Physics 218, Chapter 3 and 41 Physics 218 Review Prof. Rupak Mahapatra

Physics 218, Chapter 3 and 4

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Physics 218ReviewProf. Rupak Mahapatra

Physics 218, Chapter 3 and 4 2

Checklist for Final•Work out all past finals from webpage

•Do ALL end of chapter exercises from all chapters

–The final questions are typically text book style questions

•Look up your final schedule


Projectile Motion

The physics of the universe:

The horizontal and The horizontal and vertical Equations vertical Equations of Motion behave of Motion behave

independentlyindependentlyThis is why we use vectors in the

first place


How to Solve Problems

The trick for all these problems is

to break them up into

the X and Y directions


Firing up in the air at an angle

A ball is fired up in the air with speed Vo and angle o. Ignore air friction. The acceleration due to gravity is g pointing down.What is the final velocity here?

Maximize Range Again

• Find the minimum initial speed of a champagne cork that travels a horizontal distance of 11 meters.


Physics 218, Chater 5 & 6 7

Translate: Newton’s Second Law

The acceleration is in the SAME direction as the NET FORCE

This is a VECTOR equation

If I have a force, what is my acceleration?

More force → more acceleration

More mass → less acceleration

gmWWeight

ma F ,ma F

am F

:EquationVector

yyxx


Pulling a box

FP

A box with mass m is pulled along a frictionless horizontal surface with a force FP at angle as given in the figure. Assume it does not leave the surface. a)What is the acceleration of the box? b)What is the normal force?


2 boxes connected with a string

Two boxes with masses m1 and m2 are placed on a frictionless horizontal surface and pulled with a Force FP. Assume the string between doesn’t stretch and is massless.

a)What is the acceleration of the boxes? b)What is the tension of the strings between the boxes?

M2 M1


The weight of a boxA box with mass m is resting on a

smooth (frictionless) horizontal table.

a) What is the normal force on the box?

b)Push down on it with a force of FP. Now, what is the normal force?

c) Pull up on it with a force of FP

such that it is still sitting on the table. What is the normal force?

d)Pull up on it with a force such that it leaves the table and starts rising. What is the normal force?

FP


Atwood MachineTwo boxes with masses

m1 and m2 are placed around a pulley with m1

>m2

a)What is the acceleration of the boxes?

b)What is the tension of the strings between the boxes?

Ignore the mass of the pulley, rope and any friction. Assume the rope doesn’t stretch.


Kinetic Friction• For kinetic friction, it turns out that

the larger the Normal Force the larger the friction. We can write

FFriction = KineticFNormal

Here is a constant• Warning:

– THIS IS NOT A VECTOR EQUATION!


Static Friction• This is more complicated• For static friction, the friction

force can vary

FFriction StaticFNormal

Example of the refrigerator: – If I don’t push, what is the static friction force?

– What if I push a little?

Physics 218, Lecture IX 14

Two Boxes and a PulleyYou hold two boxes, m1

and m2, connected by a rope running over a pulley at rest. The coefficient of kinetic friction between the table and box I is . You then let go and the mass m2 is so large that the system accelerates

Q: What is the magnitude of the acceleration of the system?

Ignore the mass of the pulley and rope and any

friction associated with the pulley


An Incline, a Pulley and two Boxes

In the diagram given, m1 and m2 remain at rest and the angle is known. The coefficient of static friction is mu and m1

is known. What is the mass m2?

m2m

1

Ignore the mass of the pulley and cord and any

friction associated with the pulley


Skiing

You are the ski designer for the Olympic ski team. Your best skier has mass m. She plans to go down a mountain of angle and needs an acceleration a in order to win the race

What coefficient of friction, , do her skis need to have?


Is it better to push or pull?You can pull or push a sled with the same force magnitude, FP, but different angles , as shown in the figures.Assuming the sled doesn’t leave the ground and has a constant coefficient of friction, , which is better?

FP

Physics 218, Chapter 7 & 8 18

Work for Constant Forces

The Math: Work can be complicated. Start with a simple case

Do it differently than the bookFor constant forces, the work is:

…(more on this later)

W=F.d


Total sum

Integral

Find the work: CalculusTo find the total work, we must sum up all the little pieces of work (i.e., F.d). If the force is continually

changing, then we have to take smaller and smaller lengths to add. In the limit, this sum becomes an

integral.

b

a

xdF


Non-Constant Force: Springs

•Springs are a good example of the types of problems we come back to over and over again!

•Hooke’s Law

•Force is NOT CONSTANT over a distance

Some constantDisplacement

xkF


Work done to stretch a Spring

How much work do you do to stretch a spring (spring constant k), at constant velocity (pulled slowly), from x=0 to x=D?

