0001.pdf
0220025501
Exercises
1
2
3
4
5
6
7
8
9
Surname, First name
Applied Natural
(3NBB0)
Final exam Q3
Sciences
a
b
c
d
e
f
→b
a
b
c
d
e
f
→c
b
c
d
e
f
→a
Formal
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
4
4
4
4
4
4
4
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
7
8
8
8
8
8
8
8
9
9
9
9
9
9
9
0
0
0
0
0
0
0
Fill in your answer(s) to the multiple-choice questions as shown above (circles = one correct answer).
Particular Ans on paper exam instructions
• Write in a black or blue pen.
• You answer open-ended questions by using the text box. Provide your answers on the papers inside
the answer box underneath a question. If you need more space for your answers, use the extra
space at the end of the exam, and clearly indicate there which question you continue answering.
In the text box of the particular question, clearly state that you proceed with your answer on a
different page.
• Hand in all pages. Do not remove the staple. If you remove it anyhow, check that you hand in all
pages.
Dear student,
You’re about to take an exam. Write down your name and your student ID at the appropriate places above.
Make sure that you enter your student ID by fully coloring the appropriate boxes. On the examination
attendance card, you fill in the PDF number. You can find the correct number on the top of the first page
of your exam (e.g. 1234.pdf).
Please read the following information carefully:
Date exam: April 11th, 2023
Start time 18.00
End time: 21.00 (+30 minutes for time extension students)
Number of questions: 6
Maximum number of points/distribution of points over questions: Q1: 26p, Q2: 8p, Q3: 8p, Q4: 4p, Q5: 6p;
Q6: 6p; tot 58p
Method of determining the final grade: 10*points/58 rounded to 0.1
1 / 28
0001.pdf
0220025502
Answering style: Q1: multiple choice; Q2-Q5: reasoning/calculation.
Problem 6 is the major-dependent problem. There are 3 versions. Answer the version that corresponds to
your major. The other two versions should remain blank.
Permitted examination aids
• Scrap paper (fully blank, provided)
• Non-graphical calculator
• Equation sheet (provided)
Important:
• You are only permitted to visit the toilets under supervision
• Examination scripts (fully completed examination paper, stating name, student number, etc.) must
always be handed in
• The house rules must be observed during the examination
• The instructions of subject experts and invigilators must be followed
• Keep your work place as clean as possible: put pencil case and breadbox away, limit snacks and
drinks
• You are not permitted to share examination aids or lend them to each other
• Do not communicate with any other person by any means
During written examinations, the following actions will in any case be deemed to constitute fraud or
attempted fraud:
• using another person’s proof of identity/campus card (student identity card)
• having a mobile telephone or any other type of media-carrying device on your desk or in your clothes
• using, or attempting to use, unauthorized resources and aids, such as the internet, a mobile telephone,
smartwatch, smart glasses etc.
• having any paper at hand other than that provided by TU/e, unless stated otherwise
• copying (in any form)
• visiting the toilet (or going outside) without permission or supervision
First-year bachelor students: The final grade for this exam will be announced no later than fifteen
working days after the date of this exam, unless this exam takes place in Q4 or the interim period. For Q4
final exams, grades will be announced within five working days after the end of the Q4 final test period. For
interim period final exams, grades will be announced no later than five working days before September 1.
All other students: Generally, the final grade for this exam will be announced no later than fifteen working
days after the date of this examination. Specifically for bachelor exams administered in the interim period,
exam grades will be announced no later than five working days before September 1.
You can start the exam now, good luck!
2 / 28
0001.pdf
0220025503
Problem 1 - Multiple Choice questions
2p
The escape speed for an object at the surface of the earth is the minimum (vertical) speed the
object should get so that it will not fall back to the earth. The escape speed v is given by
√
8πGρ
v=
R
3
1a
with ρ the mass density of the earth (mass per unit volume), R the radius of the earth, and G the
gravitational constant.
