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Moment of Force Concept of Torque: In Newton s Laws of Motion we have studied that that if a body is at rest and has to be brought in motion , we must apply a force. Similarly, in order to Stop a moving body we must apply a Force. Thus Force is actually the cause of Translational Motion. On Application of force , the body attains a Linear acceleration(a). However, in order to produce Rotational Motion in the body force must be applied in a different way because not all forces have the ability to rotate the body. Rotation is always caused by a pair of forces (called couple). These pair of forces must be equal in magnitude , opposite in direction and must act on different points in the body. Together , they cause the body to rotate and this turning effect of the couple forces is termed as torque. Thus we can conclude that whatever role Force plays in Translational Motion, Torque plays the same role in Rotational Motion. For this reason , Torque is also known as the Rotational Analogue of Force. Whenever , Torque acts on a body , attains an angular acceleration. If I ask you , Does a body require force to stay in motion? . Well , of course not because the body which was already moving with uniform velocity will continue to do so even if no force acts on it due to inertia of motion. However , you will agree that if the body was initially at rest , it does require force to come into Translational Motion. Similarly, Although torque is the cause of Rotational Motion , but even in the absence of torque the body can continue rotational motion with constant angular velocity. I want to put stress on the point that A Single force can never rotate the body . But , we apply a single force with our hands to open/close a door , isn t it? Does the door rotate when we apply a single force? Well it does. Does this fact Contradict the statement I just out a stress on? No , things are perfectly fine here. Let's Understand. The doors are always hinged at one end. When we apply a force(F) at some distance from the hinge , all the points in door experience this force (F) and tend to cover some angular displacement in the direction of applied force , but the hinge prevents the motion of the points on the door at the pivoted end. These points apply a force (F) (which we had applied on the free end) on the hinge and in turn the hinge applies an equal and opposite force (F) on the fixed points. Thus , we can clearly see that there are two forces that act on the door. i) The external force (F) that we had applied at the free end of the door. ii) The Force of reaction(F) on the fixed points of the door by the hinge. These two forces constitute a couple and rotate the body. Let me Introduce to you to some Terminologies: Point of Application of Force = It is the point on the body at which the force is applied. Line of Action of Force = The imaginary line along which force is applied. ( Basically draw dotted lines in the direction of force) Point of rotation = The fixed point about which the body rotates. Axis of rotation= The real/imaginary line about which the body rotates. Lever arm/ Moment arm/ Torque arm= The perpendicular distance between the line of action of force and axis of rotation How do they define torque Mathematically? Moment of force(Torque) is defined as the product of force and perpendicular distance between the line of action of force and axis of rotation. T=Fr SI unit: Nm =kgm2s-2 (but this is not abbreviated as Joule) CGS unit: dynecm=gcm2s-2 (but this is not abbreviated as erg) 1Nm=107 dyne cm NOTE: Torque and work are two different physical quantities but they have the same units. The only difference is since work is a mode of transfer of energy , so for work the unit Nm is abbreviated as J(Joule) but for torque Nm is not Joule. Is Torque a scalar or vector? Well, Torque is a vector quantity. To be frank, it is a type of vector called Axial Vector. Axial Vectors are those vectors whose direction is along the axis of rotation and since the axis is perpendicular to the plane of rotation ,so Torque is also perpendicular to the plane. Now you might say , " But there are two directions of a perpendicular drawn to the plane?" "Inward to the plane and outward to the plane". Which is the direction of torque? Lets Understand using a diagram To find the direction of Torque , you will have to use "Right Hand Thumb Rule" which simply states " Curl the fingers of your right hand in the direction of rotation of the body and your thumb will give you the direction of torque" Since we have only two choices for direction of torque " Perpendicular to the plane inward or outward" , so in order to represent it we use sign convention. If the Applied force rotates the body in anti clockwise sense , then the torque is perpendicular to the plane and outward and we call it positive. However, If the Applied force rotates the body in clockwise sense , then the torque is perpendicular to the plane and inward and we call it negative. Moment of Couple: Qualitatively, The net torque exerted due to the couple forces acting on the body is termed as Moment of Couple. Quantitatively , The product of either force and perpendicular distance between line of action of both forces is termed as the moment of couple. Since couple forces are equal in magnitude , opposite in direction and act on the same body ( of course at different points) , so the net force on the body is zero. Thus , they can never produce Translational Motion. It must be very clear that Moment of couple depends only on the perpendicular distance