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```SURNAME: ..........................................................
STUDENT NUMBER:
INITIALS: ...........................................
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UNIVERSITY OF CAPE TOWN
DEPARTMENT OF PHYSICS
PHY1012F
Physics A for Engineers
FINAL EXAMINATION
Thursday 06 June 2019
Time: 120 min + 10 min reading time
Internal Examiner:
External Examiner:
Total marks: 72
Full marks: 70
Dr S.M. Wheaton
Dr G. Bosman
INSTRUCTIONS:
1. Complete your details at the top of the page.
2. Answer all questions in the spaces provided on the question paper neatly and legibly.
3. Use blank sheets on question paper for all calculations and/or rough work.
4. A list of constants and formulae are available on the last two (2) pages.
5. Prescribed calculators are allowed.
7. Hand in your question paper at the end of the test.
2
Question 1
a. The number 8 ball in a game of pool moves on a horizontal frictionless table. The graphs
below indicate 𝑣𝑥 and 𝑣𝑦 components of the ball’s velocity. The ball started from the origin.
i.
In which direction (angle) is the ball moving at t = 2 s? (Give your answer as an angle
from the x-axis.)

ii.
How far from the origin is the ball at t = 4 s?

PHY1012F Exam 2019-drg
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Question 1…continued
b. The number 8 ball of mass 𝑚 = 0.250 kg now strikes the cushion of the pool table at an
angle of 𝜃1 = 60.0° at a speed of 𝑣1 = 27.0 m s−1 . It bounces off at an angle of
𝜃2 = 71.0° and a speed of 𝑣2 = 10.0 m s−1.
i.
What is the magnitude of the change in momentum of the ball?

ii.
In which direction, with respect to the positive 𝑥-axis does the change in momentum
vector point?

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Question 2
a. Give an example of a particle that has constant kinetic energy but is accelerating. Can a nonaccelerating particle (constant mass) have a changing kinetic energy? If so, give an example.

b. Calculate the horizontal force 𝐹⃑ required such that 𝑚1 = 9.00 kg does not slide up or down
along the wedge 𝑚2 = 150 kg. All surfaces are frictionless and 𝜃 = 20°.

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Question 2…continued
c. A pendulum of length 𝐿 = 1.0 m with bob of mass 𝑚 = 1.0 kg is released from rest at an
angle of 𝜃 = 30° from the vertical. When the pendulum reaches the vertical position, the
bob strikes a mass 𝑀 = 3.0 kg that is resting on a frictionless table that has a height
ℎ = 0.85 m.
i.
Calculate the velocity 𝑣⃑ of the bob and the box just after they collide elastically.

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Question 2…continued
ii.
Determine how far away from the bottom edge of the table, ∆𝑥, the box will strike
the floor.

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Question 3
a. Your bicycle tires have a radius of 0.33 m. It takes you 14.2 minutes to ride 14 times around
a circular track of radius 73 m at a constant speed.
i.
What is the angular velocity (in rad/s) of the bicycle around the track?

ii.
What is the angular velocity (in rad/s) of a tire around its axis?

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Question 3…continued
b. A solid sphere of radius 𝑅 and mass 𝑀 is placed at a height ℎ0 on an inclined plane of slope
𝜃. When released, it rolls without slipping to the bottom of the incline. Next a cylinder of
the same mass and radius is released on the same incline. If both the cylinder and the
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sphere reach the same speed at the bottom of the incline, show that ℎ = ℎ0 .

14
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Question 4
a. An experiment measures the temperature of a 250 g substance while steadily applying heat
to it. The figure below shows the result of the experiment.
i.
Determine the specific heat capacity (c) of the solid phase.

ii.
Determine the specific heat capacity (c) of the liquid phase.

iii.
Determine the heat of fusion (Lf).

iv.
Determine the heat of vapourisation (Lv).

