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Exam 2013, Questions And Answers

Module

Dynamics of Mechanical System (MM1DMS)

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University of Nottingham

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####### MM1DMS-E

The University of Nottingham

####### DEPARTMENT OF MECHANICAL, MATERIALS AND MANUFACTURING ENGINEERING

####### A LEVEL 1 MODULE, SPRING SEMESTER 2013-

####### DYNAMICS OF MECHANICAL SYSTEMS

Time allowed TWO Hours

Candidates may complete the front cover of their answer bookand 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 THREE questions in Section B

Only silent, self-contained calculators with a Single-Line Display, or Dual-Line Display are permitted in this 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

Several of our exam papers require students to answer a specific number of questions from those available, e. Section A answer ALL questions and Section B answer 3 out of 5 questions. If students answer more than the required number of questions only the required number will be marked, so for the example given, only the first 8 solutions in the exam script will be marked.

If during the exam students attempt additional questions they should clearly indicate on their script which solutions they want to be marked - simply putting a line through solutions that should be disregarded is recommended.

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

####### MM1DMS-E

####### 2

####### SECTION A

Answer ALL questions in this section. You should spend about 45 minutes on this section.

1. Figure Q1 shows a particle of mass m 1 kg acted upon by 3 forces ( 5 N, 10 N

and 15 N) in a plane.

Figure Q

(a) Calculate the components of the resultant force acting in the horizontal and vertical directions. [2]

(b) Calculate the magnitude and direction of the particle acceleration. [2]

####### MM1DMS-E

####### 4

Figure Q3 shows a car travelling around a constant radius bend at a constant speed of 30 m/s. It takes 10 seconds for the car to travel around a bend that can be

approximated as a 60 segment of a circle with a radius of 300 m.

Vf= 30 m/s Vi= 30 m/s

60 o

Figure Q

(a) Calculate the change in velocity of the car and the average acceleration over this distance. [3]

(b) Compare the result for the average acceleration to the instantaneous acceleration of the car. [1]

####### MM1DMS-E

####### 5

Figure Q4 shows a simple gear train consisting of 3 gears.

Figure Q

(a) Calculate the overall gear ratio for the gear train. [3]

(b) If the input gear wheel is rotating at 45 rpm, calculate the rotational speed

of the output gear wheel. [1]

Turn over

Input Output

80 teeth 100 teeth

50 teeth

####### MM1DMS-E

####### 7

####### SECTION B

Answer THREE questions only in this section. You should spend about 1 hour 15 minutes on this section.

6. Figure Q6 shows a schematic diagram of a fairground ride in which point mass mis

supported by a light, inextensible cable of length Lthat is attached to a cylindrical hub

at point A. The cylindrical hub has radius a and rotates about a vertical axis at

constant angular velocity. As the hub rotates, the point mass moves with a constant

radius in the horizontal plane. The radius of motion dependson the rotational speed of

the hub which affects inclination angle.

Figure Q

(a) Draw and label a free body diagrams for the point mass, carefully showing all forces and accelerations. [5]

(b) Apply Newton’s 2nd law to the point mass and show that the angular speed

 is related to inclination angle as follows:

( sin )

tan



 

a L

g





####### [9]

Continued on next page Turn over

A

####### MM1DMS-E

####### 8

(c) If m 60 kg, L 5 m, a 2 m, 30 , calculate:

i) the tension in the cable ii) the angular speed of the point mass, ω, in rpm iii) the velocity of the point mass [9]

Figure Q7 shows a pulley system used to control the horizontal position of a load

mass on a rough surface having coefficient of friction . The load mass has mass

m and is attached to two light, inextensible cables either sideof the load mass.

Each cable wraps around a drum having radius r and moment of inertia J, and

each drum is free to rotate about an axis through its centre. A torqueT is applied

to the right hand drum only as shown, causing each drum to have angular

acceleration and the load mass to have horizontal acceleration a. The cables

both remain parallel to the slope and are always taut.

Figure Q

(a) Draw and label free body diagrams for the load mass and each drum, carefully showing all forces and accelerations. [8]

(b) Apply Newton’s 2nd law to the load mass and to each drum to determine the equation of motion for each body in terms of the forces/torques and accelerations/angular accelerations identified in part (a).Neglect any influence of friction in the pulley. [9]

governing the equivalent rotation system of angular acceleration,, is:

Tmgr 2( Jmr 2 )

####### [6]

g + +

####### MM1DMS-E

####### 10

9. Figure Q9 shows a 25 kg object that is launched horizontally from a platform at a

height of h 4 m above the horizontal ground. The launch mechanism applies an

average horizontal force of 200 N for a period of 05 sec. The influence of air

resistance can be neglected.