between the line of action of both forces. This makes us draw a very important conclusion i.e " Moment of couple doesn't depend upon the position of pivot". In other words , the value of moment of couple ( net torque) will be same about all points in the body whereas this is not the case for the individual torques. You didn't get that right? Let me explain Suppose a couple is acting on a body. The torque produced by individual forces depends on the position of pivot but the net torque( due to both forces) is independent of the position of pivot. Are Couple Forces Action-Reaction Pair? Do I really need to answer this? Why don t you think for a while? Yes , you are correct " Obviously they are not action-reaction pairs because they act on same body , whereas action-reaction pairs should act on different bodies" Conceptual Questions on Torque: Q) How can you change the sense of rotation of a body? Ans. i) By Changing the point of application of force, ii) By Changing the direction of applied force. Q. Why are handles of door situated at the free ends of door? Ans. By placing the handle at the free ends it is ensured that the perpendicular distance between the line of action of force and axis of rotation (moment arm) is maximum and thus we can produce the required torque by applying least effort. Q) A hand flour grinder has its handle near the rim ,why? Ans. The handle used to rotate the grinder is at the maximum possible distance from the axis of rotation,so that the required moment of force produced by applying least effort. Q.It is easier to turn a steering wheel of large diameter compared to a small one,why? Ans. If the wheel is of large diameter ,then the distance between the line of action of force and axis of rotation is maximum and consequently the required torque is produced with least effort. Q. Why do spanners have long handle? Ans. Long handles ensure large distance between line of action of force and axis of rotation due to which the required torque is produced and the nut can be loosened/tightened with least effort. Q) The wheel of a bicycle is rotated using a foot paddle ,why? Ans. The Foot peddle is situated at maximum possible distance from the axle(fixed point) which ensures that torque arm is maximum and thus the wheel can be rotated with least effort. Equilibrium of bodies: A body is said to be in equilibrium if on application of a certain number of forces , it maintains its original state(rest or motion) If it was at rest , it should remain at rest and if it was in any type of motion (Translational or Rotational) it should continue to be in motion. On the basis of type of motion exhibited by the rigid body , equilibrium is of two types: (Conditions of Equilibrium) 1. Translational Equilibrium: A body is said to be in Translational Equilibrium if the net force acting on the body is zero. 2. Rotational Equilibrium: A body is said to be in Rotational Equilibrium if the net torque on the body about the point of rotation/axis of rotation should be zero. If a body is in Translational as well as Rotational equilibrium then it is said to be in Mechanical Equilibrium . If the net torque on a body is zero, then is net force on the body necessarily zero? Well , I don t think so. Think of a situation where there is a body kept of ground. It is being pulled towards right by a force of 50N and towards left by a force of 30N. Since these forces are acting in horizontal direction , the body wont rotate( net torque is zero) , but net force is 20N. If the net force on a body is zero, then is net torque on the body necessarily zero? This time , you might have answered yes , but you are wrong. If a pair of couple forces act on a body , the net force on the body is zero but the body does rotate so net torque is not zero. However , in this situation we can draw an important conclusion. If the net force on a body is zero , then the net torque on the body may be zero or may not be zero , but the net torque is same about all points in the body Note: Both Translational and Rotational Equilibrium can be further classified as Static and Dynamic Equilibrium depending on the original state of the body. i) Static Equilibrium: When a body remains in a state of rest under the influence of several forces , it is said to be in static equilibrium. eg: A book kept of table, A beam balance in horizontal position ii) Dynamic Equilibrium: When a body remains in a state of motion under the influence of several forces , it is said to be in dynamic equilibrium. eg: i) Rain drops falling through air after attaining terminal velocity When rain drops descend first their velocity increases due to the gravitational pull of the earth but after some time their weight gets balanced by upthrust due to air and viscous force due to which the net force on them becomes zero and their velocity becomes constant. This constant velocity is termed as terminal velocity. Since , the net force on the rain drops is zero but they still move with constant velocity so they are an example of dynamic equilibrium. ii) Uniform Circular Motion: In Uniform circular motion , although centripetal force always acts towards the centre but net torque of this centripetal force is zero and thus angular velocity is constant. Thus , it is an example of dynamic equilibrium. # Principle of Moments: For a body in rotational equilibrium, the sum of all anticlockwise moments is equal to the sum of clockwise moments about the point of rotation. Note: The physical balance/ beam balance is a device that works on this principle.
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