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Question 4…continued
b. A 200 g piece of copper at temperature 450 K and a 100 g piece of aluminium at
temperature 200 K are dropped into an insulated bucket containing water at 280 K. If the
thermal equilibrium temperature of the mixture is 282.6 K, what was the mass of the water
in the bucket?

𝑐𝐴𝑙 = 900 J/kg K
𝑐𝐶𝑢 = 385 J/kg K
𝑐𝐻2 𝑂 = 4190 J/kg K
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Question 4…continued
c. A monatomic gas follows the process 1 → 2 → 3 shown below.
[1 atm = 101325 Pa]
i.
How much heat is needed for process 1 → 2?

ii.
How much heat is needed for process 2 → 3?

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Question 5
a. What is the order of magnitude of the horizontal force exerted on a passenger car travelling
at the speed limit (120 km/h) on a freeway when a large truck passes close by going in the
opposite direction, also at the freeway speed limit?

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Question 5…continued
b. A 5.00 kg air-filled, sealed, rigid float tank that has a volume of 1.00 m3 is pulled 50.4 m
down to the seafloor in order to assist in lifting a sunken object. A diver standing on the
seafloor cranks a winch to pull the tank down.
i.
How much work is required to pull the tank down? (You may safely ignore the mass
in the air tank and the mass of the winch rope.)

ii.
What is the average power required if it takes 10 min to winch down the tank? 
PHY1012F Exam 2019-drg
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Physical constants and useful formulae
Speed of light
Acceleration due to gravity (Earth)
Gas constant
Density of air
Stefan-Boltzmann constant

