# SOLID MECHANICS DYNAMICS TUTORIAL CENTRIPETAL FORCE

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 SOLID MECHANICS DYNAMICS TUTORIAL CENTRIPETAL FORCE This work coers elements of the syllabus for the Engineering Council Exam D5 Dynamics of Mechanical Systems C10 Engineering Science. This tutorial examines the relationship between inertia and acceleration. On completion of this tutorial you should be able to Explain and define centripetal and centrifugal acceleration. Explain and define centripetal and centrifugal force. Sole problems inoling centripetal and centrifugal force. Derie formulae for the stress and strain induced in rotating bodies. Sole problems inoling stress and strain in rotating bodies. Analyse problems inoling ehicles skidding and oerturning on bends. It is assumed that the student is already familiar with the following concepts. Newton s laws of Motion. Coulomb s laws of friction. Stiffness of a spring. The laws relating angular displacement, elocity and acceleration. The laws relating angular and linear motion. Basic ector theory. Basic stress and strain relationships. All the aboe may be found in the pre-requisite tutorials.

2 1. CENTRIPETAL AFFECTS 1.1 ACCELERATION AND FORCE Centripetal acceleration occurs with all rotating bodies. Consider a point P rotating about a centre O with constant angular elocity ω (fig. 1). Figure 1 The radius is the length of the line O-P. The tangential elocity of P is = ωr. This elocity is constant in magnitude but is continually changing direction. Let s remind ourseles of the definition of acceleration. change in elocity a = time taken a = t Velocity is a ector quantity and a change in direction alone is sufficient to produce a change. It follows that a point traelling in a circle is continuously changing its direction, elocity and hence has acceleration. Next let s remind ourseles of Newton s second Law of Motion which in its simplest form states Force = Mass x acceleration. It follows that anything with mass traelling in a circle must require a force to produce the acceleration just described. The force required to make a body trael in a circular path is called CENTRIPETAL FORCE and it always pulls towards the centre of rotation. You can easily demonstrate this for yourself by whirling a small mass around on a piece of string. Let s remind ourseles of Newton s Third Law. Eery force has an equal and opposite reaction. The opposite and equal force of centripetal is the CENTRIFUGAL FORCE.

3 The string is in tension and this means it pulls in both directions. The force pulling the ball towards the middle is the centripetal force and the force pulling on your finger is the centrifugal force. The deriation of the formula for centripetal force and acceleration is done by considering the elocity as a ector. Figure Consider the elocity ector before and after point P has reoled a small angle δθ. The magnitude of 1 and are equal so let s denote it simply as. The direction changes oer a small period of time δt by δθ radians. We may deduce the change by using the ector addition rule. Figure The first ector + the change = Final ector. The rule is 1 + δ =. This is illustrated below. Figure 4 δ is almost the length of an arc of radius. If the angle is small, this becomes truer. The length of an arc is radius x angle so it follows that δ = δθ This change takes place in a corresponding small time δt so the rate of change of δ δ θ elocity is = δt δt In the limit as δt dt, δ/δt δt δ dt, δt d d dθ dθ is the acceleration. a = = = ω ω = = rate of change of angle dt dt dt dt d dt

4 Since = ωr then substitute for and a = ωr and this is the centripetal acceleration. Since ω = /R then substitute for ω and a = /R Centripetal acceleration = ωr Centripetal acceleration = /R If we examine the ector diagram, we see that as δθ becomes smaller and smaller, so the direction of δ becomes radial and inwards. The acceleration is in the direction of the change in elocity and so centripetal acceleration is radial and inwards. If point P has a mass M, then the force required to accelerate this mass radial inwards is found from Newton's nd Law. Centripetal force = M ω R or in terms of elocity Centripetal force = M /R Centrifugal force is the reaction force and acts radial outwards. WORKED EXAMPLE No.1 Calculate the centripetal acceleration and force acting on an aeroplane of mass 1500 kg turning on a circle 400 m radius at a elocity of 00 m/s. SOLUTION Centripetal acceleration = /R = 00/400 = 5 m/s. Centripetal force = mass x acceleration = 1500 x 5 = 7.5 kn WORKED EXAMPLE No. Calculate the centripetal force acting on a small mass of 0.5 kg rotating at 1500 re/minute on a radius of 00 mm. SOLUTION ω= πn/60 = x π x 1500/60 = 157 rad/s Cent. acc. = ωr = (157) x 0. = 795 m/s. Cent. force = Mass x acc. = 0.5 x 795 = 697 N

