Forces Notes

 2013 Yeow Kok Han Page 1 FFoorrcceess 11 IInnttrroodduuccttiioonn This chapter is about recognising the different kinds of forces a t work in different situations and knowing the characteristics of the forces. This is important whenever one is interested in explaining the motion of objects, which is the focus of the chapter on Dynamics. Understanding the forces at work is also important in calculating work done for a given situation. 22 EEllaassttiicc FFoorrccee Most objects, e.g. a spring or a rubber block, when stretched or compressed, will exert a force op posite in direction to that applied on the object. The force exerted by the object on the external agent is an elastic force. In Fig. 2.1, the elastic force is also sometimes called tensile force or tension. When the elastic fo rce is proportional to the distance extended or compressed, the object is said to obey Hooke’s law. Hooke’s law states that where k is a positive constant and the negative sign indicates that the elast ic force is opposite to the direction of displacement. Though called a ‘law’, Hooke’s law is not universal or always obeyed. There are many objects or materials which do not obey Hooke’s law at all or may obey it for a limited range of x only. k is known as the constant of proportionality or in the case of a spring, also called the spring constant. Note that k is constant only as long as the dimensions and material of the object are not changed. For a given x, the larger the value of k, the larger the elastic force and hence the larger the external force needed to cause the displacement x. Therefore k is also called the stiffness. NNeewwttoonn’’ss TThhiirrdd LLaaww Note that in Fig. 2.1 the force exerted by hand on spring is equal in magnitude but opposite in direction to the force exerted by spring on hand . This is reflected by the arrows of equal length but opposite directions. In fact, Newton’s third law states: All forces exist in such equal and opposite pairs or action-reaction pairs. An elastic body is one which can return to its original size and shape after being deformed. In response to an external applied force, an elastic body will exert an elastic force on the external body. When the elastic force varies linearly with the extension or compression, the elastic body is said to obey Hooke’s law. Newton’s third law implies that forces always occur in pairs When body A exerts a force on body B, body B must exert an equal but opposite force on body A. Pulling forces exerted by hands on spring Elastic forces exerted by spring on hands Pushing forces exerted by hands on block Elastic forces exerted by block on hands Fig. 2.1 Fig. 2.2 xkF   or F = kx (magnitudes) Springs identical except for different lengths have different k values Fig. 2.3 Springs identical except for different materials have different k values

 2013 Yeow Kok Han Page 2 EEllaassttiicc FFoorrcceess IInnssiiddee aa SSpprriinngg In Fig. 2.1, if we focus our attention on a middle segment of the spring as shown, would the elastic for ces exerted by this segment on the neighbouring segments be the same or smaller than the elastic forces on the hands that pulled the spring? Unlike Hooke’s law, Newton’s third law is universal. As shown in Fig. 2.5, while the middle segment B exerts elasti c forces on the neighbouring segments A and C, the neighbouring segments also exert equal but opposite elastic forces on the middle segment. Hence, the elastic forces (magnitude) exerted by the whole spring on the external agent and by any segment on neighbouring segments are the same . This situation is frequently described as ‘the tension in a spring is the same at every point’. Strictly speaking, the tension is only the same everywhere in the spring if it is not accelerated or its mass is negligible. So, assuming massless springs, when different springs of different spring constants are connected together and pull ed at both ends as shown, the tensions(magnitude T) between the springs and inside them will be the same. However their extensions will be different unless they are totally identical springs. In the same way, ropes which are pulled will have the same tension throughout. PPootteennttiiaall EEnneerrggyy In general, potential energy (PE) can be stored in a system with (a) at least two bodies which (b) exert forces - on each other - that depend only on their positions. In the case of a stretched spring, energy is stored in the spring as can be seen when two masses are attached to the e nds of the spring and released. The two masses will be accelerated towards one another gaining kinetic energy ( KE) while the elastic PE (EPE) gets converted to KE. What if the stretched spring is released without any masses attached? In this case, the PE is converted to KE of t he various spring segments and perhaps some sound energy when neighbouring coils knock into each other upon release. Another type of PE you already know is the gravitational PE. When a ball is lifted higher from the ground and thus further from the Earth, PE is stored between Earth and the ball. When the ball is released, the mutual attraction between the Earth and ball will accelerate them towards one another. However due to the large mass of the Earth, it hardly moves while the ball moves much more easily and gets practically all the KE that comes from the stored PE. For a stretched or compressed elastic body at rest, elastic forces exist throughout the body and the magnitude is the same on every part of the body. Potential energy (PE) can be stored in a system with at least 2 bodies which exert forces on each other in a way that depend only on their positions. Fig. 2.6 T T Fig. 2.5 elastic forces in red by A on B by B on A A B C by B on C by C on B by external agent on spring by external agent on spring Fig. 2.4 elastic forces on neighbouring segments elastic forces on hands

 2013 Yeow Kok Han Page 3 FF--ee && FF--LL ggrraapphhss For different forces FA applied to pull a spring, it is found that the spring will settle at different final lengths. If the spring obeys Hooke’s law, then the graph of applied force FA versus extension e will look as shown in Fig. 2.8 a while the graph of FA versus length of spring is as shown in Fig. 2.8b. HHooww DDooeess EEPPEE GGeett SSttoorreedd?? Since energy cannot be created but only transferred from one body to another or converted from one form to another, where does the elastic PE come from? It will be learnt that work done by a force is equal to the energy transferred or converted from on e form to another form. When a person exerts a force to extend a spring, the person uses chemical energy to do the work which gets stored inside the spring as elastic PE. Work done is calculated by the product of force and the displacement in the direction of the force. In the case of a spring being stretched, the force needed to stretch the first mm and the force needed to stretch the second mm are different. As the extension increases, the force needed also increases, so how do we find the work done? Consider the work done in increasing the length by s. The applied force varies from F1 to F2. F1s is the area of the shaded rectangle as shown in Fig.2.9a. This area represents the work done by F1 over a distance s. Since the forc e is actually not constant, we can always line up a series of very narrow rectangles within the distance s such that the total area of the rectangles would effectively be the total work done over s. Thus for any F-e plot; whether straight line or curve; th e area under the plot represents the work done. This work done