Transcript Review for Midterm 4

Classical Mechanics
Review #4
Review for Midterm 4
Mechanics Review 4, Slide 1
Example: Pole Supported by a Wire
A pole of mass M and length L is attached to a wall by a pivot at
one end. The pole is held at an angle θ above the horizontal by a
horizontal wire attached to the pole at its other end. The moment of
inertia of the pole is ICM = ML2/12.
(a) What is the tension in the wire?
(b) What are the vertical and horizontal
components of the force R on the pole at wall
the pivot?
(c) Now the wire breaks. What is the initial
angular acceleration of the pole?
(d) Find the angular speed of the pole just
before it hits the wall.
wire
pole
θ
Mechanics Review 4, Slide 2
Example: Pole Supported by a Wire
A pole of mass M and length L is attached to a wall by a pivot at
one end. The pole is held at an angle θ above the horizontal by a
horizontal wire attached to the pole at its other end. The moment of
inertia of the pole is ICM = ML2/12.
 P  0  TL sin   Mg( L / 2) cos  0
 Fx  0  Rx  T
 Fy  0  Ry  Mg
 P  I P  Mg( L / 2) cos  I P
Ei  E f  Mg ( L / 2)(1  sin  )  12 I p 2f
wire
wall
pole
θ
Mechanics Review 4, Slide 3
Example: Bullet Collision with Rod
A bullet m is fired horizontally with speed v at a vertical rod with length l
and mass M. The rod is at rest but can rotate freely about a pivot P at
its upper end. The bullet hits the rod at its center and gets embedded in
it. The moment of inertia of the rod ICM,rod = Ml2/12.
(a) What is the magnitude of the angular momentum of the bullet about
the pivot P before it collides with the rod?
P
(b) What is the total moment of inertia of
the system about the pivot?
v
(c) What is the angular speed of the system
right after the collision?
l
(d) What is the maximum angle through
which the system will turn before it stops?
Lb  mvl / 2
IP,Total = ICM,rod + M(l/2)2 + m(l/2)2
Linitial  Lb  L final  ITotal
E  0
1
2
I P,Total 2  ( M  m) gycm
Mechanics , Slide 4
Example: Falling Rod
A rod of length l and mass M is pivoted about a horizontal,
frictionless pin through one end. The rod is released, almost
from rest in a vertical position. The moment of inertia of the rod
about its center is ICM = Ml2/12. At the instant the rod becomes
horizontal find:
(a) The rotational kinetic energy of the rod.
(b) The angular acceleration of the rod.
l
(c) The speed of the center of mass of the rod.
M
(d) The angular momentum of the rod.
m
(e) The force R that the pivot exerts on the rod.
L  I P f
Ei  E f  Mg (l / 2)  1 I P 2f
2
 P  I P  Mg(l / 2)  I P
 Fy  Macm, y  Ry  Mg  Mat ,cm
vcm   f (l / 2) at ,cm   (l / 2)
2
F

Ma

R

Mv
 x
cm, x
x
cm /(l / 2)
Mechanics Review 4, Slide 5
Example: Block-Spring system
A block of mass m connected to a spring with spring constant k
oscillates on a horizontal frictionless surface. The block is
displaced x0 from equilibrium and given an initial velocity v0.
(a) What is the period of its motion?
(b) Find A and ϕ.
(c) What is the maximum speed and acceleration of the block?
(d) Express the position, velocity and acceleration as a
function of time.
2
m
T
 2

k
vmax  A
2
amax   A
x  A cos(t + φ)
x0  A cos(φ)
v   ωA sin(t + φ) v0   ωA sin(φ)
a   ω2A cos(t + φ)
Mechanics Review 4, Slide 6
Example: Block-Spring system
A block of mass m connected to a spring with spring constant k
oscillates on a horizontal frictionless air track.
(a) Calculate the total energy of the system and the maximum
speed of the block if the amplitude is A.
(b) What is the velocity of the block when its position is x?
2
E  K  U  12 kA2  12 mvmax
Mechanics Review 4, Slide 7
Example: Rod and Disk
A solid disk of mass m1 and radius R is rotating with angular
velocity ω0. A thin rectangular rod with mass m2 and length l = 2R
begins at rest above the disk and dropped on the disk where it
begins to spin with the disk. ICM,rod = Ml2/12, ICM,disk = MR2/2.
(a) What is the final angular momentum of the rod-disk system?
(b) What is the final angular velocity of the disk?
(c) What is the final kinetic energy of the system?
m2
m1
0
Linitial  I disk o
f
Lfinal  (I disk  I rod ) f
Mechanics Review 4, Slide 8
Example: Rod and Disk
A solid disk of mass m1 and radius R is rotating with angular
velocity ω0. A thin rectangular rod with mass m2 and length l =
2R begins at rest above the disk and dropped on the disk where
it begins to spin with the disk. ICM,rod = Ml2/12, ICM,disk = MR2/2.
The rode takes a time Δt to accelerate to its final angular speed.
What average torque is exerted on the disk?
m2
 average
L