6 SELF ASSESSMENT EXERCISE No.1 1. A centrifugal clutch is similar to that shown in figure 5. The clutch must transmit a torque of 5 Nm. The coefficient of friction µ between the drum and the friction lining is 0.4. Each sliding head has a mass of 0.4 kg acting at a radius of 0.15 m. The inside radius of the drum is 0.18 m. Calculate the minimum speed required. (Answer 5.78 rad/s or 51.7 re/min). A rotating arm has a sliding mass of kg that normally rests in the position shown with the spring uncompressed. The mass is flung outwards as it reoles. Calculate the stiffness of the spring such that the mass compresses it by 0 mm when the arm reoles at 500 re/min. (Answer 8.4 N/mm) Figure 6

7 STRESS and STRAIN in ROTATING BODIES.1 STRESS Wheels experience stresses in them because the centripetal reaction (centrifugal force) tends to stretch the material along a radius. This can cause the wheel to disintegrate if it runs to fast. The effect might be catastrophic in a grinding wheel or flywheel. Een if the wheel does not disintegrate, the stress will cause it to expand. The blades on a turbine or compressor rotor will stretch slightly under this strain and might touch the casing. Consider a bar of uniform cross section A rotating about its centre as shown. Figure 7 The radius to the tip is R. Consider a small length δr at radius r. The mass of this element is δm = ρaδr. The centrifugal force produced by this mass is δf. δf = δm ωr = ρaδrωr In the limit as δr dr this becomes df = ρadrωr The centrifugal force acting on this section due to the mass between r and R is then found by integration. R [ R r ] F = ρaω r dr = ρaω r The tensile stress induced at this section (at radius r) is the force per unit area. F [ R r ] σ = = ρaω A A [ R r ] σ = ρω

8 . STRAIN Consider the same element (figure 7). Suppose the element δr increases in length by δx. In the limit as δr dr, the strain in this short length is then ε = dx dr ρω = σ ρω = [ R r ] E E [ R r ] ε = E The extension of the short length is ρω dx = σ = [ R r ] dr E For the length from r to R, the extension is found by integrating again. ρω x = E ρω x = E ρω x = E R [ R r ] r R R ρω dr = E R R r R r r R r + r r R r WORKED EXAMPLE No. 4 A bar 0.5 m long with a uniform section is reoled about its centre. The density of the material is 7 80 kg/m. The tensile stress in the material must not exceed 600 MPa. Calculate the speed of rotation that produces this stress. Go on to calculate the extension of the bar. The elastic modulus is 00 GPa. SOLUTION [ R r ] σ = 600x10 = ρω The maximum stress will be at the middle with r = 0 and R = 0.5 m σ = 600 x 10 = 780ω ω =.45 x 10 ω = 1566 Conerting to re/min N = ω x 60/π = re/min 6 [ ] rad/s The extension of one half of the bar from r = 0 to R = 0.5 m is ρω R r 780 x 1566 x 0.5 x = R r 0 0 E + 9 = x 00 x 10 + x = or 0.5mm

9 SELF ASSESSMENT EXERCISE No. 1a. The blade of a turbine rotor has a uniform cross section. The root of the blade is attached to a hub of radius r. The tip of the blade has a radius R. Show that the stress at the root of the blade due to centrifugal force is gien by σ = ρω [ R r ] 1b. The hub radius is 00 mm and the tip radius is 0 mm. The density of the blade material is 8 00 kg/m. Calculate the stress at the root when the rotor turns at re/minute. (157.8 MPa) 1c. Derie the formula for the extension of the blade tip x gien below. ρω x = E R R r r + where E is the elastic modulus. 1d. Calculate the extension of the tip gien that E = 06 GPa. ( mm)

