Work and Energy Preview In this chapter you will be introduced to the concept of work and the related concepts of kinetic energy and potential energy. After After completion of the chapter you will be able to calculate the work done by constant forces, the kinetic energy of a moving object, the gravitational potential energy of an object, and the power developed by a force. In addition, you will be able to use the principle of conservation of mechan ical energy to solve a variety of problems in a much simpler way than you have done previously.
Quick Reference
Important Terms Work
Kinetic energy Gravitational potential energy Total Total mechanical energy Conservative force Average Average power
The work done by a CO!TAT CO!TAT force acting on an object is the component of the force along the displacement of the object times the magnitude of the displacement. The energy of an object due to its its motion. motion. The energy of an object due to to its position relative relative to the earth. The sum of the kinetic and potential energies of an object. A force which does work on an object which is independent independent of the path taken by the object be tween its starting point and its ending point. The work done by a force on an object divided by the time taken to do the work.
Work The work done on an object by a force, F, is "#.$% where s is the magnitude of the displacement of the object and & is the angle between the force and the displacement. The work done by gravity on an object is
"#.'% where m is the mass of the object, the object.
is the initial height of the object, and
is the final height of
The work done by nonconservative forces acting on an object is "#.#%
"#.(a%
"#.(b%
Energy The kinetic energy of an object of mass, m, and speed, v, is "#.)% The gravitational potential energy of an object a height, h, above the reference level is "#.*% The total mechanical energy of an object is the sum of its kinetic and potential energies.
The work done by nonconservative forces produces a change in the total mechanical energy of the object. "#.+%
Power The average power developed by a force which has done work, , in time, t, is "#.$-a% The average power developed by a force of magnitude , moving an object with an average speed, is
"#.$
%$Theorems and Principles ork/0nergy Theorem / the total work done by all forces acting on an object is "#.1% 2rinciple of Conservation of 3echanical 0nergy / the total mechanical energy of an object remains constant as the object moves, provided that no net work is done by nonconservative forces. "#.4a%
Work Done by a Constant orce The concept of work plays an important role in physics since it connects, via the work/energy theorem, ewton5s second law to the important scalar 6u antities of kinetic energy and potential energy. Eample ! Positive and negative work
A constant force of '-.- is needed to keep a car traveling at constant speed as it moves *.- km down the road. 7ow much work is done8 Is the work done on or by the car8 The force needed to keep the car moving is in the direction of the car5s displacement. 06uation "#.$% gives the work.
Eample "
There must be a retarding force acting on the car in e9ample $ if it is to remain traveling at constant speed. hat is this force8 7ow much work does the force do8 Is this work done on or by the car8 ewton5s second law applied along the direction of motion of the car gives
or
The work done by the retarding force is then given by "#.$% to be
Eample #
A force is pulling an +*.- kg block across a hori:ontal surface. The force acts at an angle of )-.-; above the surface. The coefficient of kinetic friction is -.)--, and the block moves a distance of +.-- m. ind "a% the work done by the pulling force, "b% the work done by the kinetic frictional force, "c% the total work done by all the forces.
"a% The work done by the pulling force is
"b% The magnitude of the frictional force, , acting on the block is found by applying ewton5s second law to the free body diagram. In the vertical direction
then
The work done by this force is
"c% The total work done by all forces is
ote that the work due to the gravitational force and the normal force is :ero since these forces act at a 4-; angle to the motion of the block.
The Work!Energy Theorem and "inetic Energy et work done on an object always produces a change in the kinetic energy of the object according to the work/energy theorem "#.1%. The theorem is often very useful in finding the speed of an object if the net work done by the forces acting on it are known or can be calculated.
Eample $ Using the work-energy theorem to find the speed of an object
If the block in e9ample 1 starts from rest, what is the speed of the block after it has traveled +.-m8
The work/energy theorem gives
!olving for
gives
Another common use of the work/energy theorem is finding information about the forces acting on an object when information about the object5s speed changes is known. Eample % Finding an average force when the speeds are known
A baseball pitcher can throw a 4.-- ounce baseball with speed measured by a radar gun of 4-.- miles per hour "$1) ft
The work/energy theorem gives
if the ball starts at rest. The work done is
, so
#ravitational Potential Energy The gravitational force is one of a class of =special> forces called conservative forces. hat makes the gravitational force special is that the work done on an object by gravity depends only on the initial and final positions of the object. The work does OT depend on how the object got from the initial to the final position. Eample & Work done on an object by gravity
7ow much work is done by gravity when a *.- kg object moves $.* m hori:ontally and then moves 1.- m vertically upward8 ?ertically downward8 The work done by gravity during the hori:ontal motion is
The work done by gravity during the vertical motion is
If the object is moving vertically downward, the work done during the vertical motion is
As the previous e9ample shows, the magnitude of the work done by gravity on the object when it is going up is the same as the magnitude of the work done by gravity on the object when it is going down. The negative of the work done by gravity is called the potential energy of the object. The work/energy theorem then gives rise to e6uation #.#%. Eample ' Using gravitational potential energy
A 1.-- kg model rocket is launched vertically upward with sufficient initial speed to reach a height of
, even though air resistance "a non/conservative force% performs of work on the rocket. 7ow high would the rocket have gone if there were no air
resistance8 The e6uation "#.(b% written for the actual motion of the rocket gives
If the rocket were launched with the same initial kinetic energy and no air resistance acts, then "#.(b% is
or
The Conservation of $echanical Energy If no friction or other non/conservative force acts on an object then its total mechanical energy remains constant. This principle is very useful in solving problems involving the motion of an object. It is often easier to use than ewton5s laws o r the kinematic e6uations since it involves scalars rather than vectors. Also, it only re6uires that you have knowledge about the motion at the beginning and end. Eample (
A truck is descending a winding mountain road. hen the truck is $'+- m above sea level and traveling $* m
The truck would have the ma9imum possible speed if friction were ignored. In this case, the total mechanical energy of the truck is conserved as it moves down the mountain road.
Taking the :ero of potential energy at sea level we have
Power The average power is the rate at which work is done as given by e6uation "#.$-a%. Eample )
A (1 kg sprinter, starting from rest, reaches a speed of (.- m
so
"b% The force that the sprinter must e9ert to run at constant speed is e6ual in magnitude and opposite in direction to the force e9erted on him by air resistance. The power needed to run the race at a constant speed is then
2ractice 2roblems $ The driver of a $*--/kg car slams on the brakes locking the wheels. A total retarding force of $+-- acts to stop the car in a distance of (-.- m. 7ow much work is done in bringing the car to a halt8 Is the work done on or by the car8 ) A ')--/kg truck descending a *.-; hill is brought to a stop in 4* m. The driver applies the brakes so that the wheels lock. "a% If the coefficient of kinetic friction between the truck tires and the road is -.#-, how much work is done by friction in stopping the truck8 "b% 7ow much work is done on the truck by gravity8 1 In problem $, how fast was the car traveling initially8 ' In problem ), how fast was the truck traveling immediately before the brakes were applied8
* A bicyclist tops the crest of a )-.- m high hill moving with a speed of $.- m