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G.1EE1.el3 (EEE1023): Electronics IIIMechanics lecture 7Moment of a force, torque, equilibrium of a bodyDr Philip /ee1.el3/
G.2Moments, torque and general equilibrium Moments of a force– vector product equation– moment about a point Torque as a vector– properties of a couple– components of a torque Equilibrium of a body– forces in balance– torques in balance Preparation– What is the moment of a force? Give the equation for a moment– What is an example of torque? Find a definition of torque– What does direction of torque mean? Illustrate with an example
G.3Moment of a force about a point A moment is a rotating action of a force about a pivot pointIt tends to turn a body about an axis:– the axis is to the force and its displacement from the pivot– the axis does not intersect with the forceʼs line of action andis not to the force Moment of a force P about point Ois defined:MO d P e(Nm)where d r sin α It measures of the turning effect of PUsing the right-hand rule,the direction of MO is as e
G.4Moment expressed as a vector The line of action of force is defined byits point of application and direction Force F applied at position r yields amoment about O:MO r Fwhere denotes the vector product of r and F( F r, opposite!) The cross product is simply the product whendisplacement r is perpendicular to force F:i.e., M r F otherwise, it depends on the angle θ:M r F sin θ
G.5Ways of calculating moments As seen from the moment equation, there is an equivalenceM (r sin θ) F r (F sin θ)which allows two interpretations for the size of the moment:1.2.it is the perpendicular distance fromthe pivot point to the line of actionmultiplied by the magnitude of theforceit is the displacement multiplied bycomponent of the forceperpendicular to it
G.6Moments about three orthogonal axes The three directions of rotation arecommonly called pitch, roll and yawExpressing torque in vector format witha cartesian reference frame, we get: MX M MY MZ These components can be calculated directlyin the vector product, e.g.M r F Note:pitchrollyaw 3 0 140 -1 70 0 -2 0 210 i j kj k ik i jj i -kk j -ii k -j
G.7Moment of a couple: torque A couple is a pair of parallel, equal and opposite forcesThe lines of action are not co-linear, separated by a distance, dAlthough the resultant force is zero, it is not in equilibriumMomentʼs magnitude is dP and direction to the plane offorces (right-hand rule) For constant d P, the moment becomes a torque as d 0:T d P
G.8Components of a torque As a special kind of moment, torque T can be resolved intoorthogonal components tooT TX i TY j TZ k Component of T about an arbitrary axis AB through the Origin,in direction e, isTAB (T.e) emagnitudedirection
G.9Properties of torque: pivot point Equivalent force combinations can yield the same resultant: We can use this property to shift the pivot point and simplifyThe resultant moment is sum of all moments about one point:nM M1 M 2 M nM # ri " Pii 1
G.10Resultant of a force and torque system The resultant is the sum of all forces and moments on a body:nR P1 P2 P3 PnR " Pii 1nM # ri " PiM M1 M 2 M ni 1 We can substitute them without changingexternal effect on it!!
!G.11Equilibrium of a rigid body In general, equilibrium occurs when all forces and torques actingon a body are in balance The resultant force and moment are both zero There is no acceleration of the bodynR " Pi 0i 1nM # ri " Pi 0i 1
G.12Free body diagrams A free body diagram shows a body isolated from other bodiesAll external forces and torques acting on it are shownTo solve for a body in equilibrium:1. Decide on the body of interest2. Draw a diagram of the bodyisolated from other bodies incontact with it3. Show all forces and torquesacting on the body(Newtonʼs 3rd law of opposition)4. Select an appropriate referenceframe5. Establish all vectors and unit vectors6. Apply equilibrium conditions
G.13Equilibrium of a rigid body example Equilibrium occurs whenthe resultant force andthe resultant moment areboth equal to zero:
G.14Example problem: gearbox in equilibriumYou now have all the toolsto solve a problem like this!
G.15Gearbox example: torque component about OProblem:If the gearbox is a 1-mcube with CoG at itscentre, write down anexpression for theresultant torquecomponent TX about O,where TX T.iSolution:TX ½ W – RC Tin
G.16Resultant moment example A yacht sailing windward (B)experiences ʻweather helmʼ whichmakes the helmsman steer toprevent turning into the wind Calculate the torque required fromthe rudder to hold its courseTL r L 1/2 0 2 0 0 4200 8400 -2100 0
G.17Kinematics (1): rigid body statics Moments– moment of a force about a point– calculation using the vector product– torque induced by a couple– components of torque about an axis Resultant moment– sum of all moments about one point– can be used to simplify problems Equilibrium– free body diagram– resultant force and resultant momentequal to zero
G.18Kinematics (2) Motion of a rigid body How do you define angularvelocity?– How can we calculate it?– How can we express it asa vector? What are the relationships between angular displacement,angular velocity and angular acceleration?– write mathematical expressions of the relationships in bothdirections
Free body diagrams A free body diagram shows a body isolated from other bodies All external forces and torques acting on it are shown To solve for a body in equilibrium: 1. Decide on the body of interest 2. Draw a diagram of the body isolated from other bodies in contact with it 3. Show all forces and torques