10 . APPLICATION TO VEHICLES.1 HORIZONTAL SURFACE Figure 8 When a ehicle traels around a bend, it is subject to centrifugal force. This force always acts in a radial direction. If the bend is in a horizontal plane then the force always acts horizontally through the centre of graity. The weight of the ehicle is a force that always acts ertically down through the centre of graity. The diagram shows these two forces. The centrifugal force tends to make the ehicle slide outwards. This is opposed by friction on the wheels. If the wheels were about to slide, the friction force would be µw where µ is the coefficient of friction between the wheel and the road. Figure 9 The tendency would be for the ehicle to oerturn. If we consider the turning moments inoled, we may sole the elocity which makes it oerturn. If the ehicle mass is M its weight is Mg. The radius of the bend is R. Figure 10 Consider the turning moments about point O The moment of force due to the weight is W x d or M g d The moment due to the centrifugal force is C.F. x h Remember the formula for centrifugal force is C.F. = (M/R) The moment due to the centrifugal force is C.F. x h = (M/R) x h When the ehicle is about to oerturn the moments are equal and opposite. Equating the moments about point O we get Rearrange to make the subject. = (g d R/h)½ M = (M/R) x h = Mg d

11 If the ehicle was about to slide sideways without oerturning then the centrifugal force would be equal and opposite to the friction force. In this case µw = M/R µmg = M/R = (µgr) ½ Comparing the two equations it is apparent that it skids if µ<d/h and oerturn if µ>d/h WORKED EXAMPLE No.5 A wheeled ehicle traels around a circular track of radius 50 m. The wheels are m apart (sideways) and the centre of graity is 0.8 m aboe the ground. The coefficient of friction is 0.4. Determine whether it oerturns or slides sideways and determine the elocity at which it occurs SOLUTION Oerturning = (g d R/h)½ = (9.81 x 1 x 50/0.8)½ = 4.76 m/s Sliding sideways = (µgr)½ = (0.4 x 9.81 x 50)½ = 14 m/s It follows that it will slide sideways when the elocity reaches 14 m/s.

12 . BANKED SURFACE The road surface is flat and inclined at θ degrees to the horizontal but the bend is still in the horizontal plane. The analysis of the problem is helped if we consider the ehicle simply as block on an inclined plane as shown. Figure 11 The C.F. still acts horizontally and the weight ertically. The essential distances required for moments about point O are the ertical and horizontal distances from O and these are h' and d' as shown. When the ehicle is just on the point of oerturning the moments about the corner are equal and opposite as before. h' = (h d tanθ) cosθ d' = (h tanθ + d) cosθ Figure 1 Equating moments about the corner we hae By applying trigonometry to the problem you should be able to show that M h' /R = Mg d' h' = (h d tanθ) cosθ /R = g (h tanθ + d) cosθ d + tanθ Make the subject. = gr h d 1 tanθ h This is the elocity at which the ehicle oerturns. If the ehicle slides without oerturning, then the total force parallel to the road surface must be equal to the friction force. Remember the friction force is µn where N is the total force acting normal to the road surface. Resole all forces parallel and perpendicular to the road. The parallel forces are F cosθ and W sinθ as shown. The normal forces are F sinθ and W cosθ as shown. The friction force opposing sliding is µn Figure 1

13 The forces acting parallel to the surface must be equal and opposite when sliding is about to occur. Balancing all three forces we hae µn + Wsin θ = F cosθ The total normal force N is the sum of the two normal forces. N = Wcosθ + F sinθ Substituting µ Wcosθ + Fsinθ + Wsinθ = Fcosθ { } µwcosθ + µfsinθ + Wsinθ = Fcosθ µmgcosθ + µm sinθ + Mgsinθ = M cosθ R R µgcosθ + µ sinθ + gsinθ = cosθ R R µ + µ tanθ + tanθ = Rg Rg Rg Rg { µtanθ 1} { 1 µtanθ} = Rg = tanθ + µ { µ + tanθ} { 1 µtanθ} = tanθ µ This gies the elocity at which the ehicle slides.

