Work and Energy

A micro-course in work and energy in grade-12 physics that is the section three of the course in classical mechanics available on Physics 12: Classical Mechanics. This course contains lectures on different topics of work and energy, a complete discussion on work-kinetic energy theorem, conservation law of mechanical energy, and some different types of potential energies. Problems and solutions are also included to this course.

Energy 

Energy is defined as the ability to do work. There are two different kinds of energy:   
- Kinetic energy, E_k : energy of an object because of its motion. Kinetic energy is defined by this equation:   E_k =  ½  mv²
In this equation ' v ' is the speed of the object, and ' m ' is its mass.
- Potential energy, E_p : energy of an object because of its position relative to the other objects, or because of its position in a force field.

Work

Work done on an object by a constant force acting on it, is equal to the inner product of force and displacement vector of the object. When force is non-constant, the integral of the inner product of force and differential of displacement vector of the object, is the work done by the force on the object. Find more about work in this video tutorial:



Work

Work and Kinetic Energy 

According to the work- kinetic energy theorem, work done on an object by a net force acting on it, is equal to the change in the kinetic energy of the object:  
                  W = ΔE_k
In this equation "W" is the net work (or work done by a net force), and ΔE_k  is the change in kinetic energy of the object. The proof of the work- kinetic energy theorem is discussed in Grade-12 Physics: Mechanics.

Work and Gravitational Potential Energy 

By applying a vertical force on an object, you can lift it up. When the applied force is equal to the weight of the object, the object moves upward with a constant velocity (if air resistance is negligible), so the kinetic energy of the object remains constant, and according to the work- kinetic energy theorem, the net work done on the object, will be equal to zero. 

 An object is lifted up, by a force with the same magnitude as the weight of object.

 
The work done by the applied force on the object thorough its motion from height  h_i to h_f , is given by,

Since the the object moves up-ward and gravitational acceleration is down-ward, the above equation can be written as,

where "mgh_i" and "mgh_f"  are gravitational potential energy of the object at height  hi and hf, respectively. So the work done on the object changes its gravitational potential energy.

Conservative vs. Non-conservative Forces  

The work that a conservative force does on an object is path independent; but the work done by a Non-conservative (dissipative) force depends on the path taken between to points.
Electric, gravitational, and elastic (or spring) forces are examples of conservative forces; while friction and air resistance are examples of non-conservative forces.

 

Mechanical Energy and its Conservation Law 

The sum of kinetic and potential energies of an object is called mechanical energy of the object: 
                    E_m = E_k + E_p
 When a system is isolated and its internal forces are conservative, its mechanical energy is conserved over time.
 

Elastic Potential Energy

Elastic potential energy is the energy stored in a elastic material or a physical system, as a result of work performed to disturb its shape or volume.  
Elastic potential energy of a perfectly elastic spring is given by this equation:     
                       E_ep =  ½  κx²

where "κ" is the spring constant, and "x" is the amount of extension or compression of the spring as a result of work done on it by an applied force. A proof of the above equation is given in Grade-12 Physics: Mechanics.

Power

Power is the rate at which work is done; In the SI system, its unit is joule per second (J/s) or watt (W).
If Δw  is the amount of work done on an object in time interval of Δt , the average power over that time interval, is given by this equation:   

So the instantaneous power can be found as:

And when power is constant over time, we have: 
                                            
                                  P_avg = P(t)

Problems and Solutions

Problem 1
In order to move an object with mass of 'm' downward in vertical direction with a constant acceleration of 0.25g , you can use a rope. Find the work done by the rope if downward displacement of the object is 'd'; how much is the amount of net work done on the object?

Problem solving lecture in work and energy:1 

Problem 2

A student throws a ball upward in vertical direction with a velocity of 10 m/s. The mass of the ball is 20 grams.

a) Find the kinetic energy and mechanical energy of the ball at launch point.

b) If the maximum height of ball is 4 meters, how much of its mechanical energy is dissipated when it's in its maximum height.

Problem solving lecture in work and energy:2 

Problem 3


Velocity of an object with mass of m=2 kg, is given by :

 in terms of time, t ; where î and  ĵ are unit vectors on X and Y axises respectively. 
Find the net work done on the object from a initial time, t_i = 1 s, to a final time,  t_f = 3 s ,
a) directly from the definition of work.
b) using work-kinetic energy theorem.

Problem solving lecture in work and energy:3

Problem 4

A block with a mass of 2.5 kg slides on a frictionless horizontal

surface with a speed of 2 m/s, on a straight line. It is brought to rest

by compressing a perfectly elastic spring in its path. Find the spring constant if its compression is 8.0 cm. 

Problem solving lecture in work and energy:4