ECE_105_FNL_Fall_2014_final.pdf
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University of Waterloo Final Examination Fall 2014
Course Number:
ECE 105
Course Title:
Physics of Electrical Engineering I
Professors:
M. Balogh, H. Hosseinzedeh and G. Scholz
First Name: _______________________
Lecture Section __________
Last Name: _______________________
ID #: ________________________ ____________________________ ____
Signature ___________________________
Date of Exam:
Dec 10, 2014
Time Period:
12:30 pm– 3:00 pm
Duration of Exam:
2.5 hours
Number of Exam Pages: (Including this cover sheet)
13
Aids Allowed:
Calculator, formula sheet provided
Exam Type
Closed Book
Exam is printed single sided on white paper.
X
Select this box if second side of paper is to be used for rough work calculations.
1. Write your answers in the spaces provided provi ded whenever applicable. a pplicable. 2. Make sure your calculator is in either radians or degrees, as required. If you use a programmable calculator, make sure that the memory is cleared. 3. Ask a proctor to clarify a question if you think it is necessary. If the student proctor cannot answer your question satisfactorily ask him / her to call the Professor. 4. Questions in Part C are worth 12 marks each. Answer only 5 of the 6 questions. If you answer more than five, only the first five will be marked. Write full solutions directly in the exam booklet, and put your final answer in the boxes provided. Use the back of the opposite page if necessary; indicate clearly if you have done so. These will be hand-marked, with partial credit given for correct procedure. MARKING SCHEME #A1 #A2
Part A (1 each)
Part B (3 each)
Part C (12 each)
#A3
#A4
#A5
#A6
#A7
#A8
#A9
#A10
#B1
#B2
#B3
#B4
#B5
#B6
#C1
#C2
#C3
#C4
#C5
#C6
#B7
#B8
#B9
ECE 105 Final Fall 2014
Page 2 of 13
Part A: Answer all questions. Each question is worth 1 mark.
1. An object is held in place by friction on an inclined surface. The angle of inclination is increased until the object starts moving. If the surface is ke pt at this angle, the object a. slows down. b. moves at uniform speed. c. speeds up. d. impossible to tell
2. Consider a person standing in an elevator that is accelerating upward. The magnitude of the normal force N exerted by the elevator floor on the person is: a. larger than b. identical to c. smaller than the magnitude of the force of gravity on the person.
3. What force is responsible for changing a car’s direction as it rounds a frictionless banked curve? a. The car’s weight b. The vertical component of the normal force c. The horizontal component of the normal force d. The force of friction
4. A parallel translation of an object’s axis of rotation always changes the moment of inertia.
4) True
or
False
5. If an object is rotating there must be nonzero net torque acting on it at that instant.
5) True
or
False
6. A larger net force on an object always produces a larger change in its momentum compared with a smaller net force.
6) True
or
False
ECE 105 Final Fall 2014
Page 3 of 13
7. What is the direction of the velocity of a particle at point P on the following wave, which is traveling to the left? a. Left v y b. Right c. Up d. Down P e. None of the above x
8. A wheel is rolling without slipping, and the velocity of its centre of mass is v. What is the velocity of a bug that sits on the face of the wheel, directly above the centre and halfway between the centre and top of the wheel? a. v v b. 2v c. 1.5v d. 0.5v e. None of the above
9. Rank the following objects in order of decreasing magnitude of net torque. All forces (indicated with arrows) have the same magnitude, an d all rods are uniform, with the same length. In each case, X marks the pivot point. a. c > d > a = b b. b > d > a > c c. d > a > c > b d. d > c = a > b e. None of the above
A
B
C
D
10. Rank the following objects, which all have the same mass, in order of decreasing moment of inertia. The first three objects (two hoops and a uniform disk) have the same radius, R, and the fourth object (thin rod) has length 2R. In each case the X marks the pivot point. a. b > d > a > c b. b > a = c > d c. b = d > c > a d. d > b > a > c e. None of the above
A
B
C
D
ECE 105 Final Fall 2014
Page 4 of 13
Part B: Answer all nine questions. Each question is worth 3 marks.