14 WORKED EXAMPLE No.6 A motor ehicle traels around a banked circular track of radius 80 m. The track is banked at 15o to the horizontal. The coefficient of friction between the wheels and the road is 0.5. The wheel base is.4 m wide and the centre of graity is 0.5 m from the surface measured normal to it. Determine the speed at which it oerturns or skids. SOLUTION 1. OVERTURNING d 1. + tanθ + tan( 15) = gr h = 9.81X d 1. 1 tanθ 1 tan( 15) h 0.5 Hence = 76.6 m/s. SKIDDING { µ + tanθ} { 1 µ tanθ} = Rg = 80x9.81x { tan15} { tan15} Hence = 6. m/s The ehicle will skid before it oerturns.

15 SELF ASSESSMENT EXERCISE No. 1. A motor ehicle traels around a banked circular track of radius 110 m. The track is banked at 8o to the horizontal. The coefficient of friction between the wheels and the road is 0.4. The wheel base is m wide and the centre of graity is 0.6 m from the surface measured normal to it. Determine the speed at which it oerturns or skids. ( 66. m and 4.9 m ). A ehicle has a wheel base of.1 m and the centre of graity is 1.1 m from the bottom. Calculate the radius of the smallest bend it can negotiate at 10 km/h. The coefficient of friction between the wheels and the track is 0.. (For skidding R = 77.5 m and for oerturning R = m hence the answer is 77.5 m). Repeat question gien that the bend is banked at 10o. (For skidding R = 5. m and for oerturning R = 8. m hence the answer is 5. m)

### SOLID MECHANICS DYNAMICS TUTORIAL MOMENT OF INERTIA. This work covers elements of the following syllabi.

SOLID MECHANICS DYNAMICS TUTOIAL MOMENT OF INETIA This work covers elements of the following syllabi. Parts of the Engineering Council Graduate Diploma Exam D5 Dynamics of Mechanical Systems Parts of the

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.

Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.

### Mechanical Principles

Unit 4: Mechanical Principles Unit code: F/60/450 QCF level: 5 Credit value: 5 OUTCOME 3 POWER TRANSMISSION TUTORIAL BELT DRIVES 3 Power Transmission Belt drives: flat and v-section belts; limiting coefficient

### CHAPTER 15 FORCE, MASS AND ACCELERATION

CHAPTER 5 FORCE, MASS AND ACCELERATION EXERCISE 83, Page 9. A car initially at rest accelerates uniformly to a speed of 55 km/h in 4 s. Determine the accelerating force required if the mass of the car

### FLUID MECHANICS. TUTORIAL No.7 FLUID FORCES. When you have completed this tutorial you should be able to. Solve forces due to pressure difference.

FLUID MECHANICS TUTORIAL No.7 FLUID FORCES When you have completed this tutorial you should be able to Solve forces due to pressure difference. Solve problems due to momentum changes. Solve problems involving

### Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is

Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of

### SOLID MECHANICS DYNAMICS TUTORIAL PULLEY DRIVE SYSTEMS. This work covers elements of the syllabus for the Edexcel module HNC/D Mechanical Principles.

SOLID MECHANICS DYNAMICS TUTORIAL PULLEY DRIVE SYSTEMS This work covers elements of the syllabus for the Edexcel module HNC/D Mechanical Principles. On completion of this tutorial you should be able to

### SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES

SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 4. On completion of this tutorial

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

### Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

### 3 Work, Power and Energy

3 Work, Power and Energy At the end of this section you should be able to: a. describe potential energy as energy due to position and derive potential energy as mgh b. describe kinetic energy as energy

### PHY121 #8 Midterm I 3.06.2013

PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

### Angular acceleration α

Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

### Chapter 4. Forces and Newton s Laws of Motion. continued

Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting

### ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION

ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION This tutorial covers pre-requisite material and should be skipped if you are

