Work and Kinetic Energy

The curved line in Fig. 2.1 represents the path of a body of mass m (a closed system) mov
ing relative to the x–y coordinate frame shown. The velocity of the center of mass of the
body is denoted by V.1 The body is acted on by a resultant force F, which may vary in
magnitude from location to location along the path. The resultant force is resolved into a
component Fs along the path and a component Fn normal to the path. The effect of the
Forces acting on a moving system

component Fs is to change the magnitude of the velocity, whereas the effect of the component F
n is to change the direction of the velocity. As shown in Fig. 2.1, s is the instantaneous
position of the body measured along the path from some fixed point denoted by 0. Since the
magnitude of F can vary from location to location along the path, the magnitudes of Fs and
F n are, in general, functions of s.
Let us consider the body as it moves from s  s1, where the magnitude of its velocity is V1,
to s  s2, where its velocity is V2. Assume for the present discussion that the only interaction
between the body and its surroundings involves the force F. By Newton’s second law of motion,
the magnitude of the component Fs is related to the change in the magnitude of V by
 
The quantity is the kinetic energy, KE, of the body. Kinetic energy is a scalar quantity. The change in kinetic energy, KE, of the body is2
 
The integral on the right of Eq. 2.3 is the work of the force Fs as the body moves from s1 to
s2 along the path. Work is also a scalar quantity.




 
where the expression for work has been written in terms of the scalar product (dot product)
of the force vector F and the displacement vector ds. Equation 2.6 states that the work of the
resultant force on the body equals the change in its kinetic energy. When the body is accelerated by the resultant force, the work done on the body can be considered a transfer of
energy to the body, where it is stored as kinetic energy.
Kinetic energy can be assigned a value knowing only the mass of the body and the magnitude of its instantaneous velocity relative to a specified coordinate frame, without regard
for how this velocity was attained. Hence, kinetic energy is a property of the body. Since
kinetic energy is associated with the body as a whole, it is an extensive property.
UNITS. Work has units of force times distance. The units of kinetic energy are the same as
for work. In SI, the energy unit is the newton-meter, called the joule, J. In this book
it is convenient to use the kilojoule, kJ