Class 23: Work and kinetic energy (Cont) Acceleration by chain rule - PowerPoint PPT Presentation

Class 23: Work and kinetic energy (Con’t)

Acceleration by chain rule (1D) If we know the velocity as a function of time, we can differentiate it w.r.t. time and find out how the acceleration depends on time: dv x  a dt However, very often we only know the velocity as a function of position (i.e. coordinate x). What to do in this case? dv dx dv   a x dt dt dx dv   a v x dx

The answer In most cases we live in a “force field” – there is always a force acting on us and this force depends on where we are. x 2 d d      F m x F m v x x x 2 dt dx d   m v v F x x x dx x   1 1 f      2 2 mv - mv F dx xf xi x   2 2 x i

3D x   1 1 f     2 2 mv - mv F dx fx ix x   2 2 x i y   1 1 f    2 2  mv - mv F dy fy iy y   2 2 y i z   1 1 f    2 2  mv - mv F dz fz iz z   2 2 + z i   x y z   1 1 f f f           2 2 mv - mv F dx F dy F dz   f i x y z   2 2   x y z i i i

Work (abbreviation: W)  x y z f f f       Work done W by a force F F dx F dy F dz x y z x y z i i i 1. Work is a scalar (sum of definite integrals) – it has no direction. 2. Unit of work: Joule (J). Joule is not a fundamental unit, J  Nm  Kgm 2 s ‐ 2 . 3. Work done by a force can be positive, negative, or 0.

Dot product (a.k.a. scalar product) y       A B A B A B A B x x y y z z B A   x        A B | A || B | cos 1. The result is a scalar, that’s why its called the scalar product. 2. The equivalency is useful to calculate the angle between two vectors, if you know the components of these two vectors.

Work  x y z f f f       Work done W by a force F F dx F dy F dz x y z x y z i i i  x y z f f f       F F dx F dy F dz x y z x y z i i i x y z f f f       F dr F dr F dr x x y y z z y x y z i i i    r  f    Work done W by a force F F d r  r f F r i dr x dr r i F

When F is constant  x y z f f f       Work done W by force F F dx F dy F dz x y z x y z i i i x y z f f f       F dx F dy F dz x y z x y z i i i       F x F y F z x y z      ˆ ˆ ˆ           F d ( d x i y j z k r r ) y f i F F Path independent r f x d  r i

More than one force x   1 1 f     2 2 mv - mv F dx fx ix x   2 2 x i y   1 1 f     2 2 mv - mv F dy fy iy y   2 2 y i z   1 1 f     2 2 mv - mv F dz fz iz z   2 2 + z i   x y z   1 1 f f f          2 2  mv - mv F dx F dy F dz   f i x y z   2 2   x y z i i i Dot product notations x     1 1 f      2 2 mv - mv F d r One force: f i   2 2 x i x  x        1 1 f f            2 2  mv - mv F d r F d r f i i i   Many forces:   2 2 i i x x i i Total work Work done by total force

Kinetic energy (abbreviation: K)    1 1 1      2 2 2 2 m(v v v ) mv m v v Kinetic energy of a moving particle x y z 2 2 2 1. Kinetic energy is a scalar – it has no direction. 2. Unit of kinetic energy: Joule (J), the same unit as work. 3. Kinetic energy is always positive, because m>0 and v 2 >0. There is no negative kinetic energy.