Lecture 11

Physics 207: Lecture 11, Pg 1

Lecture 11Goals:Goals:

Assignment:Assignment:

Read through Chapter 10, 1st four section MP HW6, due Wednesday 3/3

Chapter 8:

Employ rotational motion models with friction or in free fall

Chapter 9: Momentum & ImpulseChapter 9: Momentum & Impulse Understand what momentum is and how it relates to forces Employ momentum conservation principles In problems with 1D and 2D Collisions In problems having an impulse (Force vs. time)


One last reprisal of the Free Body Diagram

Remember1 Normal Force is to the surface2 Friction is parallel to the contact surface3 Radial (aka, centripetal) acceleration requires a net force

Zero Gravity Ride


Zero Gravity Ride

A rider in a horizontal “0 gravity ride” finds herself stuck with her back to the wall.

Which diagram correctly shows the forces acting on her?


Banked Curves

In the previous car scenario, we drew the following free body diagram for a race car going around a curve at constant speed on a flat track.

Because the acceleration is radial (i.e., velocity changes in direction only) we need to modify our view of friction.

n

mg

Ff

So, what differs on a banked curve?


Banked Curves (high speed)

1 Draw a Free Body Diagram for a banked curve.

2 Use a rotated x-y coordinates

3 Resolve into components parallel and perpendicular to bank mar

xx

y y

N

mgFf


Banked Curves, high speed

4 Apply Newton’s 1st and 2nd Laws

mar

xx

y y

N

mg cos

Ff

mg sin

mar cos

mar sin

Fx = -mar cos = - Ff - mg sin

Fy = mar sin = 0 - mg cos + N

Friction model Ff ≤ N (maximum speed when equal)


Banked Curves, low speed

4 Apply Newton’s 1st and 2nd Laws

mar

xx

y y

N

mg cos

Ff

mg sin

mar cos

mar sin

Fx = -mar cos = + Ff - mg sin

Fy = mar sin = 0 - mg cos + N

Friction model Ff ≤ N (minimum speed when equal but

not less than zero!)


Banked Curves, constant speed

vmax = (gr)½ [ ( + tan ) / (1 - tan )] ½

vmin = (gr)½ [(tan - ) / (1 + tan )] ½

Dry pavement“Typical” values of r = 30 m, g = 9.8 m/s2, = 0.8, = 20°

vmax = 20 m/s (45 mph)

vmin = 0 m/s (as long as > 0.36 )

Wet Ice“Typical values” of r = 30 m, g = 9.8 m/s2, = 0.1, =20°

vmax = 12 m/s (25 mph)

vmin = 9 m/s(Ideal speed is when frictional force goes to zero)


Navigating a hill

Knight concept exercise: A car is rolling over the top of a hill at speed v. At this instant,

A. n > w.B. n = w.C. n < w.D. We can’t tell about n without

knowing v.

This occurs when the normal force goes to zero or, equivalently, when all the weight is used to achieve circular motion.

Fc = mg = m v2 /r v = (gr)½ (just like an object in orbit)

Note this approach can also be used to estimate the maximum walking speed.

At what speed does the car lose contact?


Orbiting satellites vT = (gr)½


Locomotion: how fast can a biped walk?


How fast can a biped walk?What about weight?

(a) A heavier person of equal height and proportions can walk faster than a lighter person

(b) A lighter person of equal height and proportions can walk faster than a heavier person

(c) To first order, size doesn’t matter


How fast can a biped walk?What about height?

(a) A taller person of equal weight and proportions can walk faster than a shorter person

(b) A shorter person of equal weight and proportions can walk faster than a taller person

(c) To first order, height doesn’t matter


How fast can a biped walk?What can we say about the walker’s

acceleration if there is UCM (a smooth walker) ?

Acceleration is radial !

So where does it, ar, come from?

(i.e., what external forces act on the walker?)

1. Weight of walker, downwards

2. Friction with the ground, sideways


Impulse & Linear Momentum

Transition from forces to conservation laws

Newton’s Laws Conservation Laws

Conservation Laws Newton’s Laws

They are different faces of the same physics

NOTE: We have studied “impulse” and “momentum” but we have not explicitly named them as such

Conservation of momentum is far more general than

conservation of mechanical energy


Forces vs time (and space, Ch. 10)

Underlying any “new” concept in Chapter 9 is (1) A net force changes velocity (either magnitude or

direction) (2) For any action there is an equal and opposite

reaction

If we emphasize Newton’s 3rd Law and emphasize changes with time then this leads to the

Conservation of Momentum Principle


Example

There many situations in which the sum of the product “mass times velocity” is constant over time

To each product we assign the name, “momentum” and associate it with a conservation law.

(Units: kg m/s or N s) A force applied for a certain period of time can be

graphed and the area under the curve is the “impulse”

F (

N)

10

0 2

Time (sec)

Area under curve : “impulse”With: m v = Favg t


Force curves are usually a bit different in the real world


Example with Action-Reaction Now the 10 N force from before is applied by person

A on person B while standing on a frictionless surface For the force of A on B there is an equal and opposite

force of B on A

F (

N)

10

0 2Time (sec)

0

-10

A on B

B on A

MA x VA = Area of top curve

MB x VB = Area of bottom curve

Area (top) + Area (bottom) = 0


Example with Action-ReactionMA VA + MB VB = 0

MA [VA(final) - VA(initial)] + MB [VB(final) - VB(initial)] = 0

Rearranging terms

MAVA(final) +MB VB(final) =

MAVA(initial) +MB VB(initial)

which is constant regardless of M or V

(Remember: frictionless surface)


Example with Action-Reaction

MAVA(final) +MB VB(final) = MAVA(initial) +MB VB(initial)

which is constant regardless of M or V

Define MV to be the “momentum” and this

is conserved in a system if and only if the system is not acted on by a net external force (choosing the system is key)

Conservation of momentum is a special case of applying Newton’s Laws


Applications of Momentum Conservation

234Th238U Alpha Decay

4Hev1v2

Explosions

Radioactive decay:

Collisions


Impulse & Linear Momentum

Newton’s 2nd Law: FF = maa

)v(v

mdtd

dtd

m

This is the most general statement of Newton’s 2nd Law

pp ≡ mvv (pp is a vector since vv is a vector)

So px = mvx and so on (y and z directions)

Definition: For a single particle, the momentum p is defined as:

dt

dp

F


Momentum Conservation

Momentum conservation (recasts Newton’s 2nd Law when net external F = 0) is an important principle

It is a vector expression (Px, Py and Pz) .

And applies to any situation in which there is NO net external force applied (in terms of the x, y & z axes).

ddtP 0

dtd

EXT

PF 0EXTF


Momentum Conservation

Many problems can be addressed through momentum conservation even if other physical quantities (e.g. mechanical energy) are not conserved

Momentum is a vector quantity and we can independently assess its conservation in the x, y and z directions

(e.g., net forces in the z direction do not affect the momentum of the x & y directions)


Exercise 2Momentum Conservation

A. Box 1

B. Box 2

C. same

Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface.

The ball hitting box 1 bounces elastically back, while the ball hitting box 2 sticks.

Which box ends up moving fastest ?

1 2


Lecture 11

Assignment:Assignment:

For Monday: Read through Chapter 10, 1st four sections

MP HW6 due Wednesday 3/3

Lecture 11

Documents

Transcript of Lecture 11