Physics: Review Problems

1
Physics Review Problems

• Kinematics
• Dynamics
• Work Energy
• Center of mass
• Momentum conservation collisions
• Rotational dynamics
• Statics
• Simple Harmonic Motion
• Waves

2
Kinematics

• Bill stands on the roof of his house, a distance
  d13m above the ground, operating a pitching
  machine. At t 0 he launches a ball with
  initial speed v0 at an angle q above horizontal.
  The ball hits the ground at the base of his
  friend Teds house, a distance D 80m away, at
  exactly tf 4s.
• What are v0 and q ?
• What is the maximum height H reached by the ball
  ?
• How fast vf is the ball moving when it hits the
  ground ?

v0
q
d13m
Bill
Ted
vf
D80m
3
Kinematics...

• The distance traveled in the x direction is
• The y position of the ball is given by
  which at t tf becomes

(a)
(b)
dividing (b) by (a)
4
Kinematics...

• Plug the given numbers into
• Now put this back into and solve for
  v0

5
Kinematics...

• The time at the top of the trajectory tt can be
  found by solving vy vy0 - gtt 0
• Plug this into the y-position equation

So H d h 26.6m
tt
v0
h
q
H
d13m
Bill
Ted
6
Kinematics...

• The y-component of the velocity at t tf can be
  found by plugging into vyf vy0 - gtf
• The x-component of velocity is constant
• The final speed is then

(could also find this usingenergy conservation)
v0
q
tf
Bill
Ted
vf
7
Dynamics

• Three blocks are connected by massless strings
  and frictionless pulleys as shown. The kinetic
  coefficient of friction between block C and the
  table is m0.5. The masses of blocks C and A are
  MC20kg and MA6kg. Block C accelerates to the
  right with a 0.7m/s2.
• What are the tensions in the strings, TA and TB?
• What is MB.

8
Dynamics...

• First find TA Consider F ma for mass A
• TA - MAg MAa
• TA MA(a g) 63N

TA
a
MAg
9
Dynamics...

• Next find TB Consider F ma for mass C
• TB - TA - mMCg MCa
• TB TA MC(a mg) 175N

a
MC
TA
TB
f mMCg
MCg
N
10
Dynamics...

• Finally find MB Consider F ma for mass B
• MBg - TB MBa
• MB(g - a) TB

TB
a
MBg
11
Work Energy

• A cart starts up a hill with initial speed v0.
  The body of the cart has mass m, and each of its
  four wheels has mass m (and radius r ). The
  wheels are uniform disks, and roll without
  slipping. Give answers in terms of m, v0 and g
  only.
• What is the initial kinetic energy K of the cart?
• How high up the hill h does the CM of the cart
  rise?

v0
h
12
Work Energy...

• The kinetic energy has both translational and
  rotational contributions
• For a disk and since the wheels dont slip

v0
13
Work Energy...

• Since energy is conserved, DK -DU

v 0
v0
h
14
Center of Mass

• Three uniform spherical masses are shown. Two
  are moving with the indicated velocities, and the
  third (at the origin) is stationary.
• What is the location and velocity of the center
  or mass?
• What is the total momentum of the system?

15
Center of Mass...

• Location of CM

16
Center of Mass...

• Velocity of CM

17
Center of Mass...

• Momentum of CM

18
Momentum conservation collisions

• Two particles collide elastically as shown below.
  The small particle has mass m 1kg and is
  initially moving in the x direction. The final
  velocity of m is 5m/s in the y direction, and the
  final momentum of M is 26.57o below horizontal.
• Find vi and P.
• What is the kinetic energy of M after the
  collision?
• Determine the mass M.

vf
y
q 26.57o
vi
x
after
before
P
19
Momentum conservation collisions...

• Conserve momentum in the x and y directions
• y
• x

vf
y
q 26.57o
vi
x
after
before
P
20
Momentum conservation collisions...

