Final examination and evaluations
Lecture 45 and last: Revision, the rest
6:30 - 9:30 PM, Monday 17th December
Last names A-H in Anderson Hall 370
I-Z in Moos Tower 2650
Make-up exam 8.00-11:00 AM Tuesday 18th Dec, in Anderson Hall 350
If you have a clash
Lecture 44: Revision Dynamics to
Circular Motion
Equations of Motion
For constant acceleration:
1
x = x0 + v0t + at 2
2
v = v0 + at
a = const
2
v2 - v0 = 2a( x - x0 )
Same for y, z.
In problems, resolve in orthogonal directions, usually x,y with
the accel
Final examination and evaluations
Lecture 43: Damped and Driven
Oscillations
m
d2x
dx
+b
+ kx = 0
dt 2
dt
=
6:30 - 9:30 PM, Monday 17th December
Last names A-H in Anderson Hall 370
k
b2
m 4m 2
I-Z in Moos Tower 2650
Make-up exam 8.00-11:00 AM Tuesday 18t
Simple Harmonic Motion
Lecture 41: Simple Harmonic Motion
z
j
g
L
=
=
k
L
dx
= 2 x
dt 2
z-axis
mg
= oscillation frequency
xCM
Period
T=
= phase
d
k
m
=
where A = amplitude
F = -ky
m
The general solution is an oscillation with x = A cos(t + )
R
y=0
m
d
S
Gyroscopic Motion:
Lecture 40 : Gyros and SHM
y
y
1
2
R
=t
R
3
4
6
2
2
-R
5
d2x
= 2 x
dt 2
4
6
5
=
k
m
=
If the left support is removed, what will happen?
3
0
x
Suppose you have a spinning gyroscope in the
configuration shown below:
1
L
pivot
support
pivo
Quiz 4, Thursday and Friday
Lecture 39: More Statics
y
= 0
F = 0
The fourth and last quiz will be
On Thursday in your discussion session, group quiz
x
On Friday at 8.00am, individual quiz
L/2
2w
Last names beginning A-R in room 150
Nw
Last names beginnin
Statics:
Lecture 38 Statics
d1
m1
d2
Statics is the study of systems that dont move.
Ladders, Stability of solid objects
Balanced objects
Buildings, Suspension bridges
All the forces have to balance, the system is in equilibrium
F = 0 = 0
+
CM
m2
Chinese
Lecture 37: Angular Momentum
EXT = I
Recap Angular Momentum
Define the rotational analogue of momentum p to be the
angular momentum L = r p
dI
+
dt
Recall that
FEXT =
dp
dt
dL
= r FEXT = EXT
dt
No external torque, EXT=0, angular momentum is conserved
Angu
Lecture 36: Angular Momentum
Recap Angular Momentum
Define the rotational analogue of momentum p to be the
angular momentum L = r p
M
D
m
F
Consider the rate of change of L
D/4
v1
v2
2
LZ = pd
initial
x
r
d
L
dL
= r FEXT = EXT
dt
dp
dt
No external torque,
Torque due to Gravity
Lecture 35 Torque and Angular Momentum
EXT =
dL
dt
For a set of particles
L = r p and EXT = r FEXT
where
where i = ri Fi
= I
i
i
In the absence of external torques
In
EXT =
Take the y-axis vertical in the direction of gravity and t
Recap Torque
Lecture 34: Rotation, Work and Energy
A torque is a force which rotates a body about an axis
1
2
K = I ( 2 i2 ) = WNET
f
P=
Newtons 2nd Law has a rotational equivalent
dW
d
=
=
dt
dt
R
M
= rF
= rF sin = rFt = Fl
m
I
A
The torque is given b
Recap: Torque
Lecture 33: Torque
I
Fi will make the disk spin
= I
R
We can divide the force into a
tangential and a radial component
The radial component does not
affect the spin
= Ft r
T
m
a
Apply a force Fi to a disk
The force that is making the disk
Lecture 32: Rolling and Torque
1
1
2
K NET = ICM 2 + MVCM
2
2
Physics 1301: Lecture 32, Pg 1
Physics 1301: Lecture 32, Pg 2
Rolling Motion
Kinetic Energy and CM motion
The kinetic energy of a wheel or ball rolling without slipping
is
Rolling ball and/or d
Recap: Comparison angular and
rotational kinematics
Lecture 31: Rotation, Kinetic Energy and
Moments of Inertia
a
I = 2mL2
m
Angular
I = mL2
m
= constant
L
R
m
m
I=
v4
m
m
m r
v1
2
i
r2
r3
r1 m1
v3
x = x0 + v 0t +
12
at
2
And for a point at a distance R
Lecture 30: Impulse and Rotation
= constant
= 0 + t
v
1
= 0 + 0 t + t 2
2
R
Collision timescales
timescales
Collisions typically involve interactions that happen
quickly.
