6161103 11.4 conservative forces


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6161103 11.4 conservative forces

  1. 1. 11.4 Conservative ForcesWork done by a force when it undergoes adifferential displacement has been defined asdU = F cosθ dsIf the force is displaced over a path that hasfinite length s, the work is determined byintegrating over the path U = ∫ F cos θds sTo evaluate the integral, obtain arelationship between F and the component ofdisplacement ds cosθ
  2. 2. 11.4 Conservative ForcesIn some instances, however, the work doneby a force will be independent of its path andinstead, will depend only on the initial andfinal locations of the force along the pathAs force with such a property is called aconservative force
  3. 3. 11.4 Conservative ForcesWeight Consider body initially at P’ If the body is moved down along arbitrary path a to second position, then for a given displacement ds along the path, the displacement component in the direction of W has a magnitude of dy = ds cos θ
  4. 4. 11.4 Conservative ForcesWeight Since both the force and displacement are in the same direction, the work is positive y U = ∫ W cosθds = ∫ Wdy or s 0 U = Wy Similarly, for work done by the weight when the body moves up a distance y back to P’, along arbitrary path A’, U = −Wy
  5. 5. 11.4 Conservative ForcesWeight Weight of a body is therefore a conservative force since the work done by the weight depends only on the body’s vertical displacement and is independent of the path along which the body moves
  6. 6. 11.4 Conservative ForcesElastic Spring Force developed by an elastic spring (Fs = ks) is also a conservative force If the spring attached to a body and the body is displaced along any path, such that it causes the spring to elongate or compress from position s1 to s2, the work will be negative since the spring exerts a force Fs on the body that is opposite to the body’s displacement
  7. 7. 11.4 Conservative ForcesElastic Spring For either extension or compression, work is independent of the path and is simply s2 s2 U = ∫ Fs ds = ∫ (−ks)ds s1 s1 1 2 1  = − ks2 − ks12 Friction 2 2  Work done by a frictional force depends on the path; longer the path, the greater the work Frictional forces are non-conservative and work done is dissipated in the form of heat