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Sean Dalton Course Format Lectures/Tutorials Laboratory Work Course Assessment Laboratory Work (20%) Assessments (20%) Final Exam (60%)Requirements Mechanical Engineering Science (Hannah+Hillier) Engineering Science (Hughes and Hughes) Scientific Calculator Science Notebook

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Course 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.

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Steps when answering Questions The following steps should be followed when solving mechanics problems The following steps should be followed when solving 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 diameter A = / 4 D 2 D = 4A = 4x1 = m Calculate the diameter of a circle of area 1m 2 Calculate the diameter of a circle of area 1m 2

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Units All Quantities in Mechanics must conform to some system of units All Quantities in Mechanics must conform to some system of units - The Imperial system - The Metric system - International system of Units SI units Base Units Vs. Derived Units Base Units Vs. Derived Units There are 7 base units (5 reqd. in Mechanic) There are 7 base units (5 reqd. in Mechanic)

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Units All remaining units are called derived units All remaining units are called derived units Achieved by combining the base units Achieved by combining the base units

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Multiples of Base Units

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Mass Vs. Weight Mass is defined as the amount of matter (material) in a body (and is expressed in kilograms) Mass is defined as the amount of matter (material) in a body (and is expressed in kilograms) Weight is the force exerted by gravity acting on a body (and is expressed in Newtons) Weight is the force exerted by gravity acting on a body (and is 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

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Pressure/stress Pressure is a measure of the force per unit area (units N/m 2 ) Pressure is a measure of the force per unit area (units N/m 2 ) - P = FForce AArea - 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) = 30 2 / x 60 = = 1170mm 2 Area (woman) = 8 2 /2 + 3 x 16 = = 148mm 2 Pressure (man) = 784.8/1170 = 0.67 N/mm 2 (MN/m 2 ) Pressure (woman) = 784.8/148 = 5.3 N/mm 2 (MN/m 2 ) Weight = 80x9.81= 784.8N

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Triangle/Polygon of Forces A Force is that which changes or tends to change a bodies state of rest or of uniform motion in a straight line. A Force is that which changes or tends to change a bodies state of rest or of uniform motion in a straight line. The unit of Force is called the Newton and is the force required to give a mass of 1kg an acceleration of 1m/s 2. The unit of Force is called the Newton and is the force required to give a mass of 1kg an acceleration of 1m/s 2. A quantity which has both magnitude and direction is referred to as a Vector quantity (Force) A quantity which has both magnitude and direction is referred to as a Vector quantity (Force) A quantity which has magnitude only is a Scalar quantity (e.g. Mass, Time) A quantity which has magnitude only is a Scalar quantity (e.g. Mass, Time)

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Vector representation of forces Since a force has both magnitude and direction it can be represented by a vector Since a force has both magnitude and direction it can be represented by a vector The arrow indicates the direction of the force The arrow indicates the direction of the force The length of the arrow indicates the magnitude of the force (drawn to a suitable scale) The length of the arrow indicates the magnitude of the force (drawn to a suitable scale) ForceVectorScale 2 N = 1 cm 10 N 20 N 5 cm 10 cm

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Resultant/Equilibrant When a number of forces act on a body the resultant force is that single force which would have the same effect When a number of forces act on a body the resultant force is that single force which would have the same effect - This can be found by representing the forces as vectors and adding them head to tail The equilibrant is that force which must be applied to a system to produce equilibrium. (equal and opposite to resultant) The equilibrant is that force which must be applied to a system to produce equilibrium. (equal and opposite to resultant) 5 N 10 N 15 N 15 cm 10 cm5 cm Resultant Equilibrant Scale: 1N = 1cm

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Resultant Find the resultant forces shown opposite Find the resultant of the forces shown below

