classical mechanics review 4: units 1-19 mechanics review 4, slide 1

Classical Mechanics Review 4: Units 1-19

Mechanics Review 4, Slide 1

Important Equations

Rolling Motion:

Kinetic Energy of a Rolling Object:

Gravitational Potential Energy of a Rigid Body:

Statics: and about ANY axis

Dynamics: and

Dynamics: .

If , then


Fr

t = rF sinq

RaRvRs cmcmcm ,,22

2

1

2

1cmcmtotal MvIK

cmg MgyU

0F

0

cmext aMF

Iext

Angular Momentum of a Particle: sin, rpLprL

Angular Momentum of a Symmetric Rigid Body

IL

td

Ldext

0ext constantLtotal

Kinetic Energy of a Rigid Body rotating about a fixed axis 2

2

1 AR IK

Example: Pole Supported by a Wire A pole of mass M and length L is attached to a wall by a pivot at

one end. The pole is held at an angle θ above the horizontal by a horizontal wire attached to the pole at its other end. The moment of inertia of the pole is ICM = ML2/12.

(a) What is the tension in the wire?

(b) What are the vertical and horizontal

components of the force R on the pole at

the pivot?

(c) Now the wire breaks. What is the initial

angular acceleration α of the pole?

(d) Find the angular speed of the pole just

before it hits the wall.


θ

wire

pole

wall

Example: Pole Supported by a Wire A pole of mass M and length L is attached to a wall by a pivot at

one end. The pole is held at an angle θ above the horizontal by a horizontal wire attached to the pole at its other end. The moment of inertia of the pole is ICM = ML2/12.


θ

wire

pole

wall

0cos)2/(sin0 LMgTLP

TRF xx 0

MgRF yy 0

PPP ILMgI cos)2/(

221)sin1)(2/( fpfi ILMgEE

Example: Rolling up a Plane

A uniform sphere of mass M and radius R rolls without slipping up a plane inclined at an angle with the horizontal. The initial center-of-mass velocity of the sphere is vcm. ICM = 2MR2/5.

(a) What is the maximum height, H, above the horizontal, the sphere can reach?

(b) What is the frictional force exerted by the plane on the sphere?


cmscmcm IRfI

cmscmx MaMgfMaF sin Racm

MgHMvI cmcm 22

2

1

2

1

RM

vcm

fs

Rvcm

Example: Collision with a Disk

A bullet of mass m = 0.2 kg is flying at the constant velocity V = 10 m/s in x direction and at a distance of d = 0.4 m from the x-axis towards a vertical solid disk of mass M = 1kg that is rotating counterclockwise about the fixed axis with the angular velocity ω0 = 10 rad/s. The bullet strikes the disk and sticks at point A. The radius of the disk is 0.5 m. For the disk: Icm = MR2/2.

(a) What is the angular momentum of the bullet about the center of the disk?

(b) What is the angular velocity ω of the disk right after the collision?

(c) What would the angular velocity of the disk be, when the bullet has rotated to point B after the collision?


V=10m/s

x

m=0.2kgM=1 kgd = 0.4 m

A

w0 = 10 rad/s

R=0.5 m

Btotfi ILL

mvdLb

mgdIIEE ftottotfi 2212

21

After the collision energy is conserved

0, cmdbi ILL

Example: Falling disk

A solid uniform disk of diameter d and mass M is initially supported by a force from the pivot and another vertical force F at the other end of the disk as shown. (ICM = MR2/2)

(a) Find the force F and the force R the pivot exerts on the disk.

(b) Now the force F is removed and the disk is released from rest. It is free to rotate about the pivot in the presence of gravity. Find the angular speed of the disk at the

lowest point.

(c) Find the speed of the center of the disk

at the lowest point.

(d) Find the force R the pivot exerts on the

disk at that point.


