Sean Daltonwww.itsligo.ie/staff/sdalton

Course FormatCourse FormatLectures/TutorialsLaboratory Work

Course AssessmentCourse AssessmentLaboratory Work (20%)

Assessments (20%)Final Exam (60%)

RequirementsRequirementsMechanical Engineering Science (Hannah+Hillier)

Engineering Science (Hughes and Hughes)Scientific CalculatorScience Notebook

Course ContentCourse Content

- Forces systems: Triangle of Forces, Bowes notation- Moments: Principle of moments, levers etc- Centroid of Area, Centre of gravity- Friction:Horizontal and Inclided surfaces- Stress and strain: Elastic/Plastic, Factor of Safey- Linear Motion: Force and Acceleration- Rotary motion and Torque- Centrifugal Force- Simple Machines

Mechanics is divided into two areas:Statics is the area of mechanics which allows us to work out forces in stationary structures.Dynamics allows us to determine to effect of a force on an object which is free to move.

Steps when answering QuestionsSteps when answering Questions

The following steps should The following steps should be followed when solving be followed when solving mechanics problems mechanics problems - State what is being calculated- Write down the formula - Rearrange the formula ……………..(if necessary)

- Insert Values- Calculate the answer- Add units- Underline the answer

Calculating diameterA = /4 D2

D = 4A = 4x1 = 1.128 m

Calculate the Calculate the diameter of a diameter of a circle of area 1mcircle of area 1m22

UnitsUnits

All Quantities in Mechanics must conform to All Quantities in Mechanics must conform to some system of unitssome system of units- The Imperial system- The Metric system

- International system of Units SI units

Base Units Vs. Derived UnitsBase Units Vs. Derived UnitsThere are 7 base units (5 reqd. in Mechanic)There are 7 base units (5 reqd. in Mechanic)

UnitsUnits

All remaining units are called derived unitsAll remaining units are called derived units

Achieved by combining the base unitsAchieved by combining the base units

Multiples of Base UnitsMultiples of Base Units

Mass Vs. WeightMass Vs. Weight

Mass is defined as the amount of Mass is defined as the amount of matter (material) in a body (and is matter (material) in a body (and is expressed in kilograms)expressed in kilograms)

Weight is the force exerted by Weight is the force exerted by gravity acting on a body (and is gravity acting on a body (and is expressed in Newtons)expressed in Newtons)

- Whether on the earth or the moon the amount of sugar in a bag of sugar is constant. This is a measure of its mass. in Kilograms

- However it will feel heavier on the earth than on the moon. This is a measure of its weight. In Newtons

Pressure/stressPressure/stress

Pressure is a measure Pressure is a measure of the force per unit of the force per unit area (units N/marea (units N/m22))- P = F Force A Area- A man and woman of equal mass (80kg) stand on a floor. If the man is wearing flat shoes and woman is wearing stiletto heals as shown calculate the maximum pressure exerted on the floor in each case.

Area (man) = 302 /2 + 12 x 60 = 1414 + 720 = 1170mm2

Area (woman) = 82 /2 + 3 x 16 = 100 + 48 = 148mm2

Pressure (man) = 784.8/1170 = 0.67 N/mm2 (MN/m2)

Pressure (woman) = 784.8/148 = 5.3 N/mm2 (MN/m2)

Weight = 80x9.81= 784.8N

Triangle/Polygon of ForcesTriangle/Polygon of Forces

A A ForceForce is that which changes or tends to change is that which changes or tends to change a bodies state of rest or of uniform motion in a a bodies state of rest or of uniform motion in a straight line.straight line.

The unit of Force is called the Newton and is the The unit of Force is called the Newton and is the force required to give a mass of 1kg an force required to give a mass of 1kg an acceleration of 1m/sacceleration of 1m/s22..