1. A 200 g block hangs from a spring with spring constant of 10 N/m. At time t=0s the block is 20cm below the equilibrium point and moving upward with a speed of 100 cm/s. What is the block’s oscillation frequency?
f=
Hz
2
2. Hail rains onto a flat 5.0 m roof. Half the stones bounce off and the other half remain 2 on the roof. If the roof can maximally withstand 500 N/m of force, at what rate can it hail onto the roof (hailstones/s) before the roof collapses? Each hailstone weighs 25 g and has a terminal speed of 25 m/s.
R=
3. A horizontal bar of non-uniform density and length L=2m is supported at both ends with a string, as shown. If the angles are ! !" and ! !" , find the location of the bar’s center of mass (com), relative to the length L. !
!
!
!
x/L =
s
-1
ECE 105 Final Fall 2014
Page 5 of 13
4. A car rounds a curve with radius R=70m, that is banked at 15 degrees. The coefficient of static friction between the car wheels and the road is 0.6. Find the maximum speed at which the car can round the corner without slipping.
vmax =
m/s
5. A transverse pulse is propagating on a 2.0 m long string of mass 50 g. The pulse is described by the equation:! ! ! , with x and y in meters, and t in seconds. !
!
!
!!
!! !
!
!!"
Find the tension in the string.
T=
N
6. Two gliders are set on a horizontal air track. A spring of spring constant k=200 N/m is attached to the back end of the second glider. As shown in the figure, the first glider, of mass m1=1.5 kg, moves to the right with speed v1=2 m/s, and the second glider, of mass m2=5kg, moves more slowly to the right with speed v2=0.5 m/s. When m1 collides with the spring attached to m2, the spring compresses by a distance xmax, and then the gliders move apart again. Find the speed v at the moment of maximum compression (before gliders start moving apart), and the amount of maximum compression, xmax.
v=
m/s
xmax =
m
ECE 105 Final Fall 2014
Page 6 of 13
7. A uniform sphere of mass M=2kg and radius r=30cm rolls perfectly down a plane with friction (µs=0.6, µk =0.2) inclined at an angle !=30 degrees with respect to the horizontal. If it starts from rest, find the magnitude of the velocity of a point at the top of the sphere, directly above the centre of mass, by the time the wheel has dropped a vertical distance of 2m.
v=
8. Two blocks lie on a frictionless surface. The top block has a mass of 0.5kg, and the lower block has a mass of 3kg. You apply a force of F=5N to the top block as shown. If the coefficient of friction between the two blocks is µs=0.6 and µk =0.2, find the magnitude of the acceleration of the bottom block.
m/s
F
a=
m/s
2
9. A continuous harmonic wave travels to the left, with speed v=3m/s, along an infinite string. A point on the string experiences an acceleration that varies with time, ranging in 2 magnitude from 0 to 25 m/s . The vertical distance between the points of maximum upward and maximum downward acceleration is 5cm. Find the wavelength of the wave.
" =
m
ECE 105 Final Fall 2014
Page 7 of 13
Part C: Answer five (5) of the following questions. If you answer more than five, please indicate which ones you want marked. Otherwise, only the first five will be marked.
1. A 0.1 kg mass is on a frictionless horizontal surface and has springs attached on both ends with constants k 1 and k 2 and undergoes simple harmonic motion (SHM). The two constants are related by k 1 = k 2 /2. When the mass is 4 cm to the right of its equilibrium position, its speed is 90 cm/s to the right, and the angular frequency of the oscillations, #, is 30 s-1. a. (1 marks) What is the period of the oscillations?
k 1
T =
k 2
s
b. (2 marks) When the system is in equilibrium, how is the amount by which spring 1 is compressed or stretched related to that of spring 2?
$x1
=
$x2
c. (4 marks) What is the amplitude of the oscillations ?
A =
cm
d. (3 marks) What is the speed of the block, 3s later?
v =
m/s
k 1 =
N/m
k 2 =
N/m
e. (2 marks) What are the values of k 1 and k 2 ?