D


Work Energy Relationship

• If net positive work is done on a stationary box it speeds up. It now has energy

•Work Equation naturally leads to derivation of kinetic energy

Kinetic Energy = ½mV2


Work-Energy Relationship

•If net work has been done on an object, then it has a change in its kinetic energy (usually this means that the speed changes)

•Equivalent statement: If there is a change in kinetic energy then there has been net work on an objectCan use the change in energy

to calculate the work


Summary of equations

Kinetic Energy = ½mV2

W= KECan use change in speed to calculate the work, or

the work to calculate the speed


Conservation of Mechanical Energy

• For some types of problems, Mechanical Energy is conserved (more on this next week)

• E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick

K2+U2 = K1+U1

Conservation of Mechanical EnergyE2=E1


Problem Solving• What are the types of examples

we’ll encounter?– Gravity– Things falling– Springs

• Converting their potential energy into kinetic energy and back again

E = K + U = ½mv2 + mgy


Problem Solving

For Conservation of Energy problems:

BEFORE and AFTER diagrams


Potential EnergyA brick held 6 feet in the air has potential energy

•Subtlety: Gravitational potential energy is relative to somewhere!

Example: What is the potential energy of a book 6 feet above a 4 foot high table? 10 feet above the floor?

• U = U2-U1 = Wext = mg (h2-h1)•Write U = mgh•U=mgh + ConstOnly change in potential energy is really meaningful


Other Potential Energies: Springs

Last week we calculated that it took ½kx2 of work

to compress a spring by a distance xHow much

potential energy does it now how

have?

U(x) = ½kx2

Physics 218, Lecture XII 31

Energy Summary

If work is done by a non-conservative force it does negative work (slows something down), and we get heat, light, sound etc.

EHeat+Light+Sound.. = -WNC

If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost)

K1+U1 = K2+U2+EHeat…

K1+U1 = K2+U2-WNC

Physics 218, Lecture XIII 32

Force and Potential EnergyIf we know the potential energy, U,

we can find the force

This makes sense… For example, the force of gravity points down, but the potential increases as you go up

dxdU

xF


Mechanical Energy

•We define the total mechanical energy in a system to be the kinetic energy plus the potential energy

•Define E≡K+U


Conservation of Mechanical Energy

• For some types of problems, Mechanical Energy is conserved (more on this next week)

• E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick

K2+U2 = K1+U1

Conservation of Mechanical EnergyE2=E1

Physics 218, Lecture XV 35

Friction and Springs

A block of mass m is traveling

on a rough surface. It reaches a

spring (spring constant k) with

speed Vo and compresses it a total distance

D. Determine

Physics 218, Lecture XV 36

Robot ArmA robot arm has a funny Force equation in 1-dimension

where F0 and X0 are constants. The robot picks up a block at X=0 (at rest) and throws it, releasing it at X=X0. What is the speed of the block?

2

0

2

0X x3x

1F F













Physics 218, Chapter 12 49

Overview: Rotational Motion

• Take our results from “linear” physics and do the same for “angular” physics

• Analogue of –Position ←–Velocity ←–Acceleration ←–Force–Mass–Momentum–Energy

Start here!

Ch

ap

ters

1-3


Velocity and Acceleration

22

2

secradians/ dtd

or dtd

onaccelerati angular the as Define

secradians/ dtd

or t

velocity angular the as Define


Right-Hand RuleYes!Define the direction to point along the axis of rotation

Right-hand Rule

This is true for and


Uniform Angular Acceleration

Derive the angular equations of motion for constant angular

acceleration

t

t21

t

0

200


Rolling without Slipping

•In reality, car tires both rotate and translate

•They are a good example of something which rolls (translates, moves forward, rotates) without slipping

•Is there friction? What kind?


Derivation

• The trick is to pick your reference frame correctly!

• Think of the wheel as sitting still and the ground moving past it with speed V.

Velocity of ground (in bike frame) = -R

=> Velocity of bike (in ground frame) = R


Centripetal Acceleration

• “Center Seeking”

• Acceleration vector= V2/R towards the center

• Acceleration is perpendicular to the velocity

)r̂(Rv

a2

R

direction r̂


Circular Motion: Get the speed!

Speed = distance/time Distance in 1 revolution

divided by the time it takes to go around once

Speed = 2r/TNote: The time to go around once is known as the Period, or T


More definitions

•Frequency = Revolutions/sec

radians/sec f = /2

•Period = 1/freq = 1/f


Ball on a String

A ball at the end of a string is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t.

What is the centripetal acceleration?


The Trick To Solving Problems

)r̂(Rv

m

amF2


Banking Angle

You are a driver on the NASCAR circuit. Your car has m and is traveling with a speed V around a curve with Radius R

What angle, , should the road be banked so that no friction is required?














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Physics 218, Chapter 3 and 41 Physics 218 Review Prof. Rupak Mahapatra