Physics I
95.141
LECTURE 10
10/6/10
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Exam Prep Problem
• Two blocks are connected by a cord/pulley system, as shown below.
mB=5kg. The surface under block B has μS=0.6 and μS=0.4.
• A) (5pts) Draw the free body diagram for the blocks. Show
coordinates for each.
• B) (10pts) The mass of mA is slowly increased from zero, at what
mass mA will the system start to move? What will its acceleration be
for this value of mA?
• C) (10pts) What is the acceleration of the system if mA is 10kg?
B
A
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Exam Prep Problem
• Two blocks are connected by a cord/pulley system, as shown below.
mB=5kg. The surface under block B has μS=0.6 and μS=0.4.
• A) (5pts) Draw the free body diagram for the blocks. Show
coordinates for each.
B
A
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Exam Prep Problem
• Two blocks are connected by a cord/pulley system, as shown below.
mB=5kg. The surface under block B has μS=0.6 and μK=0.4.
• B) (10pts) The mass of mA is slowly increased from zero, at what
mass mA will the system start to move?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Exam Prep Problem
• Two blocks are connected by a cord/pulley system, as shown below.
mB=5kg. The surface under block B has μS=0.6 and μK=0.4.
• B) (10pts) The mass of mA is slowly increased from zero, at what
mass mA will the system start to move? What will its acceleration be
for this value of mA?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Exam Prep Problem
• C) (10pts) What is the tension in the cord if mA is 10kg?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
EXAM 1 Results
40
# Students
• Results from
Mean:exam
62 were pretty good!
20 given problems ahead of
• However,STDEV:
you were
30
time, and there were only 3 problems.
• Future exams will be harder!!
20
• In order to do well, you need to put the time in to
study
10 grade is <50, you NEED to take
• If your
advantage of the resources we offer!!
0
0 10 20 30 40 50 60 70 80 90 100
Score
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Outline
•
•
•
Velocity Dependent Forces
Uniform Circular Motion
Highway curves (banked/unbanked)
•
What do we know?
–
–
–
–
–
–
–
–
–
–
–
–
Units
Kinematic equations
Freely falling objects
Vectors
Kinematics + Vectors = Vector Kinematics
Relative motion
Projectile motion
Uniform circular motion
Newton’s Laws
Force of Gravity/Normal Force
Free Body Diagrams
Problem solving
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Drag Forces
• Force acting on an object as it moves through a
Fluid or Gas.
– Boat in water
– Any motion in air
• Cars
• Skydivers
• Projectile motion
• Mathematics of velocity dependent Forces is
tricky, but a good approximations are:
FD  bv 2
FD  bv
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Terminal Velocity
• If we assume drag force is: FD  bv
• What is terminal velocity of a skydiver?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform Circular Motion (Ch. 5.2-5.4)
• It might seem counterintuitive, but another type of
constant acceleration problem comes from uniform
circular motion  moving at constant speed in a circular
path.
y
x
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform Circular Motion (Ch. 5.2-5.4)
• It might seem counterintuitive, but another type of
constant acceleration problem comes from uniform
circular motion.
• It is easiest to describe
circular motion in polar
coordinates
y
R

R, 

x
  arclength
  R
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform Circular Motion
• To find tangential speed:
y

vT
d
R
d
x
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
d  Rd
Uniform circular motion
• Example
–
–
–
–
A record spins at 45rpm
A) Do you know what a record is?
B) What is the record’s angular velocity?
C) What is the tangential speed of a bug sitting 4cm away from the
center of the record?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform Circular Motion
• Period of rotation
– Angular velocity tells us the radians/s
– Period is the time it takes to make one rotation
2R  vT T
– Frequency f is #rev/s or Hertz (Hz)
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform Circular Motion (acceleration)
• Tangential speed is constant, but velocity is not!

vT (t  dt )

vT (t )

r ( t  dt )
95.141, F2010 Lecture 9
Department of Physics and Applied Physics

r (t )
Uniform Circular Motion (acceleration)
 

 dv v (t  dt )  v (t )
a

dt
dt

vT (t  dt )

 vT (t )
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform Circular Motion (acceleration)
• What about somewhere else on the circle?

vT (t )

r (t )

r ( t  dt )

vT (t  dt )
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Acceleration (mathematically)

vT (t )

vT (t  dt )

r ( t  dt )


r v T

r
vT

r (t )

v2
a   rˆ
r
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Uniform circular motion
• Example
– A record spins at 45rpm and the bug is still sitting 4cm away from the
center of the record
– A) What is the frequency and period of the bug’s rotation?
– B) What is the acceleration of bug?
vT  0.19 m s
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Circular Motion Review

T
2
• Angular velocity [rad/s]

• Period [s]
f
• Frequency [1/s, Hz]
2

v • Centripetal acceleration. Always towards
ar  
r
center of circle!
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Centripetal Force
• Centripetal Force is the name we assign to a
Force which results in uniform circular motion.
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Example IV (Conical Pendulum)
• A small mass (m) suspended on a cord revolves in a
circle of radius r.
– A) What direction is acceleration and what causes it?
– B) Calculate speed and period of the ball.


m
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Example IV (Conical Pendulum)
• A small mass (m) suspended on a cord revolves in a circle of radius
r.
– B) Calculate speed and period of the ball.


m
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Loop-de-loops
• Imagine you want to do a loop-de-loop at
constant speed v…
• Where are the centripetal forces coming from?
y
x
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Circular Motion Problem
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Simpson’s Circular Motion
• In order to make the loop-de-loop at a constant
speed, how fast must Homer be going?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Highway Curves
• In order for the car to make a curve without
slipping/skidding, need sufficient Force from friction.
• This force is a static friction, even though the car is
moving!!
• Coordinate system!!!
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Flat Curves (unbanked)
• What is the coefficient of static friction required
to make an unbanked curve with radius R, for a
car traveling with a speed v?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Banked Curves
• Can a car make a turn on a banked frictionless surface
without skidding? For speed v, radius R, what angle is
required?
• Coordinate system!!
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Example Problem
• A car goes around an unbanked curve (R=100m) at a
speed of 50m/s. The concrete/tire interface has a
coefficient of static friction of 1. Can the car make this
turn?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics
Example Problem
• A car goes around an banked curve (R=100m) at a
speed of 50m/s. Ignoring friction, what angle should the
curve be banked at to allow the car to make the curve?
95.141, F2010 Lecture 9
Department of Physics and Applied Physics