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:

dt dv a

x 

However, very often we only know the velocity as a function of position (i.e. coordinate x). What to do in this case?

dx dv v a dx dv dt dx dt dv a

x x

   

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

dx F mv 2 1

• mv

2 1 F v dx d v m v dx d m F x dt d m F

x x x 2 xi 2 xf x x x x x 2 2 x

f i



              

3D

dz F mv 2 1

• mv

2 1 dy F mv 2 1

• mv

2 1 dx F mv 2 1

• mv

2 1

z z z 2 iz 2 fz y y y 2 iy 2 fy x x x 2 ix 2 fx

f i f i f i

  

                    

+

                

  

dz F dy F dx F mv 2 1

• mv

2 1

z z z y y y x x x 2 i 2 f

f i f i f i

Work (abbreviation: W)

dz F dy F dx F F

z z z y y y x x x

f i f i f i

  

   

Work done W by a force

• 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  Kgm2s‐2.

• 3. Work done by a force can be positive, negative, or 0.

Dot product (a.k.a. scalar product)

B A B A B A B A

z z y y x x

     



• 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.

 cos | B || A | B A      

A B x y 

Work

dz F dy F dx F F

z z z y y y x x x

f i f i f i

  

   

Work done W by a force

r d F F dr F dr F dr F dz F dy F dx F F

f i f i f i f i f i f i f i

r r z z z z y y y y x x x x z z z y y y x x x

   

 

       

      

Work done W by a force

F

x y ri rf

dr F dr

When F is constant

Work done W by force

F

x y ri rf

) r r k ˆ z j ˆ y i ˆ x d ( d F z F y F x F dz F dy F dx F dz F dy F dx F F

i f z y x z z z y y y x x x z z z y y y x x x

f i f i f i f i f i f i

                           

     

F



d

Path independent

More than one force

dz F mv 2 1

• mv

2 1 dy F mv 2 1

• mv

2 1 dx F mv 2 1

• mv

2 1

z z z 2 iz 2 fz y y y 2 iy 2 fy x x x 2 ix 2 fx

f i f i f i

  

                    

+

                

  

dz F dy F dx F mv 2 1

• mv

2 1

z z z y y y x x x 2 i 2 f

f i f i f i

r d F mv 2 1

• mv

2 1

f i

x x 2 i 2 f

         



Dot product notations One force: Many forces:

r d F r d F mv 2 1

• mv

2 1

i i x x i x x i 2 i 2 f

f i f i

                   

   

Total work Work done by total force

Kinetic energy (abbreviation: K)

v v m 2 1 mv 2 1 ) v v m(v 2 1

2 2 z 2 y 2 x

       

Kinetic energy of a moving particle

• 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

v2>0. There is no negative kinetic energy.