A quantity which has both magnitude and A quantity which has both magnitude and direction is referred to as a direction is referred to as a VectorVector quantity (Force) quantity (Force)

A quantity which has magnitude only is a A quantity which has magnitude only is a ScalarScalar quantity (e.g. Mass, Time)quantity (e.g. Mass, Time)

Vector representation of forcesVector representation of forces

Since a force has both magnitude and direction it Since a force has both magnitude and direction it can be represented by a vectorcan be represented by a vector

The arrow indicates the direction of the forceThe arrow indicates the direction of the forceThe length of the arrow indicates the magnitude of The length of the arrow indicates the magnitude of

the force (drawn to a suitable scale)the force (drawn to a suitable scale)

Force VectorScale

2 N = 1 cm10 N

20 N

5 cm

10 cm

Resultant/EquilibrantResultant/Equilibrant

When a number of forces act on When a number of forces act on a body the a body the resultantresultant force is that force is that single force which would have single force which would have the same effectthe same effect

- This can be found by representing the forces as vectors and adding them head to tail

The The equilibrantequilibrant is that force is that force which must be applied to a which must be applied to a system to produce equilibrium. system to produce equilibrium. (equal and opposite to resultant)(equal and opposite to resultant)

5 N

10 N

15 N

15 cm

10 cm 5 cm

Resultant

Equilibrant

Scale: 1N = 1cm

ResultantResultant

Find the resultant forces shown opposite

Find the resultant of the forces shown below

Triangle of ForcesTriangle of Forces

If 3 forces are acting on an If 3 forces are acting on an object but it remains stationary object but it remains stationary then it is said to be in then it is said to be in equilibriumequilibrium. (this implies ‘no . (this implies ‘no resultant’)resultant’)

In order for a body to be in In order for a body to be in equilibrium under the action of 3 equilibrium under the action of 3 coplanar forces:coplanar forces:

- the lines of action of the 3 forces must pass through a common point of concurrency.

- the forces when represented by vectors added end to end will form a closed triangle.- The principle of concurrency- The triangle of forces

Point of concurrency

Closed Triangle

ExampleExample

A mass of 10 kg is suspended by two cords from points D and E. Calculate the tension in each chord.

ExampleExample

Determine the Determine the force in the rods force in the rods AB and BC when AB and BC when carrying a load of carrying a load of 8kN. 8kN. = 60 = 60oo

W = 8000 NW = 8000 N

Fbc = 9237.6 NFbc = 9237.6 N

Fab = 4618.8 NFab = 4618.8 N

ExampleExample

The 80kg is The 80kg is supported by 2 supported by 2 rods AB and BC. rods AB and BC. Determine the Determine the force in each rod.force in each rod.

W = 784.8NW = 784.8N

FFABAB = 632.4N = 632.4N

FFBC BC = 395.2N= 395.2N

ExampleExample

A jib crane has a jib 5 m long and a tie rod 3.5 m long attached to a post 2 m vertically above the foot of the jib. Determine the force in the jib and the tie rod when a mass of 3 tonnes is suspended from it.

2m5m

3.5m

3 T 38.7o

56.9o

29.43kN73.6kN

51.5kN

ExampleExample

- Shown below is a jib crane. Determine the force in the tie and the magnitude and direction of the force at the lower wall joint

Resolution of ForcesResolution of Forces

Instead of solving forces graphically they can be Instead of solving forces graphically they can be solved mathematically (using trigonometry). Any single solved mathematically (using trigonometry). Any single force acting at a point can be replaced by a pair of force acting at a point can be replaced by a pair of forces which have the same effect.forces which have the same effect.

Where these two Where these two 'component forces''component forces' are at right are at right angles to each other they can be found easily using angles to each other they can be found easily using trigonometry. When a force is broken down into its trigonometry. When a force is broken down into its component forces the force is said to have been component forces the force is said to have been 'resolved into its components''resolved into its components'..

Fx = F Cos Fx = F Cos Fy = F Sin Fy = F Sin

Resolution of ForcesResolution of Forces

Four co-planer forces act at a point 0, the values and directions of the forces being as shown opposite. Calculate the magnitude and direction of the resultant.

Force Net horizontal force Net vertical force50 50 Cos (0) = 50 50 Sin (0) = 030 30 Cos (120) = -15 30 Sin (120) = 2620 20 Cos (210) = -17.32 20 Sin (210) = -1010 10 Cos (-90) = 0 10 Sin (-90) = -10

17.68 N 6 N

Magnitude of ResultantF = (17.682 + 62) = 18.67 N

Angle of Resultant = Tan-1 (6/17.68)

= Tan-1 (0.339) = 18.74o

Resolution of Forces (Problems)Resolution of Forces (Problems)

For the 5 forces shown determine the magnitude and nature of the resultant.

