Work & Mechanical Energy

Work:     Work is defined by the amount of energy required to move an object a certain distance.  In order to do any work, the object must have a force exerted on it; it turns out that work can be expressed by the following relationship:

W =   Fd   
W= Work done in Joules (J).

F   =   Force which does the work
D  =   distance moved  

Note:       Work can be done by any  force so it is important to be clear on which force you are talking about.

Example 1:   What is the work done by an applied force of 50N which pushes a 10 kg box along a floor a distance of 5.0 m?

W_app = F_appd
W_app = (50 N)(5.0 m)
W_app = 250 Nm         (or 250 J)

Example 2:   An applied force of 50N which pushes a 10 kg box along a floor a distance of 5.0 m is opposed by a friction force of 10N.  What is the work done by friction?          

W_f = F_fd          
W_f = (-10 N)(5.0 m)          
W_f = -50 Nm         (or 50 J)  

Note: As in the case of forces, there is a difference between the work done by the different forces acting on an object and the work done by the net force on the object or net work.

Example 3: An applied force of 50N which pushes a 10 kg box along a floor a distance of 5.0 m is opposed by a friction force of 10N.  What is the net work done on the object?            

Wnet = Fnetd          
Wnet = (40 N)(5.0 m)          
Wnet = 200 J

OR          
Wnet = Wapp + Wf          
Wnet = 250 J - 50 J          
Wnet = 200 J      

Energy:    There are two general classifications of mechanical energy that we will discuss:  Kinetic Energy, and Potential Energy.    Kinetic energy is the energy attributed to the motion of an object. Potential Energy is the energy which is stored and depends on position relative to zero potential energy (arbitrary).   Potential energy can be used to describe any type of stored energy (electric, chemical, nuclear, etc.), however, we are only interested in gravitational potential energy (for now).  

Potential Energy: Gravitational potential energy (or just potential energy for our purposes) is defined as the amount of work which gravity would do to an object if it were to fall from its position to some zero reference point (usually "the ground").            

P.E.   = F_gh          
P.E. = Potential energy in Joules   
F_g = mg           (Force of gravity)      h        = height       (from ground)  

You will find this equation expressed more often as:            

P.E. = mgh      

Kinetic Energy:   Kinetic energy is defined by its velocity and its mass.  Just like in momentum, the larger and faster moving an object is, the more energy it has.  Kinetic energy is given by the following equation:

K.E. = 1/2mv²  

Total Energy:         The total energy in a system of objects is the sum of their potential and kinetic energies:                     

E_tot = P.E. + K.E.            
OR   
E_tot = mgh + 1/2mv²            

Example:      A 10kg rock sits on the top of a 1.0 km high cliff.  How much potential energy does it possess?            

P.E.   = mgh                   
= (10kg)(9.8m/s/s)(1000m)          
P.E.   = 98 000 J  

Conservation of Energy: Energy is conserved, so the total energy before an event is the same as the total energy after the event.               

Example 1:  If the rock were to fall off the cliff, how much K.E. would it have just before it hit the bottom?            

E_tot = E_p + E_k         
E_k = E_tot - P.E.          
E_k = E_tot - mgh  (h = 0.0m)          
E_k = 98 000 J - 0          
E_k = 98 000 J  

How much work would have to be done by a person wishing to put the rock back on the top of the cliff?            

W = Fgd   (but d = h from P.E. = mgh)          
W = mgh = P.E.          
W = 98 000 J            

What was the velocity of the rock when it was half way down?            
h = 0.5 km = 500 m          
Etot = E_k + K.E.     (Etot = 98 000J)          
E_k = E_tot - E_p          
1/2mv² = Etot - mgh            
v² = 2[98 000J - ( 10kg)(9.8m/s/s)(500m)]/10kg           

v = 99 m/s   (356 km/h ???)  

Problems:   
Work Worksheet #1                    
Energy Review WS