Physics 207: Lecture 14, Pg 1 Lecture 14Goals: Assignment: l l HW6 due Wednesday, Mar. 11 l l For...

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Physics 207: Lecture 14, Pg 1

Lecture 14Goals:Goals:

Assignment: Assignment: HW6 due Wednesday, Mar. 11 For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6

do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia””

• Chapter 10 Chapter 10

• Understand spring potential energies & use energy diagramsUnderstand spring potential energies & use energy diagrams

• Chapter 11Chapter 11 Understand the relationship between force, displacement and work Recognize transformations between kinetic, potential, and thermal energies Define work and use the work-kinetic energy theorem Use the concept of power (i.e., energy per time)

Energy for a Hooke’s Law spring

Associate ½ kx2 with the “potential energy” of the spring

21 ffii mvkxmvkx

xx dvmvdxF

)( eqxxkvdxdv

mF xxxx

dxxxkdvmvdxF xxx )( eq

Energy for a Hooke’s Law spring

21 ffii mvkxmvkx

constantK UK fsfisiU

Ideal Hooke’s Law springs are conservative so the mechanical energy is constant

Energy diagrams In general:

Spring/Mass systemBall falling

Equilibrium

Example Spring: Fx = 0 => dU / dx = 0 for x=xeq

The spring is in equilibrium position

In general: dU / dx = 0 for ANY function establishes equilibrium

stable equilibrium unstable equilibrium

Comment on Energy Conservation

We have seen that the total kinetic energy of a system undergoing an inelastic collision is not conserved. Mechanical energy is lost:

Heat (friction)Bending of metal and deformation

Kinetic energy is not conserved by these non-conservative forces occurring during the collision !

Momentum along a specific direction is conserved when there are no external forces acting in this direction. In general, easier to satisfy conservation of momentum

than energy conservation.

Mechanical Energy Potential Energy (U)

Kinetic Energy (K) If “conservative” forces

(e.g, gravity, spring) then

Emech = constant = K + U

During Uspring+K1+K2 = constant = Emech

Mechanical Energy conserved

Before

During

Energy (with spring & gravity)

Emech = constant (only conservative forces) At 1: y1 = h ; v1y = 0 At 2: y2 = 0 ; v2y = ? At 3: y3 = -x ; v3 = 0 Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2

Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0

Given m, g, h & k, how much does the spring compress? Em1 = Em3 = mgh = -mgx + ½ kx2 Solve ½ kx2 – mgx +mgh = 0

mass: m

When is the child’s speed greatest?

(A) At y1 (top of jump)

(B) Between y1 & y2

(C) At y2 (child first contacts spring)

(D) Between y2 & y3

(E) At y3 (maximum spring compression)

mass: m

When is the child’s speed greatest? A: Calculus soln. Find v vs. spring displacement then maximize

(i.e., take derivative and then set to zero) B: Physics: As long as Fgravity > Fspring then speed is increasing

Find where Fgravity- Fspring= 0 -mg = kxVmax or xVmax = -mg / k

So mgh = Ug23 + Us23 + K23 = mg (-mg/k) + ½ k(-mg/k)2 + ½ mv2

2gh = 2(-mg2/k) + mg2/k + v2 2gh + mg2/k = vmax2

Inelastic Processes If non-conservative” forces (e.g, deformation, friction)

Emech is NOT constant

After K1+2 < Emech (before)

Accounting for this loss we introduce Thermal Energy (Eth , new)

where Esys = Emech + Eth = K + U + Eth

Before

During

Energy & Work

Impulse (Force vs time) gives us momentum transfer Work (Force vs distance) tracks energy transfer Any process which changes the potential or kinetic energy

of a system is said to have done work W on that system

Esys = W

W can be positive or negative depending on the direction of energy transfer

Net work reflects changes in the kinetic energy

Wnet = K

This is called the “Net” Work-Kinetic Energy Theorem

Circular Motion

I swing a sling shot over my head. The tension in the rope keeps the shot moving at constant speed in a circle.

