Nonuniform Circular Motion - University of · PDF fileNonuniform Circular Motion ... uniform...

Nonuniform Circular Motion

!! In all strictness, we have derived the expression for

radial acceleration for uniform circular motion

!! The derived formula(e) actually apply for non-

uniform circular motion, as long as the radius of the

trajectory is constant:

1

Nonuniform Circular Motion

!! In all strictness, we have derived the expression for

radial acceleration for uniform circular motion

!! The derived formula(e) actually apply for non-

uniform circular motion, as long as the radius of the

trajectory is constant:

2

These hold as long as r is constant,

regardless of what happens to vinst and !inst

3

Linear Angular

Displacement

Avg.

Velocity

Inst.

Velocity

4

Linear Angular

Displacement

Avg. Velocity

Inst. Velocity

Average

Acceleration

Instantaneous

Acceleration

5

Constant Angular Acceleration

vs Constant Linear Acceleration

linear, a is constant angular, " is constant

H-ITT #1: angular analogy

6

PHY2053 Lecture 11

Conservation of Energy

Conservation of Energy

Kinetic Energy

Gravitational Potential Energy

Symmetries in Physics

!! Symmetry - fundamental / descriptive

property of the Universe itself [“vacuum”]

!! Laws of Physics are the same at any point

in space [“translational invariance”]

!! Conservation of Momentum [Ch 7]

!! Laws of Physics are the same at any point

in time [“time invariance”]

!! Conservation of Energy [today’s lecture]

8

PHY2053, Lecture 11, Conservation of Energy!

Colloquial:!䇾Symmetric䇿!

Physics term:!䇾Parity䇿!

More practical aspect

!! There are different, mathematically equivalent ways

to formulate Newton’s laws

!! All these calculations predict certain quantities will be

conserved for a “closed” system (0 net external force)

!! Energy, momentum, angular momentum ..

!! Existence of conserved quantities simplifies otherwise

complicated calculations

9


More practically speaking..

!! Key concepts:

!! Learn to recognize and exploit conserved quantities

!! Conserved quantities derived from Newton’s laws

!! Solutions immediately satisfy Newton’s laws

10


Energy Conservation

!! Term “closed system” means: no net external force is

acting upon any element of the system

!! The total energy of a closed system does not change

over time: total energy before = total energy after

!! “The total energy in the Universe is unchanged by any

physical process” !! next: define change of energy (work), energy itself

11


Concept of Work !! Colloquial meaning of work: effort that

produces a result. In terms of mechanics:

!! Effort ! Force, F

!! Result ! Displacement #r

!! interested in displacement due to force

!! angular term cos(") projects force # displacement

!! SI unit: Joule [ J ]; relation to calorie: 1 cal = 4.2 J 12


䃗!

W = F∆r cos(θ)

Work: signed scalar quantity !! Work can be positive, negative, and zero depending

on the orientation of the force to the displacement

13


䃗!F!

䌚r!

䃗 = 90°!

F!䌚r!

䃗!F!䌚r!

䃗 < 90°!cos䃗 > 0!

W > 0!

䃗 = 90°!cos䃗 = 0!

W = 0!

䃗 > 90°!cos䃗 < 0!

W < 0!

Total Work in a Closed System !! start with total work on a particular object

14


W =X

i

Wi =X

i

Fi∆r cos(θi)

Total Work in a Closed System

!! Recall the definition of a closed system

!! Vector sum, has to be zero in all directions:

15

X

i

~Fi = 0

X

i

~Fi cos ✓i = 0

W =

X

i

Wi =

X

i

~Fi∆r cos ✓i = ∆r

X

i

~Fi cos ✓i

W =

X

i

Wi = ∆r

X

i

~Fi cos ✓i = 0

Kinetic Energy, Definition •! consider impact of work on the velocity of an object

•! start from 1D motion, works in all three (x, y, z)

16


W = Fx∆x = max∆x ax∆x =v2

f,x

2−

v2

i,x

2

W = max∆x = m

v2

f,x

2−

v2

i,x

2

!