• Conserve energy (its elastic)

and since
vf
y
vi
x
after
before
P
21
Rotational Dynamics

• A circular cylinder of radius R .2m, which can
  rotate freely on a horizontal, frictionless axle,
  has a light rope wound around it. The rope
  passes over a smooth peg and is attached to a
  hanging mass of mass m 1kg. As m falls, the
  rope unwinds from the cylinder, whose moment of
  inertia Iz is not initially known. The mass
  accelerates downward with a 0.4m/s2.
• What is the angular acceleration a of the
  cylinder about the axle?
• What is the tension T in the rope?
• What is the moment of inertia Iz of the cylinder
  about the axle?
• What is the angular speed of the cylinder after
  m falls a distance d 10m ?

22
Rotational Dynamics...

• What is the angular acceleration a of the
  cylinder about the axle?
• Since the string does not slip on the wheel, the
  magnitude of the acceleration of a point on the
  rim of the wheen is a Ra.

23
Rotational Dynamics...

• What is the tension T in the rope?
• Using F ma for the hanging mass in the down
  direction mg - T ma
• T m(g - a) 1kg (9.8 m/s2 - 0.4 m/s2)
  9.4 N

T
m
mg
24
Rotational Dynamics...

• What is the moment of inertia Iz of the cylinder
  about the axle?
• Using t Iza for the cylinder

a
T
R
Iz
25
Rotational Dynamics...

• What is the angular speed of the cylinder after m
  falls a distance d 10m ?
• Using energy conservation, DK -DU

wf
Iz
but vf Rwf
d
26
Angular Momentum

• A student wishes to close a door withour getting
  out of bed. She throws a shoe of mass m 600gm
  which hits the door exactly in the center. The
  shoe has horizontal speed v0 8m/s just before
  hitting the door and it bounces back with
  horizontal speed v1 4m/s. The door has width w
  1m. Just after the collision, the door turns
  with angular velocity wz 0.75 rad/sec.
• What is the moment of inertia of the door about
  the hinge?

hinge
wz
w1m
v0
v1
top views
27
Angular Momentum ...

• Since there are no external torques, angular
  momentum about the hinge (z-axis) is conserved

wz
w1m
v0
v1
28
Statics

• A stepladder rests on a smooth floor as shown.
  Its left-hand member has a mass 2M, and its
  right-hand member has a mass M. The members are
  connected at the top by a hinge, and are held
  together by a horizontal massless wire near the
  bottom.
• What are the normal forces FA and FB and the
  tension in the wire T.

M
2M
2d
T
d/2
d
FA
FB
29
Statics ...

• First use FTOT 0 FA FB 3Mg

Mg
2Mg
FA
FB
30
Statics ...

• Now use tTOT 0 about the pivot at A.

But we just found that FA FB 3Mg
3d/4
d/4
So
Mg
2Mg
A
d
FB
31
Statics ...

• Now use tTOT 0 for the beam on the left about
  the hinge at the top

d/4
d/4
3d/2
T
2Mg
FB
32
Simple Harmonic Motion

• A physical pendulum consists of two equal uniform
  disks attached rigidly together as shown. Each
  disk has mass M and radius R. The pendulum
  rotates freely about a pivot at the center of one
  disk. Give answers in terms of M, R and g.
• Find the moment of inertia I about the pivot.
• Find the period of oscillation Tfor small
  displacements.

pivot
R
M
33
Simple Harmonic Motion...

• The moment of inertia will have contributions
  from both disks.

top disk
bottom disk
pivot
R
M
2R
R
M
34
Simple Harmonic Motion...

• The angular frequencyof a physical pendulum is
• Where D is the distance from the pivot to the CM,
• I is the moment of inertia about the pivot,
• and M is the total mass
• so in this case

pivot
R
CM
2Mg
35
Waves

• A wave given by y 1.5 cos(0.1x - 560t ), with x
  and y in cm and t in seconds. The wire tension
  is 28 N. Find
• the amplitude,
• the wavelength,
• the period,
• the wave speed,
• the power carried by the wave.

y
x
36
Waves...
y 1.5 cos(0.1x - 560t )
Compare this to the generic form
A 1.5 cm 0.015 m
k 0.1 cm-1 10 m-1
w 560 rad/s
37
Waves...
y .015 cos(10x - 560t )
in SI units
so
w 560 rad/s
v 56 m/s
38
Waves...
y .015 cos(10x - 560t )
Finally
So we need to figure out m
39
Waves...
y .015 cos(10x - 560t )
So
40
Thanks for a great semester Good luck on the
final !!