s
I = ttif F dt
vf
vi
Vf
F
I = P
initial
final
The balls are in contact
for a ver
Quiz
Lecture 29: Elastic Collisions
Histogram
at rest
40
35
M
pi
Frequency
initial
P
M
20
15
10
Frequency
5
0
final
95
80
65
35
50
0 10 20 30 40 50 60 70 80 90 100
pf
5
m
30
25
20
m
Bin
Average 65.6
Physics 1301: Lecture 29, Pg 1
Physics 1301: Lecture 29,
Energy and Momentum Conservation
Lecture 28: Collisions
We have seen that the total kinetic energy of a system
undergoing an inelastic collision is not conserved.
Mechanical Energy can be lost or gained:
v2,f - v1,f =
Heat (friction)
Bending of metal (cra
Quiz 3, Thursday and Friday
Lecture 27: Conservation of Momentum
The third quiz will be
On Thursday in your discussion session, group quiz
On Friday at 8.00am, individual quiz
v
FEXT =
Last names beginning A-R in room 150
Last names beginning S-Z in room
Recap: Center of Mass
Lecture 26: CM & Linear Momentum
F=
dp
dt
P = MVCM
The center of mass of an object or system of objects is that
position at which the mass of the whole system can be
considered to act.
dP
= MACM
dt
For many problems the motion of the
Lever
Lever
Lecture 25: Center of Mass
m r
m
pivot
ii
RCM =
i
i
i
+
dm
A straight rod of negligible mass is
mounted on a frictionless pivot as
in the figure. Masses m1 and m2
are suspended at distances l1 and
l2.
r2
r1
RCM
r dm
rCM =
m1
m2
m1
r3
m4
m3
W
Sliding box and Spring
Lecture 24, Energy and Equilibrium
Ux =
12
kx + C
2
U y = mgy + C
U
Fx =
dU
= kx
dx
Fy =
A box of mass m = 20 kg sliding with v = 5 m/s on a horizontal
surface runs into a fixed spring of k = 10 N/m, compressing it.
Suppose there
Conservation of Kinetic + Potential
Energy
Lecture 23 Energy Examples & the
Generalised Work-Energy Theorum
Work-
Kinetic energy is energy of motion
K=1/2mv2
Potential energy is energy stored in a system
Examples:
energy due to gravity U = mgh
energy in
Lecture 22: Conservation of Energy
Review of Potential and Kinetic Energy
Potential energy is energy possessed by virtue of position or
state
Gravitational potential energy, U=mgh
Energy stored in a spring, U= kx2
Kinetic energy is energy possessed by vir
Conservation of Energy
Lecture 20: Work, Energy and Power
Wnet = K 2 K1 =
P=
v=0
1
1
2
2
mv 2 mv1
2
2
Energy cannot be created or destroyed.
Just changed from one form to another.
y
dW
dt
Energy is Conserved !
B rn
True for any isolated system.
v=0
v=0
Do
Energy & Conservation of Energy
Lecture 19: Work and Energy
Conservation laws
powerful and simple
Conservation of Energy
One of the most important concepts in physics
Alternative approach to mechanics
New and improved way to evaluate and solve problems
co
Lecture 18: Drag forces and Force Examples
Fixed Pulley
T1
FDRAG
a1
T2
j
m1
m2 a2
v
F g = mg
Physics 1301: Lecture 18, Pg 1
Quiz 2, Thursday and Friday
The second quiz will be
On Thursday in your discussion session, group quiz
On Friday at 8.00am, indiv
Lecture 17: Motion in a circle
1
sin
0
-1
/2
y
cos
3/2 2
R(x,y)
x
Physics 1301: Lecture 17, Pg 1
Uniform Circular Motion
Motion in a circle :
with constant Radius R
with constant Speed v = |v|
y
v
(x,y)
R
x
Physics 1301: Lecture 17, Pg 2
Acceleration in
Lecture 16: More on Friction
n
f
k
y
m
a
N
mg
x
T
fS
a
= 300
m
vo
mg
Physics 1301: Lecture 16, Pg 1
Friction Review
Surface friction is caused by the microscopic
interactions between the two surfaces:
A surface exerts a force on a body placed on it
We ca