= mv2

f,x

2−m

v2

i,x

2

K = mv2

2W = Kf −Ki = ∆K

Kinetic Energy, Definition

17


Work Energy Theorem

Definition of kinetic energy

K = mv2

2

W = Kf −Ki = ∆K

Example #1: Mass Driver

A mass driver is a device which uses magnetic fields to accelerate a container (mass). Predicted commercial uses include launching people and cargo to bases on the Moon. The common way to specify mass drivers is to quote the kinetic energy that an object will have when leaving the driver, if it started from rest. For a 1 MJ mass driver, compute the muzzle velocity of

a) a 0.5 kg projectile

b) a 50 kg projectile 18


Mass driver notes pt 1

19


Mass driver notes pt 2

20


Gravitational Potential Energy Near Earth

!! near Earth, the usual orientation of coordinate systems

is so that the positive y axis points “up” !! the force of gravity has only one component,

in the y-direction: Fy = $mg

!! only y displacement (#y) matters for computing work:

W = FG,y%#y = $mg % #y

21


Gravitational Potential Energy Near Earth

!! consider a vertical shot upwards, vf = 0

!! W = &K = Kf $ Ki = 0 $ 'mvf2,

!! also W = $mg % #y

!! gravity did negative work, “removing” kinetic energy

22


Energy Conservation Law

!! where did the kinetic energy go? temporarily stored

in gravitational field

!! define potential energy &Ugrav = $Wgrav = mg % #y

!! computes how much kinetic energy could be

released if we let gravity work across #y

23


Energy Conservation Law (II)

!! work-energy theorem: W = &K; &K $ W = 0

!! &K + &U = 0 ! &( K + U ) = 0

!! sum of kinetic and potential energy does not change

!! define E = K + U, then E is constant in time

24


Clicker Test #1: Rollercoaster

25


Clicker notes:

•! Velocity of an object moving down a frictionless slope only depends on the height that the object started from at rest [shown in Lecture 8, Sept 15th]!

•! Since both objects start and end at the same height, they will have the same velocity as they leave their tracks. This means that they will both fly equally far. This eliminates options 4 and 5 and leaves options 1,2 and 3!

•! Reminder: the velocity of an object only depends on the height from which the object started. Whichever ball is lower at a given point, is moving faster. The curved track ball will reach the end first because it is always below the straight track.!

26


Dissipative (Non-conservative) Forces

!! friction converts mechanical energy into heat

!! heat does not “store” mechanical energy

!! therefore, there is no point in defining a “heat” or

“frictional” potential energy

!! friction always opposes motion, so Wfriction < 0

27


Dissipative (Non-conservative) Forces

!! friction always opposes motion, so Wfriction < 0

!! extend the law of energy conservation to account for

non-conservative forces:

(Ki + Ui) + WNC = (Kf + Uf)

28


Choice of Zero Point, Near Earth

!! Due to conservation of energy, only changes in

potential energy are really relevant for kinematics

!! The absolute value of potential energy at a point in

space is arbitrary - up to an additive constant

!! We have the freedom to pick a convenient point in

space and declare that the potential energy at that

point equals 0 J

29


Choice of Zero Point, Near Earth

!! We have the freedom to pick a convenient point in

space and declare that the potential energy at that point

equals 0 J

!! All other potential energies are then computed relative

to that point, based on &U = U(y) $ U(0)

!! U(y) = &U + U(0) = mg % #y + 0 = mg % (y $ 0)

30


Example #1: Rollercoaster

A roller-coaster is barely moving as

it starts down a ramp of height h.

The first figure it encounters is a

loop of radius R. How high must the

ramp be so that the roller-coaster

never loses contact with the rails?

31


Example #1: Rollercoaster

32


R!h!

33


Rollercoaster notes pt 1

34



35



36



37



Rollercoaster, final notes

38


Comment: Given that the total height of the loop is 2R, this is not really much taller than the loop itself. The ratio of the height of the ramp and the height of the loop is 2.5R / 2R = 1.25 - the ramp has to be only 25% taller than the loop for the rollercoaster to clear the highest point in the loop and stay in contact with the rails.

Bowling Ball Demo

Gravitational Potential Energy, Planetary Scales !! derivation requires math beyond baseline calculus

!! for gravitational potential at planetary scales, there already exists a

“usual” convention:

!! potential energy infinitely far away from a planet is = 0

!! convention: an object with positive total energy can “escape” a planet

(will not fall back to the planet)

!! allows easy computation of “escape” velocities for objects starting

from any R from the planet’s center 40


Ugrav = −Gm1m2

r

Example #2: Hyperbolic Comet

A comet not bound to the Sun will only pass

by the Sun once. It will trace a hyperbolic

trajectory through the Solar system. Compute

the minimum velocity of a hyperbolic comet

when it is roughly 1 A.U. away from the Sun.

The mass of the Sun is MS = 2%1030 kg. 1

Astronomical Unit is the distance from the

Earth to the Sun, 150 million km. Does the

velocity depend on the mass of the comet?

41


Comet notes, part 1:

42



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45


Next Lecture:

Hooke’s Law,

Elastic Potential Energy

Power

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