### PHY231 Section 2, Form A March 22, 2012. 1. Which one of the following statements concerning kinetic energy is true?

1. Which one of the following statements concerning kinetic energy is true? A) Kinetic energy can be measured in watts. B) Kinetic energy is always equal to the potential energy. C) Kinetic energy is always

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects

### Physics 201 Homework 8

Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

### PHY231 Section 1, Form B March 22, 2012

1. A car enters a horizontal, curved roadbed of radius 50 m. The coefficient of static friction between the tires and the roadbed is 0.20. What is the maximum speed with which the car can safely negotiate

### VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

### ENGINEERING COUNCIL CERTIFICATE LEVEL ENGINEERING SCIENCE C103 TUTORIAL 3 - TORSION

ENGINEEING COUNCI CETIFICATE EVE ENGINEEING SCIENCE C10 TUTOIA - TOSION You should judge your progress by completing the self assessment exercises. These may be sent for marking or you may request copies

### PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

### L-9 Conservation of Energy, Friction and Circular Motion. Kinetic energy. conservation of energy. Potential energy. Up and down the track

L-9 Conseration of Energy, Friction and Circular Motion Kinetic energy, potential energy and conseration of energy What is friction and what determines how big it is? Friction is what keeps our cars moing

### Classical Physics I. PHY131 Lecture 7 Friction Forces and Newton s Laws. Lecture 7 1

Classical Phsics I PHY131 Lecture 7 Friction Forces and Newton s Laws Lecture 7 1 Newton s Laws: 1 & 2: F Net = ma Recap LHS: All the forces acting ON the object of mass m RHS: the resulting acceleration,

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 1 STRESSES IN BEAMS DUE TO BENDING. On completion of this tutorial you should be able to do the following.

MECHANICS OF SOLIDS - BEAMS TUTOIAL 1 STESSES IN BEAMS DUE TO BENDING This is the first tutorial on bending of beams designed for anyone wishing to study it at a fairly advanced level. You should judge

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS

EDEXCEL NATIONAL CERTIICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQ LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS 1. Be able to determine the effects of loading in static engineering

### Chapter 6 Circular Motion

Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

### Chapter 29: Magnetic Fields

Chapter 29: Magnetic Fields Magnetism has been known as early as 800C when people realized that certain stones could be used to attract bits of iron. Experiments using magnets hae shown the following:

### Problem Set 1. Ans: a = 1.74 m/s 2, t = 4.80 s

Problem Set 1 1.1 A bicyclist starts from rest and after traveling along a straight path a distance of 20 m reaches a speed of 30 km/h. Determine her constant acceleration. How long does it take her to

### MECHANICAL PRINCIPLES HNC/D PRELIMINARY LEVEL TUTORIAL 1 BASIC STUDIES OF STRESS AND STRAIN

MECHANICAL PRINCIPLES HNC/D PRELIMINARY LEVEL TUTORIAL 1 BASIC STUDIES O STRESS AND STRAIN This tutorial is essential for anyone studying the group of tutorials on beams. Essential pre-requisite knowledge

### Center of Gravity. We touched on this briefly in chapter 7! x 2

Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.

### SOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM

SOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM This work covers elements of the syllabus for the Engineering Council Exam D5 Dynamics of Mechanical Systems, C05 Mechanical and

### Mechanical Principles

Unit 4: Mechanical Principles Unit code: F/601/1450 QCF level: 5 Credit value: 15 OUTCOME 4 POWER TRANSMISSION TUTORIAL 2 BALANCING 4. Dynamics of rotating systems Single and multi-link mechanisms: slider

### AP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s

AP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s Answer the multiple choice questions (2 Points Each) on this sheet with capital

### Weight The weight of an object is defined as the gravitational force acting on the object. Unit: Newton (N)

Gravitational Field A gravitational field as a region in which an object experiences a force due to gravitational attraction Gravitational Field Strength The gravitational field strength at a point in

### Chapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6.

Chapter 11 11.7 A solid cylinder of radius 10cm and mass 1kg starts from rest and rolls without slipping a distance of 6m down a house roof that is inclined at 30 degrees (a) What is the angular speed

### Physics 1120: Simple Harmonic Motion Solutions

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured

### Lecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7

Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal

### PHYSICS 111 HOMEWORK SOLUTION #9. April 5, 2013

PHYSICS 111 HOMEWORK SOLUTION #9 April 5, 2013 0.1 A potter s wheel moves uniformly from rest to an angular speed of 0.16 rev/s in 33 s. Find its angular acceleration in radians per second per second.

