1. Physics 121, Sections 9, 10, 11, and 12 Lecture 7
Today’s Topics:
Homework 3: Due Friday Sept. 23 @ 6:00PM
Ch.3: # 64, 75, and 81.
Ch.4: # 4, 8, 21, 25, 36, and 40.
Chapter 4: Motion in 2-D
Projectile motion
Relative velocity
More examples of FBD’s 1

2. Typical questions : (projectile motion; for given v0 and q)
What is the maximum height the ball reaches (h) ?  h L y x v0 P
y :
h = (v0 sin q) t - 1/2 g t2
v = (v0 sin q) - g t
= 0 at P
t = (v0 sin q) / g
t = (v0 sin q) / g !
How long does it take to reach maximum height ?
Would the answers above be any different if the projectile was moving only along y-axis (1-D motion) with the initial velocity: v0 sin (q) ?
( A ) YES ( B ) NO ( C ) CAN’T TELL h y x
v0 sin(q) P

3. Typical questions : (projectile motion; for given v0 and q) h L y x v0 P
What is the range of the ball (L) ?
x :
L = vx0 t = (v0 cos q) t
How long does it take for ball to reach final point (P) ?
y :
y = (v0 sin q) t - 1/2 g t2
= 0 ! when at P
[ (v0 sin q) - 1/2 g t] t = 0
t = 0 ; t = 2 (v0 sin q) / g

4. ( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Problem 1
Suppose a projectile is aimed at a target at rest placed at the same height. At the time that the projectile leaves the cannon the target is released from rest and starts falling toward ground. Would the projectile miss or hit the target ?
( A ) MISS ( B ) HIT ( C ) CAN’T TELL y x v0
t = 0
TARGET
PROJECTILE
t = t1

5. ( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Problem 2
Suppose a projectile is aimed at a target at rest somewhere above the ground as shown in Fig. below. At the same time that the projectile leaves the cannon the target falls toward ground. Would the projectile miss or hit the target ?
( A ) MISS ( B ) HIT ( C ) CAN’T TELL  y x v0
t = 0
TARGET
PROJECTILE
t = t1

6. Inertial Reference Frames:
A Reference Frame is the place you measure from.
It’s where you nail down your (x,y,z) axes!
An Inertial Reference Frame (IRF) is one that is not accelerating.
We will consider only IRF’s in this course.
Valid IRF’s can have fixed velocities with respect to each other.
More about this later when we discuss forces.
For now, just remember that we can make measurements from different vantage points.

7. Lecture 7, ACT 1 Relative Motion
Consider an airplane flying on a windy day.
A pilot wants to fly from New Haven to Bradley airport. Having asked a friendly physics student, she knows that Bradley is 120 miles due north of New Haven and there is a wind blowing due east at 30 mph. She takes off from New Haven Airport at noon. Her plane has a compass and an air-speed indicator to help her navigate. She uses her compass at the start to aim her plane north, and her air speed indicator tells her she is traveling at 120 mph with respect to the air.
After one hour,
She is at Bradley
She is due east of Bradley
She is southeast of Bradley

8. Lecture 7, ACT 2 Relative Motion
You are swimming across a 50m wide river in which the current moves at 1 m/s with respect to the shore. Your swimming speed is 2 m/s with respect to the water. You swim across in such a way that your path is a straight perpendicular line across the river.
How many seconds does it take you to get across? a) b) c) d)
2m/s
1m/s
50m

10. Dynamics
Isaac Newton ( ) published Principia Mathematica in In this work, he proposed three “laws” of motion:
Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame.
Law 2: For any object, FNET = F = ma
Law 3: Forces occur in pairs: FA ,B = - FB ,A
(For every action there is an equal and opposite reaction.)

11. Newton’s First Law
An object subject to no external forces moves with a constant velocity if viewed from an inertial reference frame.
If no forces act, there is no acceleration.
The above statement can be thought of as the definition of inertial reference frames.
An IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”.
If one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector!

12. Is Storrs a good IRF? Is Storrs accelerating? YES!
Storrs is on the Earth.
The Earth is rotating.
What is the centripetal acceleration of Storrs?
T = 1 day = 8.64 x 104 sec,
R ~ RE = 6.4 x 106 meters .
Plug this in: aS = .034 m/s2 ( ~ 1/300 g)
Close enough to 0 that we will ignore it.
Storrs is a pretty good IRF.

13. Newton’s Second Law Units The units of force are kg m/s2 = Newtons (N)
The acceleration of an object is directly proportional to the net force acting upon it. The constant of proportionality is the mass.
Units
The units of force are kg m/s2 = Newtons (N)
The English unit of force is Pounds (lbs)

14. Lecture 7, ACT 3 Newton’s Second Law
A constant force is exerted on a cart that is initially at rest on an air table. The force acts for a short period of time and gives the cart a certain final speed.
Force
Cart
Air Track
For a second shot, I can apply a force only half as large (I’m getting tired). To reach the same final speed, for how long must I apply the force ?
A) 4 x as long B) 2 x as long C) Same time
D) 1/2 as long E) 1/4 x as long

15. Lecture 7, ACT 4 Newton’s Second Law
I push with a force of 2 Newtons on a cart that is initially at rest on an air table with no air. I push for a second. Because there is no air, the cart stops after I finish pushing. It has traveled a certain distance.
Force
Cart
Air Track
For a second shot, I push just as hard but keep pushing for 2 seconds. The distance the cart moves the second time versus the first is,
A) 4 x as long B) 2 x as long C) Same
D) 1/2 as long E) 1/4 x as long

16. Lecture 7, ACT 5 Newton’s Second Law
A constant force is exerted on a cart that is initially at rest on an air table. This force is applied for a short period of time and the cart acquires a certain final speed, which I call vf1.
Force
Cart
Air Track
I repeat the experiment, but this time the cart is already moving with constant speed when I start applying the force. After exerting the same constant force for the same time interval, the cart’s final speed is,
A) vf1 B) 2vf1 C) vf12
D) cannot be determined from the information given.