t
 average 
I rod f
f
t
Mechanics Review 4, Slide 9
Example: Person on a Beam
A uniform horizontal beam with a length L and mass M is
attached to a wall by a pin connection. Its far end is supported by
a cable that makes an angle θ with the beam.
If a person of mass m stands at a distance d from the wall, find
the tension in the cable T and the force R exerted by the wall on
the beam.
 P  0  TL sin   (ML / 2  md ) g  0
 Fx  0  Rx  T cos  0
 Fy  0  Ry  T sin   ( M  m) g  0
Mechanics Review 4, Slide 10
Example: Leaning Beam
A uniform beam of length L and mass M is leaning against a
frictionless vertical wall. The bottom of the beam makes an
angle θ with the horizontal ground. ICM = ML2/12.
(a) Assuming the beam is in static equilibrium, what is the
magnitude of the frictional force F between the beam and the
ground?
(b) What is the minimum coefficient of static friction required so
that the beam does not slip?
  0  N wall L sin   Mg ( L / 2) cos  0
 Fx  0  F  N wall
 Fy  0  N ground  Mg
F   s N ground
θ
F
Mechanics Review 4 , Slide 11
Example: Girl Playing with Gun
A girl of mass mg sits on the edge of a merry-go-round of mass M,
radius R and moment of inertia I. The merry-go-round is initially at rest
and is free to rotate about the its center. The girl shoots a gun in a
horizontal direction tangent to the edge of the merry-go-round. The
bullet has mass mb and speed v.
(a) What is the angular momentum of the bullet about the axle?
(b) What is the moment of inertia of the merry-go-round and girl system
(c) What is the angular velocity just after the gun has been shot?
(d) If there is a constant frictional torque  exerted at the axle, how long
would it take for the merry-go-round to stop?
mg
mb
v
M
R
Mechanics Review 4, Slide 12
Example: Using Angular Momentum
A sphere of mass m1 and a block of mass m2 are connected by a
light cord that passes over a pulley of radius R and mass M on its
thin rim. The block slides on a horizontal frictionless surface. The
blocks move at velocity v.
(a) Find the angular momentum L of the system about the axis of
the pulley.
(b) Using τexternal = dL/dt find the acceleration of the blocks.
L  (m1  m2 )vR  MR 2
v  R
m1 gR  (m1  m2 ) Ra  MR 2
a  R
Mechanics Review 4 , Slide 13
Example: Oscillating Hoop
A pendulum is made by hanging a thin hoola-hoop of radius R and
mass M on a small nail.
a) What is the period of oscillation of the hoop for small
displacements?
b) What is the maximum angular speed of the hoop if its is
displaced a small angle θmax from the vertical position and
released from rest?
pivot (nail)
IP = ICM + MR2
MgR

IP
 (t )   max cost
d
  max sin t
dt

  Mgycm
D
 dt 
 MgR(1  cos max )
d
1
I
2 P
2
c) Repeat part b) using energy.
d) Find the angular speed of the hoop as a function of time.
Mechanics Unit 22, Slide 14
Example: Simple Pendulum
A simple pendulum with mass m and length L hangs from the
ceiling. It is pulled back to an small angle of θ from the vertical and
released at t = 0.
a) What is the magnitude of the force on the pendulum bob
perpendicular to the string at t = 0?
b) What is the maximum speed of the mass m?

c) What is the angular displacement as a function
of time?
d) What are tangential and radial accelerations of
m as the pendulum passes through the equilibrium
position?
L
Mechanics Lecture 22, Slide 15