### PHYS 211 FINAL FALL 2004 Form A

1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each

### 3600 s 1 h. 24 h 1 day. 1 day

Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Solution Derivations for Capa #11

Solution Derivations for Capa #11 1) A horizontal circular platform (M = 128.1 kg, r = 3.11 m) rotates about a frictionless vertical axle. A student (m = 68.3 kg) walks slowly from the rim of the platform

### Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

### Chapter 9 Circular Motion Dynamics

Chapter 9 Circular Motion Dynamics 9. Introduction Newton s Second Law and Circular Motion... 9. Universal Law of Gravitation and the Circular Orbit of the Moon... 9.. Universal Law of Gravitation... 3

### Chapter 3.8 & 6 Solutions

Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

### Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

### ENGINEERING COUNCIL CERTIFICATE LEVEL

ENGINEERING COUNCIL CERTIICATE LEVEL ENGINEERING SCIENCE C103 TUTORIAL - BASIC STUDIES O STRESS AND STRAIN You should judge your progress by completing the self assessment exercises. These may be sent

### Physics 1A Lecture 10C

Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

### Lecture-IV. Contact forces & Newton s laws of motion

Lecture-IV Contact forces & Newton s laws of motion Contact forces: Force arises from interaction between two bodies. By contact forces we mean the forces which are transmitted between bodies by short-range

### COMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN

COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject

### Torque and Rotary Motion

Torque and Rotary Motion Name Partner Introduction Motion in a circle is a straight-forward extension of linear motion. According to the textbook, all you have to do is replace displacement, velocity,

### circular motion & gravitation physics 111N

circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would

### Practice Exam Three Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES OUTCOME 2 ENGINEERING COMPONENTS TUTORIAL 1 STRUCTURAL MEMBERS

ENGINEERING COMPONENTS EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES OUTCOME ENGINEERING COMPONENTS TUTORIAL 1 STRUCTURAL MEMBERS Structural members: struts and ties; direct stress and strain,

### F N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26

Physics 23 Exam 2 Spring 2010 Dr. Alward Page 1 1. A 250-N force is directed horizontally as shown to push a 29-kg box up an inclined plane at a constant speed. Determine the magnitude of the normal force,

### Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6

Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.

### MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES

MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES Simple machines: lifting devices e.g. lever systems, inclined plane, screw jack, pulley blocks, Weston differential

### Acceleration due to Gravity

Acceleration due to Gravity 1 Object To determine the acceleration due to gravity by different methods. 2 Apparatus Balance, ball bearing, clamps, electric timers, meter stick, paper strips, precision

### Simple Harmonic Motion

Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights

### Rotational Inertia Demonstrator

WWW.ARBORSCI.COM Rotational Inertia Demonstrator P3-3545 BACKGROUND: The Rotational Inertia Demonstrator provides an engaging way to investigate many of the principles of angular motion and is intended

### Chapter 6 Work and Energy

Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system

### C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N

Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a

### Slide 10.1. Basic system Models

Slide 10.1 Basic system Models Objectives: Devise Models from basic building blocks of mechanical, electrical, fluid and thermal systems Recognize analogies between mechanical, electrical, fluid and thermal

### D Alembert s principle and applications

Chapter 1 D Alembert s principle and applications 1.1 D Alembert s principle The principle of virtual work states that the sum of the incremental virtual works done by all external forces F i acting in

### ex) What is the component form of the vector shown in the picture above?

Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by

### Bending Stress in Beams

936-73-600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending

4 Impulse and Impact At the end of this section you should be able to: a. define momentum and impulse b. state principles of conseration of linear momentum c. sole problems inoling change and conseration