ECE 105 Final Fall 2014
Page 8 of 13
2. A 10kg pulley of constant thickness has a radius of R=30cm. The mass distribution is uniform, and it rotates about a frictionless pivot through the centre. Blocks of masses m=5kg and M=8kg are suspended from the pulley as shown. The string attached to mass M is wound around an axis that is a distance r=20cm from the pivot.
R r
m
M
a. (1 mark) Find the moment of inertia of the pulley.
2
I =
kgm
b. (6 marks) Find the acceleration of each block, and the angular acceleration of the pulley.
am =
m/s
2
aM =
m/s
2
%
rad/s
=
2
c. (3 marks) If the system starts from rest, how long does it take for the pulley to make one full rotation?
t =
s
d. (2 marks) What is the displacement of each mass after that time? (Choose up to be positive displacement)
ym =
m
yM =
m
ECE 105 Final Fall 2014
Page 9 of 13
3. A ruler of mass m = 100 g and length L = 30 cm is fixed to a table via a frictionless pivot, and is released from rest in its vertical position. At the instant just before the ruler is exactly horizontal find: a. (3 marks) The angular velocity !
#
=
rad/s
b. (3 marks) The angular acceleration "
%
=
rad/s
2
c. (3 marks) The tangential acceleration of the centre of mass.
a =
2
m/s
d. (3 marks) The magnitude of the total acceleration of the centre of mass
a =
m/s
2
ECE 105 Final Fall 2014
Page 10 of 13
4. A pulley of mass M=10kg has an outer radius 0.5m, and a central hole of radius 0.1m. Massless struts (not shown) allow the pulley to rotate about a frictionless pivot at its centre. Two masses (m1=2kg and m2=5kg) are hung from the pulley as shown, and are attached to two springs with spring constants k 1=250 N/m and k 2=350 N/m.
a. (3 marks) Find the moment of inertia of the pulley M2 M1
k 2
k 1
I =
kg m
2
b. (2 marks) When the system is in equilibrium, the spring k 1 is unstretched. At equilibrium, where is m2 located, relative to the unstretched position of k 2? In other words, by how much is k 2 stretched or compressed?
x =
m
c. (2 marks) The system is now displaced from equilibrium. When m1 is a distance y1 above its equilibrium position, how far from equilibrium is m2?
y2 = d. (5 marks) Find the frequency of oscillations. Hint: Write down an expression for the total energy of the system in part c, in terms of the variable y. Differentiate with respect to time.
f =
Hz
ECE 105 Final Fall 2014
Page 11 of 13
5. A model rocket with a mass of 7kg is launched from the ground, at an angle of 30 degrees from the vertical. You know that its engine provides a constant thrust of 2 50 N as it burns for a total of 4.5s. By neglecting air resistance, calculate: a. (2 marks) The net impulse on the rocket in first 4.5s
I =
Ns
b. (4 marks) Maximum height the rocket can reach?
hmax =
m
c. (3 marks) The maximum horizontal distance traveled by the rocket by the time it hits the ground.
hmax =
m
d. (3 marks) The total time for which the rocket is in the air
t =
s
ECE 105 Final Fall 2014
Page 12 of 13
6. A lineman (mass m = 75 kg) sits on top of a vertical telephone pole (10.0 m tall and mass of 175 kg) that is starting to tip because it hasn’t been anchored in the ground properly. To minimize the speed at which he’ll hit the ground, he decides to hold unto the pole until his tangential acceleration equals the acceleration of gravity, g, and then let go. a. (4 marks) Find the height above the ground at which the lineman has to jump from the top of the telephone pole ?
h =
m
b. (2 marks) What will his speed be when he lets go of the pole ?
v =
m/s
c. (3 marks) What is the speed with which he hits the ground if he lets go of the pole?
v =
m/s
d. (3 marks) With what speed would he have hit the ground had he not let go of the pole?
