1

3Hz

(d) T = 3 s, f =1

3Hz

(e) T = 3 s, but f cannot be determined

f =1

T

1Hz =1

s

Period and Angular Velocity Exercise


A ball on a string takes 3 s to go around a circle. What is the ball’sangular velocity?




(a) ω = 3 rad/s




(a) ω = 3 rad/s

(b) ω =1

3rad/s




(a) ω = 3 rad/s

(b) ω =1

3rad/s

(c) ω =2π

3rad/s = 2.1 rad/s




(a) ω = 3 rad/s

(b) ω =1

3rad/s

(c) ω =2π

3rad/s = 2.1 rad/s

(d) ω = (2π)× 3 rad/s = 18.8 rad/s




(a) ω = 3 rad/s

(b) ω =1

3rad/s

(c) ω =2π

3rad/s = 2.1 rad/s

(d) ω = (2π)× 3 rad/s = 18.8 rad/s

(e) Intentionally left blank




(a) ω = 3 rad/s

(b) ω =1

3rad/s

(c) ω =2π

3rad/s = 2.1 rad/s

(d) ω = (2π)× 3 rad/s = 18.8 rad/s


ω =1 rev

T




(a) ω = 3 rad/s

(b) ω =1

3rad/s

(c) ω =2π

3rad/s = 2.1 rad/s

(d) ω = (2π)× 3 rad/s = 18.8 rad/s


ω =1 rev

T

ω =2π

T= 2πf

Relating Linear and Angular Velocity


It now becomes important to distinguish angular velocity (ω) fromlinear velocity (v).




r




r∆θ




r∆θ

∆s = arclength




r∆θ

∆s = arclength

∆s is the linear distance traveled




r∆θ

∆s = arclength


v =∆s

∆t




r∆θ

∆s = arclength


v =∆s

∆tω =

∆θ

∆t




r∆θ

∆s = arclength


v =∆s

∆tω =

∆θ

∆t

When using radians, ∆s = r∆θ




r∆θ

∆s = arclength


v =∆s

∆tω =

∆θ

∆t


v =r∆θ

∆t




r∆θ

∆s = arclength


v =∆s

∆tω =

∆θ

∆t


v =r∆θ

∆t= r

(

∆θ

∆t

)




r∆θ

∆s = arclength


v =∆s

∆tω =

∆θ

∆t


v =r∆θ

∆t= r

(

∆θ

∆t

)

⇒ v = rω ←−ω must be in rad/s

Example


v = rω

Example: A ball on a string takes 3 s to go around a circle. If the ballis 0.5m from the center, what is its linear velocity?

June 26, Week 4Objects in ContactContact-Exercise IContact-Exercise IIUniform Circular MotionAngular PositionAngular Position IIAngular Position IIIAngular VelocityRelated QuantitiesPeriod and Frequency ExercisePeriod and Angular Velocity ExerciseRelating Linear and Angular VelocityExample

June 26, Week 4 Today: Chapter 6, Circular Motion Homework...

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