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MM1MS1-E1 STATICS AND DYNAMICS
Creation date：2024-06-18 15:46:08
Belonging category：工程力学engineering mechanics
Tag MM1MS1
STATICS AND DYNAMICS

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MM1MS1-E1

STATICS AND DYNAMICS
Time allowed TWO hours
Candidates may complete the front cover of their answer book and sign their desk card but
must NOT write anything else until the start of the examination period is announced.
Answer ALL questions in Section A and ALL questions in Section B.
Please answer Section A and Section B in separate answer booklets.
Only silent, self contained calculators with a Single-Line Display or Dual-Line Display are
permitted in the examination.
Dictionaries are not allowed with one exception. Those whose first language is not English
may use a standard translation dictionary to translate between that language and English
provided that neither language is the subject of this examination. Subject specific translation
dictionaries are not permitted.
No electronic devices capable of storing and retrieving text, including electronic dictionaries,
may be used.
DO NOT turn examination paper over until instructed to do so
ADDITIONAL MATERIAL: Formula Sheet, Graph paper
INFORMATION FOR INVIGILATORS:
Question papers should be collected in at the end of the exam – do not allow candidates to
take copies from the exam room.
Turn over
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SECTION A
Answer ALL questions in this section
1. The structure shown in Figure 1a is a preliminary design for an overhead crane
support truss in which the position B at which the load P is applied is at a
variable distance from support A, given as a.
Figure 1a
(a) Determine the reaction forces as a function of the variables.
[8]
(b) Show that in region AB the bending moment is given by
  1
a
M x xP
L
 
  
 
where x is the distance from A, and determine the bending moment
function as a function of position y measured from C for the region BC.
[10]
(c) The dimensions and loads are now fixed, but the load is now applied to the
crane through a rigid arm, as shown in Figure 1b. Determine the equivalent
forces and moments acting on the beam ABC at point B, and hence the new
support reactions. There is no need to calculate shear forces and bending
moments along the beam.
Figure 1b
[7]
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2. A thin-walled pipe 600 mm in diameter and 4 mm in thickness is used to carry
pressurised gas at a pressure of 2 MPa, leading to a state of plane stress.
(a) Explain the meaning of the term plane stress.
[4]
(b) Show that the principal in-plane stresses are 150 MPa and 75 MPa, and
make a sketch of an element of stress on the pipe to illustrate the directions
of the stresses.
[7]
(c) For this state of plane stress, sketch the Mohr’s circle, and determine the
maximum in-plane shear stress.
[6]
(d) If the maximum in-plane shear stress allowable in the material is 100 MPa,
determine the maximum pressure that can safely be carried by the pipe.
[8]
Turn over
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4
SECTION B
Answer ALL questions in this section
3. A 4kg object is launched vertically up into the air. The launch mechanism applies
constant 1kN vertical force to the object for a period of 0.25 seconds.
An observer watches the launch from a position 30m away from the launch point,
at the same height as the launch point.
(a) Carefully draw and label a free body diagram for the object during the 0.25
second launch period. What is the constant acceleration applied to the mass
during the launch? What is the velocity of the object just after the launch
period?
[5]
(b) Assuming that there is no air resistance, what is the maximum height that
the object reaches?
[5]
(c) Sketch a diagram showing the position of the object in Cartesian coordinates
(x,y) and polar coordinates (r,θ) from the observer’s point of view, i.e.
showing how x,y,r and relate to the geometry shown in Figure 3a What is
the height of the object, and the distance from the observer to the object
when the observer is looking up at an angle of 70°?
[5]
(d) Sketch diagrams of the polar components of the velocity and acceleration of
the object. What are the values for , , andr r   when the observer is looking
up at an angle of 70° and the object is travelling upwards?
[10]
Figure 3a
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4. A geared hoist system for raising and lowering a load is shown in Figure 4a. It
consists of a motor driving a compound gearbox – gear A has 20 teeth, gear B
has 200 teeth and is rigidly fixed to gear C, gear C has 25 teeth, gear D has 200
teeth and is rigidly attached to a hoist pulley of diameter 200 mm. The gearbox
efficiency has been found to be 95%. The hoist pulley is 250mm long and is
made of steel (ρ=7800Kg/m3). Ignore the thickness of the cable and the inertia
of the gearbox itself.
(a) What is the gear ratio from the input (motor) to the output (hoist pulley)?
What is the output torque applied to the hoist pulley if the motor drive
torque is 2Nm?
[5]
(b) The winch is required to lift a 120Kg mass from the ground to a height of 5m.
If a constant torque of 2Nm is applied by the motor, how long will it take to
raise the mass to 5m? What is the speed of the motor (in revolutions per
minute) when the mass is 2cm off the ground?
[8]
(c) If the cable of the hoist system snaps and the 120Kg mass drops from the
rest at a height of 56m, how long will it take reach the floor?
[6]
(d) If the mass does not rebound from the floor, what is the impulse applied to
the mass during impact? What is the average impact force applied to the
mass if the impact lasts 0.1 seconds?
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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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