PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 5
Main points of last lecture
•
vi2
Range (yi=yf): R  sin 2
g
•
Relative velocity:
•
Newton’s Laws:
vbe = vbr + vre
S F = 0,
r
2.  F  ma
r
3. F12   F21
1. If
velocity doesn’t change.
Newton’s Third Law
r
F12   F21
Force on “1” due to “2”
• Single isolated force cannot exist
• For every action there is an equal and opposite
reaction
• Action and Reaction Forces act on different
objects
Free-body Diagram
• Draw arrows for all
forces acting ON object
• If  F  0 the object is
in “equilibrium” (at rest
or constant v)
• Otherwise, find
r
acceleration by  F  ma
Ignore rotational motion for now.
Treat object as a particle.
Example 4.1
Two blocks sit on a frictionless table. The masses are
M1=2 kg and M2=3 Kg. A horizontal force F=5 N
is applied to Block 1.
1. What is the acceleration of the blocks?
2. What is the force of block 1 on block 2?
F
1. a = 1 m/s2
2. F21= 3 N
M1 M2
Mechanical Forces
•
•
•
•
•
Gravity: w=mg (down)
Normal forces
Strings, ropes and Pulleys
Friction
Springs (later)
Rules for Ropes and Pulleys
• Force from rope points AWAY from
object
• (Rope can only pull)
• Magnitude of the force is Tension
• Tension is same everywhere in the rope
• Tension does not change when going
over pulley
Approximations: Neglect mass of rope and pulley,
neglect friction in pulley
Example 4.2
I pull a 5 kg mass up with a rope, so that it
accelerates 2 m/s2. What is the tension in the rope?
T = 59 N
Example 4.3 - Atwood Machine
a) Find acceleration
b) Find T, the tension in the
string
c) Find force ceiling must exert
on pulley
5 kg
a) a = g/6 = 1.635 m/s2
b) T = 57.2 N
c) Fpulley=2T = 114.5 N
7 kg
Example 4.4a
Which statements are correct?
Assume the objects are in static
equilibrium.
T1 is _____ T2
A) Less than
B) Equal to
C) Greater than
cos(10o)=0.985
sin(10o)=0.173
Example 4.4b
Which statements are correct?
Assume the objects are static.
T2 is ______ T3
A) Less than
B) Equal to
C) Greater than
cos(10o)=0.985
sin(10o)=0.173
Example 4.4c
Which statements are correct?
Assume the objects are static.
T3 is ______ Mg
A) Less than
B) Equal to
C) Greater than
cos(10o)=0.985
sin(10o)=0.173
Example 4.4d
Which statements are correct?
Assume the objects are static.
T1+T2 is ______ Mg
A) Less than
B) Equal to
C) Greater than
cos(10o)=0.985
sin(10o)=0.173
Example 4.4
Given that Mlight = 25 kg, find all three tensions
T3 = 245.3 N, T1 = 147.4 N, T2 = 195.7 N
Cable Pull Demo
Inclined Planes
• Choose x along the
incline and y
perpendicular to incline
• Replace force of gravity
with its components
Fg,x  mgsin 
Fg,y  mg cos
30.0
Example 4.5
Find the acceleration and the tension
a = 4.43 m/s2, T= 53.7 N
Example 4.6
M
Find M such that the box slides at constant v
M=15.6 kg
Frictional Forces
• RESISTIVE force between object and neighbors
or the medium
• Examples:
• Sliding a box
• Air resistance
• Rolling resistance
Sliding Friction
• Parallel to surface,
opposing direction of motion
• ~ independent of
the area of contact
• Depends on the surfaces in contact
• Object at rest: Static friction
• Object in motion: Kinetic friction
Static Friction, ƒs
fs  m s N
• Just enough force to
keep object at rest.
• ms is coefficient of
static friction
• N is the normal force
f
F
Kinetic
Friction, ƒk
f k  mk N
 • mk is coefficient of
kinetic friction
• Friction force opposes
direction of motion
• N is the normal force
f
F
Coefficients
of Friction
f  ms N
f  mk N
ms  mk
Example 4.7
The man pushes/pulls with a force of 200 N. The
child and sled combo has a mass of 30 kg and the
coefficient of kinetic friction is 0.15. For each case:
What is the frictional force opposing his efforts?
What is the acceleration of the child?
f=59 N, a=3.80 m/s2
/
f=29.1 N, a=4.8 m/s2
Example 4.8
Given m1 = 10 kg and m2 = 5 kg:
a) What value of ms would stop the block from sliding?
b) If the box is sliding and mk = 0.2, what is the
acceleration?
c) What is the tension of the rope?
a) ms = 0.5
b) a=1.96 m/s2
c) 39.25 N
Example 4.9
What is the minimum ms required to
prevent a sled from slipping down a
hill of slope 30 degrees?
ms = 0.577
Other kinds of friction
• Air resistance, F ~ Area  v2
• Rolling resistance, F ~ v
Terminal velocity:
Fresistance  CAv 2
 mg at terminal velocity
Coffee Filter Demo
Example 4.9
An elevator falls with acceleration a = 8.0 m/s2.
If a 200-lb person stood on a bathroom scale
during the fall, what would the scale read?
36.9 lbs
Accelerating Reference Frames
• Equivalent to “Fictitious” gravitational force
g fictitious  a frame
Fictitious Force: Derivation
1 2
x  v0 t  at
2
1F 2
 v0 t 
t
2m
Eq. of motion in fixed frame
1
x0 (t)  a f t 2
2
1 (F  ma f ) 2
x  x0 (t)  v0 t 
t
2
m
F-maf looks like force in new frame,
maf acts like fake gravitational force!
Example 4.10
You are calibrating an accelerometer so that you can
measure the steady horizontal acceleration of a car by
measuring the angle a ball swings backwards.
If M = 2.5 kg and the acceleration, a = 3.0 m/s2:
a) At what angle does the ball swing backwards?
b) What is the tension in the string?

= 17 deg
T= 25.6 N
Example 4.11a
A fisherman catches a 20 lb trout (mass=9.072 kg),
and takes the trout in an elevator to the 78th floor to
impress his girl friend, who is the CEO of a large
accounting firm. The fish is hanging on a scale, which
reads 20 lb.s while the fisherman is stationary. Later,
he returns via the elevator to the ground floor with the
fish still hanging from the scale.
In the instant just after the elevator begins to move
upward, the reading on the scale will be
______________ 20 lbs.
a) Greater than
b) Less than
c) Equal to
Example 4.11b
A fisherman catches a 20 lb trout (mass=9.072 kg), and takes
the trout in an elevator to the 78th floor to impress his girl friend,
who is the CEO of a large accounting firm. The fish is hanging
on a scale, which reads 20 lb.s while the fisherman is stationary.
Later, he returns via the elevator to the ground floor with the fish
still hanging from the scale.
On the way back down, while descending at
constant velocity, the reading on the scale will be
________________ 20 lbs.
a) Greater than
b) Less than
c) Equal to
Example 4.11c
A fisherman catches a 20 lb trout (mass=9.072 kg), and takes
the trout in an elevator to the 78th floor to impress his girl friend,
who is the CEO of a large accounting firm. The fish is hanging
on a scale, which reads 20 lb.s while the fisherman is stationary.
Later, he returns via the elevator to the ground floor with the fish
still hanging from the scale.
In the instant just before the elevator comes to a
stop on the 78th floor, the mass of the fish will be
______________ 9.072 kg.
a) Greater than
b) Less than
c) Equal to