Figure Q

(a) Calculate the potential energy of the object relative to the ground prior to launch. [2]

(b) Determine the magnitude and direction of the impulse applied by the launcher to the object. [4]

(c) Calculate the kinetic energy of the object immediately after launch and the work done by the launch mechanism on the object during the launch process. [8]

(d) Determine the magnitude and direction of the velocity of theobject just before hitting the ground. [9]

Launcher

####### MM1DMS-E

####### 11

Figure Q10 shows a centrifugal blower driven by a motor. Thecentrifugal blower

has a moment of inertia of 4 kgm 2 and has the following (approximate) load-speed

characteristic in the operating range:

L 100  25 b

whereb is the angular speed of the blower in rad/s.

The blower is driven via a 3:1 reduction gearbox (100% efficiency) by an induction motor having the following (approximate) torque-speed characteristic:

Linput 750  5 m

wherem is the angular speed of the motor in rad/s.

Figure Q

(a) Calculate the moment of inertia of the blower referred to the motor axis. [6]

(b) Determine the load-speed characteristic for the blower referred to the motor axis. [9]

(d) Calculate the mechanical power supplied by the motor when it runs. [3]

End

Motor

3:

reduction

ratio

Blower

####### SECTION A

Answer ALL questions in this section. You should spend about 45 minutes on this section.

1.)

Figure Q1 shows a particle of mass m 1 kg acted upon by 3 forces ( 5 N, 10 N and 15 N) in

a plane.

Figure Q

i) Calculate the components of the resultant force acting in the horizontal and vertical directions. [2] ii) Calculate the magnitude and direction of the particle acceleration. [2]

####### SOLUTION:

i) 

 F N

 x  15  10 cos( 30 ) .6 34 [1 mark]



 F N

 y  10 sin( 30 ) 5  .0 00 [1 mark]

ii) Due to Fy=0 there is no acceleration towards direction y. [1 mark] 



F m F /m .6 34 1/ 63 4. m/sec 2

 x  x x  x   [1 mark]

####### 2.)

Figure Q2 shows a mass that is initially held at rest at height h 2 m. The mass is dropped

vertically under the action of gravity and rebounds upwards after impacting with the floor.

Figure Q

i) Calculate the velocity of the mass just before impact with the floor. [2]

ii) Assuming the coefficient of restitution e 5 , calculate the upwards velocity of the

mass immediately after impact. [2]

####### SOLUTION:

i)The distance sytravelled by the mass under the action of gravity is related to its initial velocity v0,yand time t by the relation:



 .0 495 sec

.9 81

22 * 2.

2

12  ,0     

g

s

s v t gt t

y y y

The velocity of the mass just before hitting the floor vb,ycan be derived by its relation to the initial velocity v0,ythe gravitational acceleration g and the time during which themass is constantly accelerated t: 

 v ,yb v,0ygt 0  .9 81 * .0 495  .4 86 m/sec [2 marks]

ii) The coefficient of restitution is defined as:

Velocityof separation *Velocityofapproach

Velocityofapproach

Velocityof separation

e  e

[1 mark]

Before impact After impact

####### 3.)

Figure Q3 shows a car is travelling around a constant radius bend at a constant speed of 30 m/s. It takes 10 seconds for the car to travel around a bend that can be approximated as a

60 segment of a circle with a radius of 300 m.

