Angularmomentum(one)

Angularmomentum(system,fixedaxis)

Newton’ssecondlaw(system)

Conserva<onLaw(closed&isolated)

€

l = r × p

L = I ω

τ net =

d L

dt τ net = I α

Δ L = 0

Force

Linearmomentum(one)

Linearmomentum(system)

Linearmomentum(system)

Newton’ssecondlaw(system)

Conserva<onLaw(closed,isolated)

€

F

m v = p p ∑ = P

M v com = P

d P

dt= F net

Macom = F net

0 = Δ P

Torque

Angularmomentum(one)

Angularmomentum(system)

Angularmomentum(system,fixedaxis)

Newton’ssecondlaw(system)

Conserva<onLaw(closed,isolated)

€

τ = r × F

l = r × p

L =

l ∑

L = I ω

τ net =

d L

dt τ net = I α

Δ L = 0

€

L

€

τ nethasnomeaningunlessthenettorque,andthe

totalrota2onalmomentum,aredefinedwithrespecttothesameorigin

€

L

€

τ net

€

τ net = d

L dt

Problem 11-55: A uniform thin rod of length L and mass M can rotate in horizontal plane about a vertical axis through the COM. The rod is at rest when a bullet of mass m is fired at the rod and its path makes angle θ withrespecttotherod. The bullet lodges in the rod and the angular velocity of the rod is ω immediately after the collision. Findtheini<alvelocityofthebullet:

Conservation of angular momentum

Li = m(r

x v)i

Li = mL2vsinθ

Conservation of angular momentum

Lf = Iω + mL2

4ω

Lf =ML2

12+mL2

4

ω

Conservation of angular momentumLf = LimLv

2sinθ = ML2

12+mL2

4

ω

v =M6m

+12

Lωsinθ

Problem 11-60: The uniform rod of length L and mass M rotates about an axis at the end. As it rotates through its lowest position, it collides with a putty wad of mass m that sticks to the end. If the angular velocity right before the collision was ω0, whatistheangularvelocityrightaHerthecollision?

Conservation of Angular Momentum

Li = Iω =ML2

12+ML2

4

ω =

ML2

3ω

Conservation of Angular MomentumL f = L f (rod) + L f (putty)

Lf =ML2

3ω f + mL

2ω f

Li = Lf

ML2

3ω =

ML2

3ω f + mL

2ω f

ω f =M

M + 3mω

Problem 11- 97: A particle of mass M is dropped from rest from a point that is at a height h, and a distance s from the point O. What is the magnitude of the angular momentum of the particle with respect to point O when the particle has fallen h/2?

First lets find the velocity of the object.

v f2 = 2g h

2= gh

vf = gh

L=

rmvh4s2 + h2

−k( )

Lets find the angle between r and v

r x v= rvsinθ −k( )

sinθ =

h2

s2 +h2

2=

h4s2 + h2

Chapter 12: Equilibrium

Equilibrium

Equilibrium when:

Remember Newton’s 2nd law:

€

F net =

F i∑ = m a = d

P

dt τ net =

τ i∑ = I α =

d L

dt

€

P = constant L = constant

1)  Linear Momentum of its center of mass is constant 2)  Angular Momentum about any point is constant

€

P = 0 L = 0

Static Equilibrium when:

€

a = 0 = α

d P

dt= 0 =

d L

dt

€

⇒ v com = 0⇒ ω = 0

Stability Unstable static equilibrium

Non-equilibrium Stable static equilibrium

€

F(x) = −dU (x)dx

Is the driving force towards or away from stable static equilibrium?

Stab

le st

atic

equ

ilibr

ium

U

nsta

ble

static

equ

ilibr

ium

Tower of Pisa- When will it fall?

Checkpoint 12-1 The figure gives an overhead view of a uniform rod in static equilibrium.

a)  Can you find the magnitudes of F1 and F2 by balancing the forces?

b)  If you wish to find the magnitude of F2 by using a single equation, where should you place a rotational axis?

1) Draw a free-body diagram showing all forces acting on body and the points at which these forces act. 2) Draw a convenient coordinate system and resolve forces into components.

3) Using letters to represent unknowns, write down equations for: ∑Fx=0, ∑Fy=0, and ∑Fz=0

4) For ∑τ = 0 equation, choose any axis perpendicular to the xy plane. But choose judiciously! Pay careful attention to determining lever arm and sign! [for xy-plane, ccw is positive & cw is negative]

5) Solve equations for unknowns.

Checkpoint 12-4 A stationary 5 kg rod AC is held against a wall by a rope and friction between the rod and the wall. The uniform rod is 1 m long and θ = 30°.

(a)  If you are to find the magnitude of the force T on the rod from the rope with a single equation, at what labeled point should a rotational axis be placed?

(b)  About this axis, what is the sign of the torque due to the rod’s weight and the tension?

Problem 12-7

1) Draw a free-body diagram showing all forces acting on body and the points at which these forces act.

2) Draw a convenient coordinate system and resolve forces into components.

3) Using letters to represent unknowns, write down equations for: ∑Fx=0, ∑Fy=0, and ∑Fz=0

4) For ∑τ = 0 equation, choose any axis perpendicular to the xy plane. But choose judiciously! Pay careful attention to determining lever arm and sign! [for xy-plane, ccw is positive & cw is negative]

5) Solve equations for unknowns.

A diver of weight 580 N stands on the end of a 4.5 m diving board of negligible mass. The board is attached to two pedestals 1.5 m apart. What is the magnitude and direction of the force on the board at the left and right pedestals?

1) Draw a free-body diagram showing all forces acting on body and the points at which these forces act. 2) Draw a convenient coordinate system and resolve forces into components. 3) Using letters to represent unknowns, write down equations for: ∑Fx=0, ∑Fy=0, and ∑Fz=0

4) For ∑τ = 0 equation, choose any axis perpendicular to the xy plane. But choose judiciously! Pay careful attention to determining lever arm and sign! [for xy-plane, ccw is positive & cw is negative] 5) Solve equations for unknowns.

A small aluminum bar is attached to the side of a building with a hinge. The bar is held in the horizontal via a fish line with 14 lb test. The bar has a mass of 3.2 kg.

a)  Will the 14 lb (62.4 N) hold the bar? b) A nut is placed on the end and a squirrel of mass 0.6 kg tries to climb out and get the nut. Does the squirrel get to the nut before the string breaks? If not, how far does he get?

Example: �Squirrel trap