r ( ) −

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∆ r ∆

t

= Momentum

f

t

v i

) = r

net

alternative statement of Newton’s second law The time rate of change of momentum of an object is equal to the net force acting on it p t

Impulse

„

I

r When a single, constant force F acts on the object for time ∆ t, there is an

impulse

I

r delivered to the object: = r

F

t

= area under F-t curve F

I

r Using = r

F

∆ r = ∆ r = ∆

t m

(

v

r

f

v

r

i

) = Impulse r

F

t

t

Collision of 2 objects

Conservation of Momentum

„ Momentum is conserved for the system of objects. You must define the isolated system „ Assumes only internal forces are acting during the collision „ Can be generalized to any number of objects

Types of Collisions

„ „ Momentum is conserved in any collision Inelastic collisions „ Kinetic energy is not conserved „ Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object „ Perfectly inelastic collisions objects stick together „ occur when the

Not all of the KE is necessarily lost

Average Angular Speed

• The average angular speed, ω , of a rotating rigid object is the ratio of the angular displacement to the time interval 

av

 

t f f

  

t i i

   

t

Unit: rad/s

Average Angular Acceleration

 object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: 

av

 

t f f

  

i t i

   

t

Unit: rad/s 2

Relationship Between Angular and Linear Quantities

• Displacements 

s

  

r

• Speeds

v t

 

r

• Every point on the rotating object has the same angular motion • Not every point on the rotating object has the same linear motion • Accelerations

a t

 

r

Analogies Between Linear and Rotational Motion

Period and Frequency

T

 2    2 

r v f

 1

T

 2   Time to go around once

v

Centripetal Acceleration

– The magnitude of the centripetal acceleration is given by

a c

v r

t 2 – This direction is toward the center of the circle   The angular velocity and the linear velocity are related (

v t = ω r

) The centripetal acceleration can also be related to the angular velocity

a c

  2

r

• General equation

F c

m a c

m v

2

r

• If the force vanishes, the object will move in a straight line tangent to the circle of motion

Nonuniform circular motion

to changing speed

a t

a c

a

a t a

to changing direction

c

• Total acceleration can be found from these components

a

a t

2 

a C

2 Don’t confuse:

a t

 

r a c

v

2

r

  2

r

Newton’s Law of Universal Gravitation

• Every particle in Universe attracts every other particle with force: directly proportional to product of masses inversely proportional to square of distance between them.

F  G m 1 m 2 2 r

inverse square law

G = universal gravitational constant = 6.673 x 10 -11 N m² /kg²

Applications of Universal Gravitation Gravitational force of uniform sphere on particle outside sphere same as force exerted if entire mass of the sphere concentrated at its center--Gauss’ Law • Acceleration due to gravity • g will vary with altitude g  G M r 2 E

Gravitational Potential Energy • PE = mgy is valid only near the earth’s surface • For objects high above the earth’s surface, an alternate expression is needed PE   G M E m r – Zero reference level is infinitely far from the earth

Escape Speed

• speed needed for an object to “escape” from planet • For the earth, v esc 11.2 km/s is about • Note, v is independent of the mass of the object v esc  2 GM E R E

Ch. 8 General Torque Formula Component of F  r OR  r F sin  • The

lever arm

,

d

, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin  so  = F d = F r sin 

Torque and Equilibrium • First Condition of Equilibrium  

F

x

0 

or

0

and

 

F

y

 0 •The Second Condition of Equilibrium states 0

Center of mass and of gravity

• If an object is “symmetrical’’ then the center of mass is just where you think that it should be.

• If an object is “complicated” then 

r

 

m

M

R CoM

The torque from gravity is “as if” the force of gravity acts only at the CoM

Torque and Angular Acceleration Newton’s Second Law for a Rotating Object 

I

 analogous to

F = ma

I = moment of inertia

For Uniform Ring

I

  2 

MR

2 moment of inertia depends on quantity of matter

and

its distribution

and

location of axis of rotation

Other Moments of Inertia

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