The unit of G is
a
b
c
d
e
f
g
h
2p
2p
kg ⋅ s3 /m2
kg ⋅ s2 /m3/2
m3 /(kg ⋅ s2 )
kg ⋅ s2 /m3
m2 /(kg ⋅ s3 )
m3/2 /(kg ⋅ s2 )
kg ⋅ s3 /m2/3
m3/2 /(kg ⋅ s3 )
A racing car starts from rest and reaches a final speed v in a time t. If the acceleration of the car is
constant during this time, which of the following statements is true?
1b
a
The car travels a distance 21 vt.
b
The car travels a distance vt.
c
The velocity of the car remains constant.
d
The acceleration of the car is 21 vt2 .
e
The acceleration of the car is vt.
An unstretched, massless spring has length L. An object with mass m is suspended by the spring,
stretching it over a distance ∆L.
Next, the spring with the mass is brought into vertical oscillation. The frequency of oscillation is
measured to be f .
How can the acceleration of gravitation be calculated?
1c
a
b
c
d
e
g = (2πf )2 L
g = (2πf )2 ∆L
g = f 2 ∆L
g = f 2L
None of the other answers is correct.
3 / 28
0001.pdf
2p
0220025504
1d
Block A is sandwiched between plank B and surface S that makes an angle α with the
horizontal. Plank B is connected to the wall with a massless rope.
Between block A and plank B and also between block A and surface S there is friction.
Block A is just at rest, so the static friction forces are maximum and they are just large enough to
prevent block A from sliding.
In the diagrams, WA and WB are the weights of A and B or their reaction forces, NA and
NB are the normal forces on A and B or their reaction forces, FsA and FsB are the friction forces or
their reaction forces, and T is the tension force.
Which of the diagrams is the free body diagram of the block A?
a
1
b
2
c
3
d
4
4 / 28
e
5
f
6
g
7
0001.pdf
2p
0220025505
A position-dependent force F⃗ (x, y) = αxy 2 ı̂ (with α a constant) acts on an object with mass m. The
object is released from rest at position (x0 , y1 ) (with y1 a nonzero constant). When the object is at
position (x1 , y1 ) its speed is v1 . What was the original position x0 where the object was released
from rest?
1e
a
√
mv 2
x0 = x21 − αy21
1
b
c
mv 2
x0 = αy21
√1 2
mv1
x0 =
− x21
αy 2
1
d
e
f
g
h
2p
x0 = 0
x0 = x1 +
√ m v1
α y
√ v11
x0 = x1 − m
α y1
√
mv 2
x0 = x21 + αy21
1
√ v1
−
x
x0 = m
1
α y1
1f
A ball falls straight down onto a block that is wedge-shaped and is sitting on frictionless
ice. The block is initially at rest. Assume the collision is perfectly elastic. Is the total momentum of
the block/ball system conserved and is the total kinetic energy of the total block/ball system exactly
the same immediately before and immediately after the collision?
a
The total momentum of the ball/block system is conserved but the total energy of the ball/block
system is not conserved.
b
The total momentum of the ball/block system is conserved and the total energy of the ball/block
system is conserved.
c
The total momentum of the ball/block system is not conserved and the total energy of the
ball/block system is not conserved.
d
The total momentum of the ball/block system is not conserved but the total energy of the
ball/block system is conserved.
5 / 28
0001.pdf
2p
0220025506
The standing wave on a string that is fixed at both ends is described by the equation
1g
y(x, t) = ASW sin(kx) sin(ωt).
The number of antinodes of the standing wave is n.
What is the length L of the string?
a
b
c
d
e
f
g
h
2p
π
L = n kω
L = n kω
2π
L = n 2π
k
L = n πk
k
L = n 2π
2π
L = n kω
L = n πk
L = n kω
π
1h
The two weights (made from different materials) in the picture have equal mass and are hanging
from a weight balance, in equilibrium, in a glass bell that is pumped vacuum.