𝟏
𝒙(𝒕) = 𝒙𝒐 + 𝒗𝒐𝒙 𝒕 + 𝒂𝒙 𝒕𝟐
𝟐
𝟏
𝒚(𝒕) = 𝒚𝒐 + 𝒗𝒐𝒚 𝒕 + 𝒂𝒚 𝒕𝟐
𝟐
⃑⃑𝒏𝒆𝒕 = 𝑚𝒂
⃑⃑
𝑭
𝝎=
𝒅𝜽
𝒅𝒕
𝒗𝒕 = 𝝎𝒓
3.00 × 108 𝑚/𝑠
9.80 𝑚/𝑠 2
6.022 × 1023 /𝑚𝑜𝑙
8.314 𝐽/𝑚𝑜𝑙 𝐾
1.225 𝑘𝑔/𝑚3
5.67 × 10−8 𝑊/𝑚2 𝐾 4
c
g
NA
R
𝝆𝒂𝒊𝒓
𝒗𝒙 (𝒕) = 𝒗𝒐𝒙 + 𝒂𝒙 𝒕
𝒗𝒚 (𝒕) = 𝒗𝒐𝒚 + 𝒂𝒚 𝒕
𝒇𝒔 (𝒎𝒂𝒙) = 𝝁𝒔 𝑵
𝜶=
𝒅𝝎
𝒅𝒕
𝒔 = 𝒓𝜽
𝒗𝟐
𝒂𝒓 =
= 𝝎𝟐 𝒓
𝒓
𝒇𝒌 = 𝝁𝒌 𝑵
𝑻=
𝟐𝝅𝒓 𝟐𝝅
=
𝒗
𝝎
𝒂𝒕 = 𝜶𝒓
𝟏
𝜽𝒇 = 𝜽𝒊 + 𝝎𝒊 ∆𝒕 + 𝜶(∆𝒕)𝟐
𝟐
𝝎𝒇 = 𝝎𝒊 + 𝜶∆𝒕
𝝎𝟐𝒇 = 𝝎𝟐𝒊 + 𝟐𝜶∆𝜽
PHY1012F Exam 2019-drg
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⃑⃑ =
𝑭
⃑⃑𝒇 = 𝒑
⃑⃑𝒊
𝒑
⃑⃑
𝒅𝒑
𝒅𝒕
𝒕𝒇
⃑⃑ = 𝑱⃑
∆𝒑
⃑⃑ (𝒕)𝒅𝒕
𝑱⃑ = ∫ 𝑭
𝒕𝒊
𝒔𝒇
⃑⃑𝒔 𝒅𝒔
𝑾=∫ 𝑭
⃑⃑ ∙ ∆𝒓
⃑⃑ = 𝑭∆𝒓 𝒄𝒐𝒔 𝜽
𝑾=𝑭
𝑷=
𝒔𝒊
𝑼𝒔𝒑 =
𝟏
𝒌(∆𝒔)𝟐
𝟐
𝑭𝒔 = −
∆𝑬𝒔𝒚𝒔 = ∆𝑲 + ∆𝑼 + ∆𝑬𝒕𝒉 = 𝑾𝒆𝒙𝒕
𝑾
∆𝒕
𝒅𝑼
𝒅𝒔
𝑾𝑵𝑪 = ∆𝑲 + ∆𝑼
𝑲𝒊 + 𝑼𝒊 + 𝑾𝒆𝒙𝒕 = 𝑲𝒇 + 𝑼𝒇 + ∆𝑬𝒕𝒉
⃑𝝉⃑ = ⃑𝒓⃑ × ⃑𝑭⃑
𝝉 = 𝒓𝑭 𝒔𝒊𝒏 𝝓 = 𝒓𝑭𝒕 = 𝒅𝑭
𝑰 = ∑ 𝒎𝒊 𝒓𝟐𝒊 = ∫ 𝒓𝟐 𝒅𝒎
𝝉𝒏𝒆𝒕 = 𝑰𝜶
𝒊
⃑⃑ = 𝒓
⃑⃑ × 𝒑
⃑⃑
𝑳
⃑⃑ = 𝑰𝝎
⃑⃑⃑⃑
𝑳
⃑⃑𝒏𝒆𝒕 =
𝝉
𝑬(𝒑𝒖𝒓𝒆 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏) = 𝑲𝒓𝒐𝒕 + 𝑼𝑮 =
𝑬(𝒓𝒐𝒍𝒍𝒊𝒏𝒈) = 𝑲𝒓𝒐𝒕 + 𝑲𝒄𝒎 + 𝑼𝑮 =
𝒑=
𝑭
𝑨
⃑⃑
𝒅𝑳
𝒅𝒕
𝟏 𝟐
𝑰𝝎 + 𝑴𝒈𝒚𝒄𝒎
𝟐
𝟏 𝟐 𝟏
𝑰𝝎 + 𝑴𝒗𝟐𝒄𝒎 + 𝑴𝒈𝒚𝒄𝒎
𝟐
𝟐
𝒑 = 𝒑𝟎 + 𝝆𝒈𝒉
𝑭𝑩 = 𝝆𝒇 𝑽𝒇 𝒈
𝒗𝟏 𝑨𝟏 = 𝒗𝟐 𝑨𝟐 = 𝑸
𝟏
𝟏
𝒑𝟏 + 𝝆𝒗𝟐𝟏 + 𝝆𝒈𝒚𝟏 = 𝒑𝟐 + 𝝆𝒗𝟐𝟐 + 𝝆𝒈𝒚𝟐 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝟐
𝟐
𝑸 = 𝑴𝒄∆𝑻
𝑸 = 𝒏𝑪𝑽 ∆𝑻
𝑸 = ±𝑴𝑳𝒇
𝑸 = 𝒏𝑪𝑷 ∆𝑻
𝒅𝑸 𝒌𝑨
=
∆𝑻
𝒅𝒕
𝑳
𝑸 = ±𝑴𝑳𝒗
𝑾 = −𝒏𝑹𝑻𝒍𝒏(𝑽𝒇 ⁄𝑽𝒊 )
𝒅𝑸
= 𝒆𝝈𝑨𝑻𝟒
𝒅𝒕
PHY1012F Exam 2019-drg
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