ES190
THE UNIVERSITY OF WARWICK
First Year Examinations: Summer 2018
DYNAMICS AND THERMODYNAMICS
This test is split into two sections. Section A covers Dynamics. Section B covers Thermodynamics.
Candidates should answer all parts in both Sections. Each Section is worth 50 marks.
Time allowed: 2 hours.
Only calculators that confirm to the list of models approved by the School of Engineering may be
used in this examination. The Engineering Data Book and Thermodynamic Tables will be provided.
Read carefully the instructions on the answer booklets and ensure the details required are written
on every booklet submitted. You must use a separate answer booklet for each Section.
, ES190
_________________________________________________________________________________________
SECTION A: DYNAMICS
_________________________________________________________________________________________
1. Answer all parts. The following equations may be helpful for this Section:
Velocity and acceleration in ℝ𝟐 (non-Cartesian components):
Normal-tangential Radial-transverse
Velocity 𝐯 = 𝑠̇ 𝐞𝒕 𝐯 = 𝑟̇ 𝐞𝒓 + 𝑟𝜃̇ 𝐞𝜽
Acceleration 𝐚 = 𝑣̇ 𝐞𝒕 + (𝑣 2 ⁄𝜌)𝐞𝒏 𝐚 = (𝑟̈ − 𝑟𝜃̇ 2 )𝐞𝒓 + (2𝑟̇ 𝜃̇ + 𝑟𝜃̈)𝐞𝜽
Curvature for 𝒚 = 𝒇(𝒙):
𝑑 2 𝑦⁄𝑑𝑥 2
𝜅=
(1 + (𝑑𝑦⁄𝑑𝑥)2 )3/2
(a) (i) Define the terms ‘rectilinear’ and ‘curvilinear’ motion.
(2 marks)
(ii) A particle moves with an acceleration given by 𝐚(𝑡) = 2𝑡𝑒 −𝛽𝑡 𝐢 + 𝑡 cos 𝑡 𝐣. Find
the velocity of the particle at time 𝑡 if 𝐯(0) = 2𝐣.
(8 marks)
(iii) A mass attached to a spring is travelling at a speed of 𝑣0 m s1 as it passes through
the un-extended position (𝑥 = 0) in the direction of positive displacement (𝑥 > 0).
The acceleration of the particle is given by −𝑘 2 𝑥 m s2. By finding the displacement
and speed for any arbitrary time 𝑡, show that:
𝑣(𝑡)
𝑥(𝑡) = tan(𝑘𝑡)
𝑘
(10 marks)
Question 1 continues overleaf
3
THE UNIVERSITY OF WARWICK
First Year Examinations: Summer 2018
DYNAMICS AND THERMODYNAMICS
This test is split into two sections. Section A covers Dynamics. Section B covers Thermodynamics.
Candidates should answer all parts in both Sections. Each Section is worth 50 marks.
Time allowed: 2 hours.
Only calculators that confirm to the list of models approved by the School of Engineering may be
used in this examination. The Engineering Data Book and Thermodynamic Tables will be provided.
Read carefully the instructions on the answer booklets and ensure the details required are written
on every booklet submitted. You must use a separate answer booklet for each Section.
, ES190
_________________________________________________________________________________________
SECTION A: DYNAMICS
_________________________________________________________________________________________
1. Answer all parts. The following equations may be helpful for this Section:
Velocity and acceleration in ℝ𝟐 (non-Cartesian components):
Normal-tangential Radial-transverse
Velocity 𝐯 = 𝑠̇ 𝐞𝒕 𝐯 = 𝑟̇ 𝐞𝒓 + 𝑟𝜃̇ 𝐞𝜽
Acceleration 𝐚 = 𝑣̇ 𝐞𝒕 + (𝑣 2 ⁄𝜌)𝐞𝒏 𝐚 = (𝑟̈ − 𝑟𝜃̇ 2 )𝐞𝒓 + (2𝑟̇ 𝜃̇ + 𝑟𝜃̈)𝐞𝜽
Curvature for 𝒚 = 𝒇(𝒙):
𝑑 2 𝑦⁄𝑑𝑥 2
𝜅=
(1 + (𝑑𝑦⁄𝑑𝑥)2 )3/2
(a) (i) Define the terms ‘rectilinear’ and ‘curvilinear’ motion.
(2 marks)
(ii) A particle moves with an acceleration given by 𝐚(𝑡) = 2𝑡𝑒 −𝛽𝑡 𝐢 + 𝑡 cos 𝑡 𝐣. Find
the velocity of the particle at time 𝑡 if 𝐯(0) = 2𝐣.
(8 marks)
(iii) A mass attached to a spring is travelling at a speed of 𝑣0 m s1 as it passes through
the un-extended position (𝑥 = 0) in the direction of positive displacement (𝑥 > 0).
The acceleration of the particle is given by −𝑘 2 𝑥 m s2. By finding the displacement
and speed for any arbitrary time 𝑡, show that:
𝑣(𝑡)
𝑥(𝑡) = tan(𝑘𝑡)
𝑘
(10 marks)
Question 1 continues overleaf
3
