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Physics 211

Physics 211 . 6: Work and Energy. Work done by a constant force Work done by a varying force Kinetic energy and the Work-Energy theorem Power. If a total non zero force acts on an object it effects the state of the object The longer it acts the more it effects the object

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Physics 211

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  1. Physics 211 6: Work and Energy • Work done by a constant force • Work done by a varying force • Kinetic energy and the Work-Energy theorem • Power

  2. If a total non zero force acts on an object it effects the state of the object • The longer it acts the more it effects the object • If the object is initially at rest it can cause the object to move • If the force keeps acting on the object then it acts while the object is at different positions

  3. A measure of how much it effects the object is the distance over which the force acts on the object • We describe this effect by the quality WORK • We quantify it by the amount of work done by the force • The force does work on the object

  4. The work done by an agent exerting a constant force is the product of the force in the direction of displacement times the magnitude of the displacement d F

  5. If the force is not pointing in the direction of displacement then it is ONLY the component of that force in the direction of displacement that ”does” the work F d 

  6. recall ( ) ( ) 3 i + 2 j - 4 k · 2 i - 5 j + 2 k = 3 ´ 2 + 2 ´ ( - 5 ) + ( - 4 ) ´ 2 = 12 = 3 i + 2 j - 4 k 2 i - 5 j + 2 k q cos SI units of work [ ] [ ] [ ] W = F d = Nm = Joules ( J )

  7. Work done by a varying force For an object moving along x- axis å å å ( ) W = d W = F d = F x D x i x i i i i In order that portions of force are constant over the portions of the displacement D x i å x ò final ì ü ( ) ( ) W = lim F x D x = F x dx í ý î þ ( ) x i i x D ¯ x 0 i x initial ( ) = area under the curve of the function F x x between x and x initial final

  8. Force exerted by spring on a mass m F ( x ) = - kx where x is the displacement from the equilibrium length, and k is the force constant of the spring ( x > 0 if spring is extended; x < 0 if spring is compressed ) Work done by spring on the mass x x f f ò ò 1 x f ( ) W = F x dx = - kx dx = - kx 2 2 x i x x i i 1 1 = - kx + kx 2 2 2 f 2 i \ by Newtons 3 rd law work done by mass on spring 1 1 æ ö = - - kx 2 + kx 2 è ø 2 f 2 i

  9. Kinetic Energy and the Work-Energy theorem Consider constant net force F acting over distance d W = Fd = mad net using the kinetic equations of motion v - v and noting that a = , we obtain f i t v - v æ æ 1 1 1 ö ( ) ö = m v + v t = m v - m v f i 2 2 è ø è ø f i f i t 2 2 2 1 1 i . e . W = m v - m v 2 2 f i 2 2 net 1 Define Kinetic Energy K = m v 2 2 Þ W = K - K = D K net f i

  10. Work - Energy theorem is valid for variable force x x x f f f ò ò ò d v W = F ( x ) dx = ma ( x ) dx = m dx net dt x x x i i i x x v f f f ò ò ò d v dx d v = m dx = m v dx = m v d v dx dt dx x x v i i i 1 1 = m v - m v = D K 2 2 2 2 f i

  11. Power dW Definition P = dt For a force F that is constant over a displacement d ( ) d F · d d F d d P = = · d + F · dt dt dt Þ P = F · v SI units of power [ ] J W [ ] [ ] [ ] P = F v = = = Watts ( W ) [ ] T s

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