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Triangle of Forces If 3 forces are acting on an object but it remains stationary then it is said to be in equilibrium. (this implies no resultant) If 3 forces are acting on an object but it remains stationary then it is said to be in equilibrium. (this implies no resultant) In order for a body to be in equilibrium under the action of 3 coplanar forces: In order for a body to be in equilibrium under the action of 3 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

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

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Example Determine the force in the rods AB and BC when carrying a load of 8kN. = 60 o Determine the force in the rods AB and BC when carrying a load of 8kN. = 60 o W = 8000 N Fbc = N Fab = N

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Example The 80kg is supported by 2 rods AB and BC. Determine the force in each rod. The 80kg is supported by 2 rods AB and BC. Determine the force in each rod. W = 784.8N F AB = 632.4N F BC = 395.2N

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Example 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. 2m 5m 3.5m 3 T 38.7 o 56.9 o 29.43kN 73.6kN 51.5kN

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Example - 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

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Resolution of Forces Instead of solving forces graphically they can be solved mathematically (using trigonometry). Any single force acting at a point can be replaced by a pair of forces which have the same effect. Instead of solving forces graphically they can be solved mathematically (using trigonometry). Any single force acting at a point can be replaced by a pair of forces which have the same effect. Where these two 'component forces' are at right angles to each other they can be found easily using trigonometry. When a force is broken down into its component forces the force is said to have been 'resolved into its components'. Where these two 'component forces' are at right angles to each other they can be found easily using trigonometry. When a force is broken down into its component forces the force is said to have been 'resolved into its components'. Fx = F Cos Fx = F Cos Fy = F Sin Fy = F Sin

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Resolution 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. ForceNet horizontal forceNet vertical force 5050 Cos (0) = 5050 Sin (0) = Cos (120) = Sin (120) = Cos (210) = Sin (210) = Cos (-90) = 010 Sin (-90) = N 6 N

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Resolution of Forces (Problems) For the 5 forces shown determine the magnitude and nature of the resultant. o ]

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Resolution of Forces Question 2 (Huges P38 Q22) The following forces act at a point. 20 N due north 30 N 20 degrees south of east 10 N south west 16 N 5 degrees south of west Calculate the magnitude and direction of the resultant. [5.36N at 13.8 o north of east] Question 3 (Hughes P38 Q21) The following horizontal forces act at a point: 50N in a direction due east 80N in a direction due south and 30 N in a direction 20 o north. Calculate the direction of the fourth force necessary to maintain equilibrium.[73N at 72.6 north of west]

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Moments The moment of a force about a point is a measure of its turning effect and is defined as the product of:- 1 The force 2 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

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Principle 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.

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

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

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

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Moments

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

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Friction 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.

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Laws of Friction If a force P is applied to the book an equal but opposite force called friction is exerted by the table to maintain equilibrium. If a force P is applied to the book an equal but opposite force called friction is exerted by the table to maintain equilibrium. If the value of P is increased the friction force must increase accordingly. If the value of P is increased the friction force must increase accordingly. There is a limit beyond which the friction force cannot increase. If P exceeds this value sliding starts. The maximum force is called the limiting friction force. There is a limit beyond which the friction force cannot increase. If P exceeds this value sliding starts. The maximum force is called the limiting friction force. The value of the limiting friction force is proportional to the normal force Rn and is independent of the area of contact. The value of the limiting friction force is proportional to the normal force Rn and is independent of the area of contact. = F= limiting friction where: = coeff. of friction R n Normal reaction F = R n It will also be noticed that it takes a larger force to initiate movement than it does to maintain it. It will also be noticed that it takes a larger force to initiate movement than it does to maintain it. st > sl st = coefficient of static friction sl = coefficient of sliding friction

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Angle of Repose The angle of inclination of the surface just prior to slippage is called the angle of repose. The angle of inclination of the surface just prior to slippage is called the angle of repose. Force up slope= Force down slope Force up slope= Force down slope F= W Sin R n = W Sin W Cos = W Sin = W Sin W Cos = Tan

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Angle 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.

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