Dynamics of Rigid Bodies
A rigid body is one in which the distances between constituent particles is constant
throughout the motion of the body, i.e. it keeps its shape.
There are two kinds of rigid body motion:
1. Translational
Rectilinear forces acting.
Section 1.
Dynamics (Newtons Laws of Motion)
Two approaches:
1) Given all the forces acting on a body, predict the subsequent (changes in)
motion.
2) Given the (changes in) motion of a body, infer what forces act upon it.
Review of Newtons Laws:
First Law
Section 2.
Curvilinear Motion
The study of the motion of a body along a general curve.
We define
uT
the unit vector at the body, tangential to the curve
u N
the unit vector normal to the curve
Clearly, these unit vectors change with time, uT (t ), u N (t
Exercise Class Week 1 Vectors - Answers
1. Write down the length r and the angle measured anticlockwise from the x-axis of
each of the vectors a to e in the diagram. Not much to explain here note that angles are
measured anticlockwise, because this is ang
Exercise Class Week 5 Rigid Bodies
1. A bicycle wheel weighs 1kg, all of which is at the rim (i.e. neglect the hub). Its
diameter 2r is 0.7m. If it rotates at = 10 rads1,
a: Calculate the speed of the rim, v: r = 3.5ms1
b: Calculate the momentum, p: mv =
Exercise Class Week 2 Statics and Kinetics
1. A block is resting on an inclined ramp which is at the angle of 30 to the horizontal.
Friction is keeping the block stationary. The block weighs 1kg.
a: Sketch the ramp and block.
FF
FN
mg
b: On your sketch, d
Oscillations and Waves
Question 1
For oscillations of a mass m suspended from a massless spring of strength k, the displacement from
equilibrium can be expressed as
x(t) = Asin(0 t + ),
p
where A and are unkwnown constants and the angular frequency 0 is d
Trigonometrical Equations
General solution
Example1:
Find the general (ALL) solution to the equation 2sin(2 x) + 1 = 0 :
In terms of radians and degrees.
2sin(2 x) + 1 = 0
1
sin(2 x) =
2
1
The first solution, x1 (principle solution) related to sin = is (
Exercise Class Week 3 Dynamics
1. State Newtons First Law of Motion. A body remains at rest or in a state of uniform
motion unless acted upon by an external force.
What principle does it express? The Principle of Relativity
2. State Newtons Second Law of
Exercise Class Week 4 Circular Motion
1. A space station consists of a cylindrical shell of inner diameter 20m. It rotates around
its axis to give artificial gravity of 1g at the floor (the inner surface of the shell).
a: Calculate the speed v of the floo
Objective
ObservethetransfercharacteristicsofanidealOpAmp,andidentifygainandmaximum
voltageswing
ObservethefrequencycharacteristicsofanidealOpAmpanddeterminethe3dB
frequencygainbandwidthproduct
Procedure
The circuit needed in this laboratory session is th
Objective
Use operational amplifier as comparator
Observe duty cycle changes by controlling the operational amplifier with DC source
Procedure
First of all, the power supply was connected to the operational amplifier by using two resistors with 10
k to fo
Question 5
For this question LTSpice simulation was used.
A. y=5*x1+5*x2
(1)
The circuit function will be of the type
V out =5 V 1+5 V 2
(2)
In order to implement this function the summing and inverting amplifier will be used in the
circuit. The gain of t
Question 6
For this question LTSpice simulation was used.
A. y=x
(1)
The circuit function will be of the type
V out =
dVin
dt
(2)
In order to implement this function the differentiator amplifier will be used in the circuit. The
output voltage of the diffe
Part 1: Filter responses and pole/zero pattern
Exercise 1.1: I/O equation
(a) Transfer function
H (z )
:
y ( n )=1.5 y ( n1 ) 0.9 y ( n2 ) + x ( n )+ 0.7 x ( n1 )+ 0.6 x ( n2 )
Y ( Z )=
1.5 Y ( z) 0.9 Y ( z)
0.7 X ( z) 0.6 X (z)
+ X ( z )+
+
2
z
z
z
z2
H
Part 1: Spatial Domain Methods
Exercise 1.1: Pixel operations
(a) Luminance conversion
image_Lena_id114 =
imread('Lena.bmp'); %uploading
the original image in RGB
format
figure;
subplot(121) % fistly we show
the input image which is
subjected to luminance
Part 2: Signal generation, sampling and quantization
Exercise 2.1: Basic digital signals
% Dirac
N = 10;
% number of
samples
n = -N/2:N/2;
% vector
d = [zeros(1,N/2) 1
zeros(1,N/2)];
figure;
% display
stem(n,d);
xlabel('n');
ylabel('\delta(n)');
title('Di
Part 1: Block processing method
Exercise 1.1: Filtering a random signal by direct convolution
% Laboratory 2
% Exercise 1.1
x = 5*rand(1,50)-2; %
here we create 50 random
numbers in range (-2,3)
h = [1/8 1/4 1/4 1/4
1/8]; % entering filter
impulse respons
Part 1: Window Method
Exercise 1.1: Spectral analysis
(a) Magnitude spectrum
for
H( f )
L 50
f1_id114=50;
f2_id114=60;
f3_id114=80;
Fs_id114=1000;
Ts_id114=1/Fs_id114;
L_id114=50;
n_id114=0:(L_id114-1);
for count_id114=0:length(n_id114)-1
%here we will de