How much work is done after the ball makes one full revolution?

v(A) W > 0

(B) W = 0

(C) W < 0

(D) need more info

Examples of “Net” Work (Wnet)

Examples of No “Net” Work

K = Wnet

Pushing a box on a rough floor at constant speed Driving at constant speed in a horizontal circle Holding a book at constant height

This last statement reflects what we call the “system”

( Dropping a book is more complicated because it involves changes in U and K, U is transferred to K )

K = Wnet

Pushing a box on a smooth floor with a constant force; there is an increase in the kinetic energy

Changes in K with a constant F

xx dvmvdxF

If F is constant

xx dvmvdxF

KmvmvxFxxF xixfxifx 2212

Net Work: 1-D Example (constant force)

Net Work is F x = 10 x 5 N m = 50 J= 10 x 5 N m = 50 J 1 Nm ≡ 1 Joule and this is a unit of energy Work reflects energy transfer

A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m.

F = 0° Start Finish

Units:

N-m (Joule) Dyne-cm (erg)

= 10-7 J

BTU = 1054 J

calorie = 4.184 J

foot-lb = 1.356 J

eV = 1.6x10-19 J

cgs Othermks

Force x Distance = Work

Newton x

[M][L] / [T]2

Meter = Joule

[L] [M][L]2 / [T]2

Net Work: 1-D 2nd Example (constant force)

Net Work is F x = -10 x 5 N m = -50 J= -10 x 5 N m = -50 J

Work reflects energy transfer

A force F = 10 N is opposite the motion of a box across a frictionless floor for a distance x = 5 m.

= 180° Start Finish

Work in 3D….

21)( zizfzifz mvmvzFzzF

x, y and z with constant F:

21)( yiyfyify mvmvyFyyF

21)( xixfxifx mvmvxFxxF

with zyx

KmvmvzFyFxF

Work: “2-D” Example (constant force)

(Net) Work is Fx x = F cos(= F cos(-45°) x = 50 x 0.71 Nm = 35 J = 50 x 0.71 Nm = 35 J Work reflects energy transfer

A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m

= -45°

Start Finish

Useful for performing projections.

A î = Ax

î î = 1 î j = 0

A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )

Calculation can be made in terms of components.

Calculation also in terms of magnitudes and relative angles.

Scalar Product (or Dot Product)

A B ≡ | A | | B | cos

You choose the way that works best for you!

A · B ≡ |A| |B| cos()

Scalar Product (or Dot Product)

Compare:

A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )

with A as force F, B as displacement r

and apply the Work-Kinetic Energy theorem

Notice:

F r = (Fx )(x) + (Fy )(z ) + (Fz )(z)

Fx x +Fy y + Fz z = K

So here

F r = K = Wnet

More generally a Force acting over a Distance does Work

Definition of Work, The basics

Ingredients: Force ( F ), displacement ( r )

“Scalar or Dot Product”

displace

FWork, W, of a constant force F

acts through a displacement r :

W = F · r (Work is a scalar)(Work is a scalar)

If we know the angle the force makes with the path, the dot product gives us F cos and r

If the path is curved at each point

and rdFdW

Remember that a real trajectory implies forces acting on an object

Only tangential forces yield work! The distance over which FTang is applied: Work

vpathand time

Two possible options:

Change in the magnitude of v

Change in the direction of v

aaa= +

aradialatanga= +

FradiallFtangF= +

ExerciseWork in the presence of friction and non-contact forces

A box is pulled up a rough ( > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces (including non-contact ones) are doing work on the box ? Of these which are positive and which are negative? Use a Free Body Diagram Compare force and path

Lecture 14

Assignment: Assignment: HW6 due Wednesday, March 11HW6 due Wednesday, March 11 For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6

do do notnot concern yourself with the integration process concern yourself with the integration process

Physics 207: Lecture 14, Pg 1 Lecture 14Goals: Assignment: l l HW6 due Wednesday, Mar. 11 l l For...

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