physics 218, chapter 3 and 41 physics 218 review prof. rupak mahapatra
TRANSCRIPT
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
Physics 218, Chapter 3 and 4 3
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
Physics 218, Chapter 3 and 4 4
How to Solve Problems
The trick for all these problems is
to break them up into
the X and Y directions
Physics 218, Chapter 3 and 4 5
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, Chapter 3 and 4 6
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
Physics 218, Chater 5 & 6 8
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?
Physics 218, Chater 5 & 6 9
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
Physics 218, Chater 5 & 6 10
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
Physics 218, Chater 5 & 6 11
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.
Physics 218, Chater 5 & 6 12
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!
Physics 218, Chater 5 & 6 13
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
Physics 218, Lecture IX 15
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
Physics 218, Lecture IX 16
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?
Physics 218, Chater 5 & 6 17
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:
W=F.d
Physics 218, Chapter 7 & 8 20
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
Physics 218, Chapter 7 & 8 21
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
Physics 218, Chapter 7 & 8 22
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
Physics 218, Chapter 7 & 8 23
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
Physics 218, Chapter 7 & 8 24
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
Physics 218, Chapter 7 & 8 25
Summary of equations
Kinetic Energy = ½mV2
W= KECan use change in speed to calculate the work, or
the work to calculate the speed
Physics 218, Chapter 7 & 8 26
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, Chapter 7 & 8 27
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
Physics 218, Chapter 7 & 8 28
Problem Solving
For Conservation of Energy problems:
BEFORE and AFTER diagrams
Physics 218, Chapter 7 & 8 29
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
Physics 218, Chapter 7 & 8 30
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
Physics 218, Lecture XIII 33
Mechanical Energy
•We define the total mechanical energy in a system to be the kinetic energy plus the potential energy
•Define E≡K+U
Physics 218, Lecture XIII 34
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
Physics 218, Chapter 12 50
Velocity and Acceleration
22
2
secradians/ dtd
or dtd
onaccelerati angular the as Define
secradians/ dtd
or t
velocity angular the as Define
Physics 218, Chapter 12 51
Right-Hand RuleYes!Define the direction to point along the axis of rotation
Right-hand Rule
This is true for and
Physics 218, Chapter 12 52
Uniform Angular Acceleration
Derive the angular equations of motion for constant angular
acceleration
t
t21
t
0
200
Physics 218, Chapter 12 53
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?
Physics 218, Chapter 12 54
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
Physics 218, Chapter 12 55
Centripetal Acceleration
• “Center Seeking”
• Acceleration vector= V2/R towards the center
• Acceleration is perpendicular to the velocity
)r̂(Rv
a2
R
direction r̂
Physics 218, Chapter 12 56
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
Physics 218, Chapter 12 57
More definitions
•Frequency = Revolutions/sec
radians/sec f = /2
•Period = 1/freq = 1/f
Physics 218, Chapter 12 58
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?
Physics 218, Chapter 12 60
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?