If air is let into the bell, what will happen?
a
The sphere will go down.
b
Nothing.
c
The sphere will go up.
d
You cannot tell because it depends on the materials the weights are made of.
6 / 28
0001.pdf
2p
2p
Which of the following assumptions is NOT made in the derivation of Bernoulli’s equation?
1i
a
There is negligible gravity
b
The fluid is non-compressible
c
There is negligible viscosity
d
There is no turbulence
e
There is negligible friction
A container with negligible heat capacity contains an amount of water with mass mw initially at
temperature Tw . You add an amount of ice with mass mi at temperature Ti with Ti < Tw . The melting
temperature of ice is T0 , the heat of fusion is Lf , the heat of vaporization is Lv , the specific heat of
water is cw , and the specific heat of ice is ci .
If the final temperature (after equilibrium has been established) is T0 , what was the minimum mass
of the ice and what was the maximum mass of the ice that has been added?
1j
a
b
c
d
e
f
g
h
2p
0220025507
(Tw −T0 )cw +Lf
(Tw −T0 )cw
(T0 −Ti )ci mw ≤ mi ≤ (T0 −Ti )ci +Lv mw
(Tw −T0 )cw +Lv
(Tw −T0 )cw
mw
(T0 −Ti )ci +Lv mw ≤ mi ≤
(T0 −Ti )ci
(Tw −T0 )cw
(Tw −T0 )cw +Lf
(T0 −Ti )ci mw ≤ mi ≤ (T0 −Ti )ci +Lf mw
(Tw −T0 )cw
(Tw −T0 )cw +Lf
(T0 −Ti )ci +Lv mw ≤ mi ≤
(T0 −Ti )ci mw
(Tw −T0 )cw +Lv
(Tw −T0 )cw
mw ≤ mi ≤ (T0 −T
mw
(T0 −Ti )ci
i )ci +Lv
(Tw −T0 )cw
(Tw −T0 )cw +Lv
mw ≤ mi ≤ (T0 −Ti )ci +Lf mw
(T0 −Ti )ci
(Tw −T0 )cw
(Tw −T0 )cw +Lf
(T0 −Ti )ci +Lf mw ≤ mi ≤
(T0 −Ti )ci mw
(Tw −T0 )cw
(Tw −T0 )cw +Lv
mw
(T0 −Ti )ci +Lf mw ≤ mi ≤
(T0 −Ti )ci
1k
The lens in the figure above is made of glass with an index of refraction n.
What type of lens is it?
a
positive lens
b
no lens
c
negative lens
d
a negative lens for small n and a positive lens for large n.
e
a negative lens for large n and a positive lens for small n.
7 / 28
0001.pdf
2p
0220025508
1l
A police car is driving with speed vp towards a wall and uses a siren with frequency fS .
The sound waves are reflected by the wall and after reflection the police car observes a frequency
fL .
There is no wind and the speed of sound is v.
The observed frequency fL is given by
a
b
c
d
e
f
g
2p
v
fL = v−v
fS
p
v−vp
v fS
v+vp
fL = v fS
v
fL = v+v
fS
p
v−vp
fL = v+vp fS
v+v
fL = v−vpp fS
fL =
fL = fS
1m
Fully unpolarized light with intensity I0 is incident (from the left) on a polarizer with a polarization
axis that makes an angle φ with the vertical direction. The intensity of the polarized light (behind the
polarizer) is I1 . The light then passes an analyzer with a vertical polarization axis. After passing the
analyzer, the intensity of the light is I2 (measured by a photocell).
