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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

*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

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

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**

• 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

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

• 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*

*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

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

• 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

**∑**

**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