### The Effects of Wheelbase and Track on Vehicle Dynamics. Automotive vehicles move by delivering rotational forces from the engine to

The Effects of Wheelbase and Track on Vehicle Dynamics Automotive vehicles move by delivering rotational forces from the engine to wheels. The wheels push in the opposite direction of the motion of the

### Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

### PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

### Rotational Motion: Moment of Inertia

Experiment 8 Rotational Motion: Moment of Inertia 8.1 Objectives Familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body

### Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### AP Physics C. Oscillations/SHM Review Packet

AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete

### Lecture 7 Force and Motion. Practice with Free-body Diagrams and Newton s Laws

Lecture 7 Force and Motion Practice with Free-body Diagrams and Newton s Laws oday we ll just work through as many examples as we can utilizing Newton s Laws and free-body diagrams. Example 1: An eleator

### Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.

### Tennessee State University

Tennessee State University Dept. of Physics & Mathematics PHYS 2010 CF SU 2009 Name 30% Time is 2 hours. Cheating will give you an F-grade. Other instructions will be given in the Hall. MULTIPLE CHOICE.

### Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel

Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel Name: Lab Day: 1. A concrete block is pulled 7.0 m across a frictionless surface by means of a rope. The tension in the rope is 40 N; and the

### Hand Held Centripetal Force Kit

Hand Held Centripetal Force Kit PH110152 Experiment Guide Hand Held Centripetal Force Kit INTRODUCTION: This elegantly simple kit provides the necessary tools to discover properties of rotational dynamics.

### Lab 7: Rotational Motion

Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125

### Linear Motion vs. Rotational Motion

Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a

### Physics 41 HW Set 1 Chapter 15

Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,

### Lecture Presentation Chapter 7 Rotational Motion

Lecture Presentation Chapter 7 Rotational Motion Suggested Videos for Chapter 7 Prelecture Videos Describing Rotational Motion Moment of Inertia and Center of Gravity Newton s Second Law for Rotation Class

### Experiment 9. The Pendulum

Experiment 9 The Pendulum 9.1 Objectives Investigate the functional dependence of the period (τ) 1 of a pendulum on its length (L), the mass of its bob (m), and the starting angle (θ 0 ). Use a pendulum

### AP Physics: Rotational Dynamics 2

Name: Assignment Due Date: March 30, 2012 AP Physics: Rotational Dynamics 2 Problem A solid cylinder with mass M, radius R, and rotational inertia 1 2 MR2 rolls without slipping down the inclined plane

### So if ω 0 increases 3-fold, the stopping angle increases 3 2 = 9-fold.

Name: MULTIPLE CHOICE: Questions 1-11 are 5 points each. 1. A safety device brings the blade of a power mower from an angular speed of ω 1 to rest in 1.00 revolution. At the same constant angular acceleration,

### Uniform Circular Motion III. Homework: Assignment (1-35) Read 5.4, Do CONCEPT QUEST #(8), Do PROBS (20, 21) Ch. 5 + AP 1997 #2 (handout)

Double Date: Objective: Uniform Circular Motion II Uniform Circular Motion III Homework: Assignment (1-35) Read 5.4, Do CONCEPT QUEST #(8), Do PROBS (20, 21) Ch. 5 + AP 1997 #2 (handout) AP Physics B

### G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD

### EXPERIMENT: MOMENT OF INERTIA

OBJECTIVES EXPERIMENT: MOMENT OF INERTIA to familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body as mass plays in

### Forces on Large Steam Turbine Blades

Forces on Large Steam Turbine Blades RWE npower Mechanical and Electrical Engineering Power Industry INTRODUCTION RWE npower is a leading integrated UK energy company and is part of the RWE Group, one

### Examples of Scalar and Vector Quantities 1. Candidates should be able to : QUANTITY VECTOR SCALAR

Candidates should be able to : Examples of Scalar and Vector Quantities 1 QUANTITY VECTOR SCALAR Define scalar and vector quantities and give examples. Draw and use a vector triangle to determine the resultant