17. Newton’s Third Law:
If object 1 exerts a force on object 2 (F2,1 ) then object 2 exerts an equal and opposite force on object 1 (F1,2)
F1,2 = -F2,1
For every “action” there is an equal and opposite “reaction”
This is among the most abused concepts in physics.
REMEMBER:
Newton’s 3rd law concerns force pairs which
act on two different objects (not on the same object) !

18. Consider the forces on an object undergoing projectile motion
An Example
Consider the forces on an object undergoing projectile motion
FB,E = - mB g
FB,E = - mB g
FE,B = mB g
FE,B = mB g
EARTH

19. Lecture 7, ACT 6 Newton’s Third Law
A fly gets smushed onto the windshield of a speeding bus. 
The force exerted by the bus on the fly is,
A) greater than
B) the same as
C) less than
that exerted by the fly on the bus.

20. Lecture 7, ACT 7 Newton’s Third Law
A fly gets smushed onto the windshield of a speeding bus. 
The acceleration due to this collision of the bus is,
A) greater than
B) the same as
C) less than
that of the fly.

21. Newton's Third Law...
FA ,B = - FB ,A an example,
Fm,w
Fw,m
Ff,m
Fm,f

22. Example of Bad Thinking
Since Fm,b = -Fb,m why isn’t Fnet = 0, and a = 0 ?
Fm,b
Fb,m
a ??
ice

23. Example of Good Thinking
Consider only the box as the system!
Free Body Diagram
Fm,b
Fb,m
ice

24. Example of Good Thinking
Consider only the box as the system!
Free Body Diagram
abox = Fb,m/mbox
Fb,m
abox Fg FN

25. Free Body Diagram Eat at Bob’s What are the forces on the sign ?
A heavy sign is hung between two poles by a rope at each corner extending to the poles.
Eat at Bob’s
What are the forces on the sign ?

26. Free Body Diagram T2 T1 Eat at Bob’s mg T2 T1 Add vectors mg q1 q2 q1

27. Free Body Diagram T1 T2 Eat at Bob’s mg Vertical :q1 q2
Eat at Bob’s mg
Vertical :
mg = T1sinq1 + T2sinq2
Horizontal :
T1cosq1 = T2cosq2

28. Normal Forces and String Tension
Certain forces act to keep an object in place.
These have what ever force needed to balance all others (until a breaking point).
FB,T
FT,B

29. Force Pairs
Newton’s 3rd law concerns force pairs. Two members of a force pair cannot act on the same object.
Don’t confuse gravity (the force of the earth on an object) and normal forces. It’s an extra part of the problem.
FB,T
FB,E = -mg
FE,B = mg
FT,B

30. Lecture 7, ACT 8 Newton’s 3rd Law
Two blocks are being pushed by a finger on a horizontal frictionless floor. How many action-reaction pairs of forces are present in this system? a b
(a) (b) 4 (c) 6

31. Consider the following two cases
An Example
Consider the following two cases

32. An Example The Free Body Diagrams mg FB,T= N mg Ball Falls
For Static Situation
N = mg

33. The action/reaction pair forces
An Example
The action/reaction pair forces
FB,E = -mg
FB,T= N
FT,B= -N
FB,E = -mg
FE,B = mg
FE,B = mg

34. Lecture 7, ACT 9 Gravity and Normal Forces
A woman in an elevator is accelerating upwards
The normal force exerted by the elevator on the woman is,
A) greater than
B) the same as
C) less than
the force due to gravity acting on the woman

35. Lecture 7, ACT 9b Gravity and Normal Forces
A woman in an elevator is accelerating upwards
The normal force exerted by the elevator on the woman is,
A) greater than
B) the same as
C) less than
the force the woman exerts on the elevator.

36. Exercise: Inclined plane
A block of mass m slides down a frictionless ramp that makes angle  with respect to horizontal. What is its acceleration a ? m a 

37. Inclined plane...
Define convenient axes parallel and perpendicular to plane:
Acceleration a is in x direction only. i j m a 

38. Inclined plane... Consider x and y components separately:
i: mg sin  = ma a = g sin 
j: N - mg cos  = N = mg cos  ma i j
mg sin 
mg cos  N mg 

39. Angles of an Inclined plane
ma = mg sin  mg N 
 + f = 90 f 

40. Recap of today’s lecture
Homework 3: Due Friday Sept. 23 @ 6:00PM
Ch.3: # 64, 75, and 81.
Ch.4: # 4, 8, 21, 25, 36, and 40.
Chapter 4: Motion in 2-D
Projectile motion
Relative velocity
More examples of FBD’s, etc. 27