v =
m/s
ECE 105 Final Fall 2014
f r i c t oi n :
k s
f f
= ≤
µ µ k s F F N N
H o ok e’ s L a w :
T h i r d :
F
s
S e c on d :
F~ a~
=
A
−
o n B
k ∆ s
=
−
= m 1 ~ n e t
F
F~ = B m 1 o n d d t P
N e w t on ’ s L a w s :
v A
D
=
Page 13 of 13
R el a t i v e V el o v s v ci f 2 f f s t y s : = = =
v A B
+
v B C
+
v C
D
u n i of r m l i n e ar a c c el er a v s v t i si 2 i si + + + on 2 v a a si a ∆ s∆ = s ∆ t s + t c on 2 1 s t a : s ( ∆
c m
I
+
M d 2
= 9 . 8 0 m s −
g
2
t )
2
A
P ar al el l ax i I s s t h ph e or e er em = I : 5 m2 =
M e ch an i c s
a b o u t an ax i s t I h h r o o o u p = gh m t r2 r eh 2 g e I I om t d h i i e s n k t r = cr o i d a = 2 1 l m c 1 r2 en 1 2 t m er l 2 :
m om en t of i I enr = t i m a:
r2 or
I b
o d y
= P
i
T or q u e:
Σ
~ τ e x α ~ τ ~ t
= ==
~ d L~ e τn~ r× / t / d t I F~
F u n d am en t al C on s t an t s
I m p u l s e:
×
θ
=
= =
ω
θ
ω
i 2 i
i + + + 2 i ω α α ∆ θ
p~
∆
+ 2 1 α ( ∆
t t
t ) 2
2 w T A S u t r a p i h c v e o r v el s p ( i = l n ∆ o s i λ g φ t i / w / 2 on T a ) of = v s e i n t λ : y “ w k o f x w = = A a v a p g v s T i − e n s / ( ω t µ k t ar x + v ∓ el ( ω φ i n t ◦ g + ) a i φ v n g ◦ ” t h ) e s am e d i r e c t oi n :
ω
s i m p el h ar m on ci o s ci l l a t or ,
f k
k x , U
( = ==
a v x
− −A
A A c
o s s i c n o = s ( ω A ( ω t t + c + φ o s φ ◦ ( ω ◦ ) t ) = + ω ω 2 φ
−
φ
= 1 /
f
T
= 2 π
◦ = ) x 2 λ π
ω 2
ω
f
F
=
−
=
2 1
k x
2
W a v e s an d S o u n d
k i n em a t i c s of u n i of r m an g u al r a c c el er a t oi n :
2 1 2 1
2
+
b x
F o r m u l a S h e e t , E C E 1 0 5
√ +
= 4 0 a c :
R o t a t oi n al m o t oi n :
r
a =
t
a
α r
v
ω θ
== == =
α d ω d s r v2 r ω r θ / r / / d d t t
E n er g K K U U y: r s g = =
a x
M a t h em a t i c s
a2 b 2 c
θ
ω
∆
±
b
q u a d r a t ci e q u a t oi n ,
−
c o s
f 2 f
m i r i 2
S 2 A S t u an s p er d i n i λ n ( p o m g k x s i w ) t i c o = a o v s on e s : ( ω f m 2 t t L A ) w ( , o x w ) w h = a er v 2 e e s a m s t i ar n = k v x el 1 i ,2 n g , 3 i n .. o . p ( fi p x o e s i d t e en d d i r s e ) c t oi n s :
2 a b
α α
m v~
∆
−
s c i n o s
=
t
−
+
b 2
A A BB
=
=
x
a2
==
F~ ∆
=
~
B~ · B~
p~
c o s i n e l a w :
c2
A~ A
L i n e ar m om en t u m :
J ~ =
v e c t or s :
=
2 1
=
k m m ( I ∆ g ω v 2 2 s y ) 2
( ( ( r ( t o r s p g r a t a r a n i v n i t s l g t oi a n i p a t al t on o oi t k a en an i l t l n k ai p e t l o n ) i ei t c en ) t i c t ) ai l )
W or k :
d W = = = −
F~ F ∆ · s d U r∆ s ~ ( ( i f i f
F~ F~
i s i s c c on on s er s t an v a t t i ) v e )
1
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