[36.87@20.7o]

Resolution of ForcesResolution of Forces

Question 2 (Huges P38 Q22)The following forces act at a point.

20 N due north30 N 20 degrees south of east10 N south west16 N 5 degrees south of west

Calculate the magnitude and direction of the resultant. [5.36N at 13.8o north of east]

Question 3 (Hughes P38 Q21)The following horizontal forces act at a point:

50N in a direction due east80N in a direction due southand 30 N in a direction 20o north.

Calculate the direction of the fourth force necessary to maintain equilibrium. [73N at 72.6 north of west]

MomentsMoments The moment of a force about a point is a measure of its

turning effect and is defined as the product of:- 1 The force2 The shortest distance (perpendicular distance) to the line of action of the force

The point about which a body is free to rotate is called the fulcrum.

The leverarm is the shortest distance (perpendicular distance) from the fulcrum to the line of action of the force.

F

F

x

x

Principle of MomentsPrinciple of Moments

When a body is in equilibrium under the action of any number of forces, the sum of the clockwise moments about any point in the body is equal to the sum of the anticlockwise moments about the same point.

Clockwise Moments = Anticlockwise Moments

A lever is a simple machine, which applies the principle of moments to give a mechanical advantage.

MomentsMoments

A lever is pivoted at its mid point C. 5kg A lever is pivoted at its mid point C. 5kg is suspended at E 180mm from C. is suspended at E 180mm from C. Calculate the mass required at D to Calculate the mass required at D to maintain balance.maintain balance.

MomentsMoments

Determine the position of the suspension Determine the position of the suspension point in order for the lever to be balanced.point in order for the lever to be balanced.

MomentsMoments

For the Beam shown determine the For the Beam shown determine the reactions at the supportsreactions at the supports

MomentsMoments

Centroid of AreaCentroid of Area

Determine the location of the centroid of Determine the location of the centroid of area of the shape shown below.area of the shape shown below.- [300mm]- [137.2mm]

FrictionFriction

When an object is placed on a surface the object will exert a normal force(W) on the surface.

The surface prevents the object sinking into it by exerting a normal reaction force(Rn) upwards.

In order to move the object by applying a force P, a tangential force (called friction F acting in the opposite direction) must be overcome.

Laws of FrictionLaws of Friction

If a force P is applied to the book an equal but opposite force If a force P is applied to the book an equal but opposite force called friction is exerted by the table to maintain equilibrium.called friction is exerted by the table to maintain equilibrium.

If the value of P is increased the friction force must increase If the value of P is increased the friction force must increase accordingly.accordingly.

There is a limit beyond which the friction force cannot increase. If P There is a limit beyond which the friction force cannot increase. If P exceeds this value sliding starts. The maximum force is called the exceeds this value sliding starts. The maximum force is called the limiting friction force.limiting friction force.

The value of the limiting friction force is proportional to the normal The value of the limiting friction force is proportional to the normal force Rn and is independent of the area of contact.force Rn and is independent of the area of contact.

= F = limiting friction where: = coeff. of friction

Rn Normal reaction

F = Rn

It will also be noticed that it takes a larger force to initiate It will also be noticed that it takes a larger force to initiate movement than it does to maintain it.movement than it does to maintain it.

st > sl st = coefficient of static frictionsl = coefficient of sliding friction

Angle of ReposeAngle of Repose

The angle of inclination of the The angle of inclination of the surface just prior to slippage surface just prior to slippage is called the angle of repose.is called the angle of repose.

Force up slopeForce up slope = Force down slope= Force down slopeF = W Sin Rn = W Sin W Cos = W Sin = W Sin

W Cos = Tan

Angle of reposeAngle of repose

Box A weighs 100 N and box B weighs and the coefficient of friction between the box and the ramp is 0.3. For what range of weights B will the block neither

- (a) slide up the slope or - (b) slide down the slope.