Vf= 30 m/s Vi= 30 m/s

60 o

Figure Q

i. Calculate the change in velocity of the car and the average acceleration over this distance. [3] ii. Compare the result for the average acceleration to the instantaneous acceleration of the car. [1]

####### SOLUTION:

i. Calculate the change in velocity of the car and the average acceleration over this distance. [3]

V Vi=30 m/s

f= 30 m/s

Change in velocity, Δv

Δθ = 60o

Using the cosine rule (or based on equilateral triangle):

ݒ߂ଶ=ݒ௜ ଶ+ݒ௙ଶ− 2ݒ௜ ݒ௙ ݏ݋ܿ(60)

####### ݒ߂ଶ= 2ݒ௜ ଶ− 2ݒ௜ ଶ

####### 1

####### 2

####### =ݒ௜ ଶ

####### ݒ߂=ݒ௜ = 30݉/ ݏ

Orrecognise that the triangle can be split into two equivalent RH triangle:

ݒ߂ 2 =ݒ௜ ݊݅ݏ(30) =

####### 30

####### 2

[1 mark] for equation and

[1 mark] for calculating Δv

aavg = Δv/Δt = 30/((10/3)π) = 2 m/s

[1 mark] for aavg

iii. Compare the result for the average acceleration to the instantaneous acceleration of the car. [1]

ainst= v 2 /r = 30 2 /300 = 3 m/s

[1 mark]

####### 5.)

Figure Q5 shows the dimensions of a fly wheel made from steel. The density of steel is 7800 kg m-3. The fly wheel is manufactured from a large disc, of diameter 300 mm and thickness 50 mm, which has a moment of inertia of 0 kg m 2. Two smaller discs of diameter of 275 mm and a thickness of 15 mm are then cut out. Both the disc, and its cut out sections, are centred on the axis of rotation

Figure Q

i) Calculate the moment of inertia for one of the smaller discs that have been removed.

[4]

ii) Using the above results calculate the moment of inertia for the completed fly wheel.

[1]

SOLUTION:

i) Calculate the moment of inertia for one of the smaller discs that have been removed. [3]

####### ܫ௖௨௧௢௨௧ =

####### 1

####### 2

####### ܴܯଶ

####### [1]

####### ܯ =ߩ∗݁݉ݑ݈݋ݒ= 7800 ∗ߨ∗ 0ଶ∗ 0.

####### ܯ= 6݃݇

####### [2]

####### ܫ௖௨௧௢௨௧ =

####### 1

####### 2 ∗ 6 ∗ 0.

####### ଶ= 0݉݃݇ଶ

####### [1]

ii) Calculate the moment of inertia for the completed fly wheel.

####### [1]

####### ܫ௧௢௧௔௟ = ܫ௟௔௥௚௘௪௛௘௘௟ − 2 ܫ௖௨௧௢௨௧ = 0݉݃݇ଶ

50 mm

275 mm

20 mm

300 mm

Removed sections

####### SECTION B

Answer THREE questions only in this section. You should spend about 1 hour 15 minutes on this section.

6.)

Figure Q6 shows a schematic diagram of a fairground ride in whichpoint mass mis supported

by a light, inextensible cable of length Lthat is attached to a cylindrical hub at point A. The

cylindrical hub has radiusa and rotates about a vertical axis at constant angular velocity.

As the hub rotates, the point mass moves with a constant radius in the horizontal plane. The

radius of motion depends on the rotational speed of the hub which affects inclination angle .

Figure Q

i) Draw and label a free body diagrams for the point mass, carefully showing all forces and accelerations. [5]

ii) Apply Newton’s 2nd law to the point mass and show that the angular speed is

related to inclination angle as follows:

( sin )

tan



 

a L

g





####### [9]

iii) Ifm 60 kg, L 5 m, a 2 m, 30 , calculate

a) the tension in the cable, b) the angular speed of the point mass, ω, in rpm, c) the velocity of the point mass [9]

####### A

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Exam 2013, Questions And Answers

Module: Dynamics of Mechanical System (MM1DMS)

5 Documents

Students shared 5 documents in this course

University: University of Nottingham

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MM1DMS-E1

The University of Nottingham

DEPARTMENT OF MECHANICAL, MATERIALS AND MANUFACTURING ENGINEERING

A LEVEL 1 MODULE, SPRING SEMESTER 2013-2014

DYNAMICS OF MECHANICAL SYSTEMS

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 THREE questions in Section B

Only silent, self-contained calculators with a Single-Line Display, or Dual-Line Display are

permitted in this 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

Several of our exam papers require students to answer a specific number of

questions from those available, e.g. Section A answer ALL questions and Section B

answer 3 out of 5 questions. If students answer more than the required number of

questions only the required number will be marked, so for the example given, only

the first 8 solutions in the exam script will be marked.

If during the exam students attempt additional questions they should clearly

indicate on their script which solutions they want to be marked - simply putting a

line through solutions that should be disregarded is recommended.

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