What are the relations between I0 , I1 , and I2 ?
a
b
c
d
e
I1 = I0 cos2 φ and I2 = I0 cos2 φ sin2 φ
I1 = I0 sin2 φ and I2 = I0 cos2 φ sin2 φ
I1 = 12 I0 and I2 = 12 I0 cos2 φ
I1 = I0 and I2 = I0 cos2 φ
I1 = I0 cos2 φ and I2 = I0 cos4 φ
8 / 28
0001.pdf
0220025509
Problem 2
A small object with mass m is placed on a conical surface that rotates about its vertical axis (see
figure above). The distance of the object to the vertical axis of rotation is R; the surface makes an angle
φ with the horizontal.
The coefficient of static friction between the object and the surface is µs . Initially the object does not slip
(at zero or low angular frequency ω of the cone).
The angular frequency ω is increased very gradually until the object starts to slip.
The acceleration of gravity is g.
1p
2a Draw a free body diagram of the object when it is not slipping.
9 / 28
0001.pdf
0220025510
2p
2b Reason why the normal force that the surface exerts on the object (before slipping) is NOT equal to
mg cos φ (when φ ≠ 0).
2p
2c Show via calculation that the static friction force Fs and the normal force N on the object before
slipping, can be calculated from a set of two equations of form
⎧
⎪
⎪ ±Fs cos φ ± N sin φ = Ψ
⎨
⎪
⎪
⎩ ±Fs sin φ ± N cos φ = Φ
↶
Determine the + and − signs in these equations and give expressions for Ψ and Φ.
Hint: What is the direction of the net force acting on m? Choose the x-axis along this direction.
10 / 28
0001.pdf
0220025511
↷
2d Calculate expressions for the friction force Fs and the normal force N acting on the object when it
is not (yet) sliding. Do not express Fs in N or vice versa.
Hint: solve the set of equations from question (c) by multiplying one of the equations with cos φ and
the other one with sin φ and then add or substract the two new equations.
Note that the static friction force might not yet be at its maximum.
↶
2p
11 / 28
0001.pdf
0220025512
↷
1p
2e Calculate the maximum angular frequency ωmax for which the object will not slip.
12 / 28
0001.pdf
0220025513
Problem 3
2p
3a Write down the equations for conservation of energy and of conservation of momentum during the
collision.
3b Express x2 as a function of v0 , v1 , m1 , m2 , and k but NOT of v2 .
↶
2p
Mass m2 is at rest on a frictionless surface. Connected to m2 is a massless spring with force constant k.
Mass m1 is moving with initial speed v0 and collides head-on with the spring.
The collision of m1 and the m2 /spring-system is fully elastic.
You will calculate the maximum compression xmax of the spring during collision.
At time t during the collision, the compression of the spring is x(t) which we denote as x, the speed of
m1 is v1 (t) which we denote as v1 , and the speed of m2 is v2 (t) which we denote as v2 .
13 / 28
0001.pdf
0220025514
↷
3c Show that x2 has its maximum value when
v1 =
Hint: x2 is a function of v1 . v0 is a constant.
m1
v0
m1 + m2
↶
2p
14 / 28
0001.pdf
0220025515
↷
2p
3d Show that the maximum compression xmax is given by
√
1 m1 m2
xmax = v0
k m1 + m2
15 / 28
0001.pdf
0220025516
Problem 4
A beam of light is incident in air (nair = 1) at an angle θa on the upper surface of a transparent plate,
the surfaces of the plate being plane and parallel to each other. The beam is partially reflected by the
upper surface and partially refracted. The refracted beam is partially reflected by the bottom surface and
refracted a second time at the upper surface. The plate has thickness t and refractive index n.
The distance between the two emergent beams that leave the plate at the top surface is d (see figure).
4
Calculate the distance d between the two emergent beams. Express your answer in the given
parameters (θa , t, and n).
Do not use θb and inverse trigonometric functions in your answer.
↶
4p
16 / 28
0001.pdf
0220025517
↷
17 / 28
0001.pdf
0220025518
Problem 5
A container filled with boiling water is connected to a second container that is filled with a mixture of ice
and water via a uniform rod with length L and cross-sectional area A. The thermal conductivity of the rod
material is k.
