1. Physics 218 Review
Prof. 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
Physics 218, Chapter 3 and 4

3. The horizontal and vertical Equations of Motion behave independently
Projectile Motion
The physics of the universe:
The horizontal and vertical Equations of Motion behave independently
This is why we use vectors in the first place
Physics 218, Chapter 3 and 4

4. The trick for all these problems is to
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 Qo. Ignore air friction. The acceleration due to gravity is g pointing down.
What is the final velocity here?
Physics 218, Chapter 3 and 4

6. 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

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
Physics 218, Chater 5 & 6

8. Pulling a box
A box with mass m is pulled along a frictionless horizontal surface with a force FP at angle Q as given in the figure. Assume it does not leave the surface.
What is the acceleration of the box?
What is the normal force? FP Q
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.
What is the acceleration of the boxes?
What is the tension of the strings between the boxes? M2 M1
Physics 218, Chater 5 & 6

10. The weight of a box
A box with mass m is resting on a smooth (frictionless) horizontal table.
What is the normal force on the box?
Push down on it with a force of FP. Now, what is the normal force?
Pull up on it with a force of FP such that it is still sitting on the table. What is the normal force?
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 Machine
Two boxes with masses m1 and m2 are placed around a pulley with m1 >m2
What is the acceleration of the boxes?
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. FFriction = mKineticFNormal
Kinetic Friction
For kinetic friction, it turns out that the larger the Normal Force the larger the friction. We can write
FFriction = mKineticFNormal
Here m is a constant
Warning:
THIS IS NOT A VECTOR EQUATION!
Physics 218, Chater 5 & 6

13. FFriction  mStaticFNormal
Static Friction
This is more complicated
For static friction, the friction force can vary
FFriction  mStaticFNormal
Example of the refrigerator:
If I don’t push, what is the static friction force?
What if I push a little?
Physics 218, Chater 5 & 6

14. Two Boxes and a Pulley
You 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 m. 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 Q is known. The coefficient of static friction is mu and m1 is known. What is the mass m2?
Ignore the mass of the pulley and cord and any friction associated with the pulley m2 m1 Q
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 Q and needs an acceleration a in order to win the race
What coefficient of friction, m, do her skis need to have? Q
Physics 218, Lecture IX

17. Is it better to push or pull?
You can pull or push a sled with the same force magnitude, FP, but different angles Q, as shown in the figures.
Assuming the sled doesn’t leave the ground and has a constant coefficient of friction, m, which is better? FP
Physics 218, Chater 5 & 6

18. Work for Constant Forces
The Math: Work can be complicated. Start with a simple case
Do it differently than the book
For constant forces, the work is:
…(more on this later)
W=F.d
Physics 218, Chapter 7 & 8

19. 2 Dim: Force Parallel to Displacement
W = F||d = F.d = Fdcosq where q is the angle between the net Force and the net displacement. You can think of this as the force component in the direction of the displacement.
Force
Force
Rotate
Displacement
F|| = Fcosq
Displacement
Physics 218, Chapter 7 & 8

20. Find the work: Calculus
To 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.
Total sum  Integral
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 constant
Displacement
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 object
Can use the change in energy to calculate the work
Physics 218, Chapter 7 & 8

25. W= DKE Kinetic Energy = ½mV2
Summary of equations
Kinetic Energy = ½mV2
W= DKE
Can 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 Energy
E2=E1
Physics 218, Chapter 7 & 8

27. Problem Solving E = K + U = ½mv2 + mgy
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. BEFORE and AFTER diagrams
Problem Solving
For Conservation of Energy problems:
BEFORE and AFTER
diagrams
Physics 218, Chapter 7 & 8

29. Potential Energy A 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?
DU = U2-U1 = Wext = mg (h2-h1)
Write U = mgh
U=mgh + Const
Only 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 x
How much potential energy does it now how have?
U(x) = ½kx2
Physics 218, Chapter 7 & 8

31. EHeat+Light+Sound.. = -WNC
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 XII

32. Force and Potential Energy
If 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
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 Energy
E2=E1
Physics 218, Lecture XIII

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 m
Physics 218, Lecture XV

36. Robot Arm A 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?
Physics 218, Lecture XV

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48. Physics 218, Chapter 3 and 4

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!
Chapters 1-3
Physics 218, Chapter 12

50. Velocity and Acceleration
Physics 218, Chapter 12

51. Right-Hand Rule Right-hand Rule Yes!
Define the direction to point along the axis of rotation
Right-hand Rule
This is true for Q, w and a
Physics 218, Chapter 12

52. Uniform Angular Acceleration
Derive the angular equations of motion for constant angular acceleration
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) = -wR
=> Velocity of bike (in ground frame) = wR
Physics 218, Chapter 12

55. Centripetal Acceleration
“Center Seeking”
Acceleration vector= V2/R towards the center
Acceleration is perpendicular to the velocity R
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 = 2pr/T
Note: The time to go around once is known as the Period, or T
Physics 218, Chapter 12

57. w radians/sec  f = w /2p More definitions Frequency = Revolutions/sec
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

59. The Trick To Solving Problems
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, Q, should the road be banked so that no friction is required?
Physics 218, Chapter 12

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