F

Example: Falling disk

A solid uniform disk of diameter d and mass M is initially supported by a force from the pivot and another vertical force F at the other end of the disk as shown. (ICM = MR2/2)


F

0)2/(0 dMgFd

MgRFF yy 0 00 xx RF

221)2/( fpfi IdMgEE

)2/(dv fcm

)2//(2 dMvMgRMaF cmycmy

00 xx RF

Example: Falling Rod

A rod of length L and mass M is pivoted about a horizontal, frictionless pin through one end. The rod is released, almost from rest in a vertical position. The moment of inertia of the rod about its center is I = ML2/12. At the instant the rod makes an angle of θ with the vertical find:

(a) The rotational kinetic energy of the rod.

(b) The angular acceleration of the rod.

(c) The speed of the center of mass of the rod.


L

m

M

θ2

21)cos1)(2/( fPfi ILMgEE

P PPP ILMgI sin)2/(

)2/(Lv fcm

Example: Leaning Beam

A uniform beam of length L and mass M is leaning against a frictionless vertical wall. The bottom of the beam makes an angle θ with the horizontal ground. ICM = ML2/12.

(a) Assuming the beam is in static equilibrium, what is the magnitude of the frictional force F between the beam and the ground?

(b) What is the minimum coefficient of static friction required so that the beam does not slip?

Mechanics Review 4 , Slide 10

θF

0cos)2/(sin0 LMgLNwall

wallx NFF 0

MgNF groundy 0

grounds NF

m2

m1

w0

Example: Rod and Disk A solid disk of mass m1 and radius R is rotating with angular velocity ω0. A thin rectangular rod with mass m2 and length l = 2R begins at rest above the disk and dropped on the disk where it begins to spin with the disk. ICM,rod = Ml2/12, ICM,disk = MR2/2.

(a) What is the final angular momentum of the rod-disk system? (b) What is the final angular velocity of the disk? (c) What is the final kinetic energy of the system?


odiskinitial IL froddiskfinal IIL

wf

finalinitial LL

221

fTotalf IE

m2

Example: Rod and Disk

A solid disk of mass m1 and radius R is rotating with angular velocity ω0. A thin rectangular rod with mass m2 and length l = 2R begins at rest above the disk and dropped on the disk where it begins to spin with the disk. ICM,rod = Ml2/12, ICM,disk = MR2/2.

The rode takes a time Δt to accelerate to its final angular speed. What average torque is exerted on the disk?


wf

t

Laverage

t

I frodaverage

Example: Person on a Beam


A uniform horizontal beam with a length L and mass M is attached to a wall by a pin connection. Its far end is supported by a cable that makes an angle θ with the beam. If a person of mass m stands at a distance d from the wall, find the tension in the cable T and the force R exerted by the wall on the beam.

0cos0 TRF xx

0)(sin0 gmMTRF yy

0)2/(sin0 gmdMLTLP

Example: Falling Disk

A disk of radius R and mass M has a string wrapped around it. The string is suspended from a fixed point and the sphere is released from rest. The moment of inertia of a disk about its center of mass is I = (1/2) MR2. Calculate:

(a) The angular acceleration of the disk.

(b) The tension in the string.

(c) What is the speed of the center of mass of the disk after it has fallen a distance H?


R

w=15rad/s

M

cmcmcm ITRI

cmcmy MaTMgMaF

Racm

Hav cmcm 22

Example: Using Angular Momentum

A sphere of mass m1 and a block of mass m2 are connected by a light cord that passes over a pulley of radius R and mass M on its thin rim. The block slides on a horizontal frictionless surface. The blocks move at velocity v.

(a) Find the angular momentum L of the system about the axis of the pulley.

(b) Using τexternal = dL/dt find the acceleration of the blocks.

Mechanics Review 4 , Slide 15

221 )( MRvRmmL

Rv 2211 )( MRRammgRm

Ra

Example For Fun: Bowling


vfk

RM

a

II

I

Rfk2

52

RM

RgMkR

gk

2

5


vfk

RM

a

M

Mgk gkM

Fa MaF


vw

a RM

t

Once vwR it rollswithout slipping

t

v0wR a Rt

0v v at w w = at


t

v

v0

atvv 0

Rv Rtatv 0

ga

R

g

2

5

tR

ggtv

2

50

07

2v

gt


classical mechanics review 4: units 1-19 mechanics review 4, slide 1

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