The boiling water is kept at constant temperature Tb via a heater (not drawn in the figure). The temperature
of the ice/water mixture is T0 . The rod and the container are isolated so that no heat is lost to the
surroundings.
The initial amount of water in the second container is mw ; the initial amount of ice is mi .
The specific heat of water is cw ; the heat of fusion of water is Lf .
5a Calculate the time tm that is needed to melt all the ice. Express tm in the given parameters.
↶
3p
18 / 28
0001.pdf
0220025519
↷
3p
5b Give an expression for the time dependent temperature T (t) of the second container after all the ice
has melted. Take t = 0 the moment that all the ice has melted. Express T (t) in the given parameters.
dQ dT
Hint: Use that dQ
dt = dT dt and use separation of variables.
19 / 28
0001.pdf
0220025520
Problem 6v1 for BAU/BBT/BCS/BDS/BEI/MWT/PT/BPT/BTB
6a Give an expression of the total energy Etot of the total system. Express Etot in the given parameters
and in the speed v(t) of the masses.
↶
2p
Blocks A and B are suspended by a cord that passes over a frictionless pulley C. The masses are
mA , mB , and mC respectively with mA > mB . The pulley can be regarded as a solid cylinder with radius R.
Between the cord and the surface of the pulley there is no slipping. The acceleration of gravity is g.
Let yA (t) and yB (t) the time dependent positions of A and B respectively.
20 / 28
0001.pdf
0220025521
↷
1p
6b What is the relation between the speed v(t) of the masses at time t and the positions yA (t) and
yB (t) of the masses?
21 / 28
0001.pdf
3p
0220025522
6c Take the time derivative dEtot /dt of the total energy and from this, calculate the acceleration a of
blocks A and B.
Hint: when energy is conserved, it is time-independent, so dEtot /dt = 0 and dv 2 /dt = 2v dv/dt.
22 / 28
0001.pdf
0220025523
Problem 6v2 for BST/BTN/BTW
An object (not particularly a sphere) with mass m and radius R rolls down a ramp from initial height y0
without initial velocity. The static friction is large enough to avoid slipping. The moment of inertia of the
object is I. The ramp makes an angle φ with the horizontal (see figure). The acceleration of gravity is g.
You will calculate the time at which the object reaches the end of the ramp (so height 0).
1p
7a
Give an expression of the total energy Etot of the object when it is at height y (with 0 ≤ y ≤ y0 ).
Express Etot in m, v, I, and y with v the speed at that instant.
23 / 28
0001.pdf
0220025524
3p
7b
Use that energy is conserved to calculate the total acceleration a.
Hint: when energy is conserved, it is time-independent, so dEtot /dt = 0 and dv 2 /dt = 2v dv/dt.
2p
7c
Calculate the time at which the object arrives at the bottom of the ramp.
24 / 28
0001.pdf
0220025525
Problem 6v3 for BW/BSI
8a Calculate the readings IA1 of Ammeter A1 and IA2 of Ammeter A2 .
Express IA1 and IA2 in R1 , R2 , and ε.
↶
5p
Consider the electrical circuit in the figure above.
The emf source (indicated as ε in the figure) is an ideal emf source (with zero internal resistance). In the
circuit R1 and R2 are known resistances.
A1 and A2 are ideal Ammeters that has negligible resistance and measures the current that flows from +
to −. V is an ideal Voltmeter that has an infinitely large resistance.
25 / 28
0001.pdf
0220025526
↶
↷
26 / 28
0001.pdf
0220025527
↷
1p
8b Calculate the reading V of the Voltmeter V.
Express IV in R1 , R2 , and ε.
27 / 28
0001.pdf
0220025528
Additional space
9
This page provides extra space in case you ran out of space answering earlier questions. Clearly
indicate the number of the problem you are solving or continuing.
28 / 28