Physics in Motion

Energy Conservation and Work

When only conservative forces are acting on a body, then its energy is conserved. Its energy is made of kinetic energy and potential energy, and the sum of these two types of energies is known as the mechanical energy of the body. Gravity, electricity, and magnetism are all examples of important forces that are also conservative. The conservation of energy is useful for understanding the motion of a particle under their influence.

But there are other forces, such as friction and drag, which are not conservative, as they cannot be written in terms of a potential energy. What do we do in those cases? Is energy still a useful quantity?

Over the last couple of centuries the following empirical observation has been made: when a body is slowed down by friction or drag (or other dissipative forces), heat is transferred to the stopping medium. If the body itself is made of many smaller parts then heat may also be transferred to itself in the process. If this heat quantity is included in the balance of energy, then the energy of the system as a whole, body + medium, is conserved.

Thus, conservation of energy is still a useful notion, but we must include heat in the balance of energy. Importantly, while the mechanical energy is associated with the body under consideration, heat is the energy quantity transferred to the medium the body is interacting with or to the body's own internal components.

But, whether the forces are conservative or dissipative, the kinetic energy of a body is something that we can measure directly. Conservation of energy tells us that the change in the kinetic energy is either associated with a change in potential energy or the transfer of heat, either one is important.

So, let us de fine a quantity which we call work that is the change in a body's kinetic energy,

$$W = \frac{1}{2} m v^2_f - \frac{1}{2} m v^2_i$$

Where \( m \) is the mass of the body, and \( v_i \) and \( v_f \) are the initial and final velocities, respectively. When the kinetic energy of a body increases, the work is positive and we say that work was done on the body. Alternatively, when the body's kinetic energy decreases, the work is negative and we say that work was done by the body.

The notion of work can be directly related to the forces acting on a body. Work can also be de fined as the time integral over the force acting on a body in the direction of the body's velocity,

$$W = \int^{t_f}_{t_i} \vec{F} \cdot \vec{v} dt$$

Where the integral is over the total force acting on the body at every point along its motion from the initial moment \( t_i \) to the final moment \( t_f \). The dot product, \( \vec{F} \cdot \vec{v} \) is defined as:

$$\vec{F} \cdot \vec{v} = |F||v| \cos \theta_{Fv}$$

where \( |F| \) and \( |v| \) are the magnitudes of the force and velocity, respectively, and \( \theta_{Fv} \) is the angle between these two vectors. We can see that these two definitions of work above are equivalent using Newton's second law:

$$W = \int^{t_f}_{t_i} \vec{F} \vec{v} dt = \int^{t_f}_{t_i} (m \vec{a}) \cdot \vec{v} dt = m \int^{t_f}_{t_i} \frac{d \vec{v}}{dt} \cdot \vec{v} dt$$

$$W = m \int^{t_f}_{t_i} \frac{1}{2} \frac{d}{dt} |v^2| dt$$

$$W = \frac{1}{2} m v^2_f - \frac{1}{2} m v^2_i$$

In the second line above, the chain rule was used, will in the third line, the fundamental theorem of calculus was used. This result is very important as it tells us how to relate the forces acting on a body to the change in its kinetic energy, or in other words, the work.

Another way to understand work, which is closely related to the integral definition, but without reference to the velocity of the body:

$$W = \int^{\vec{x}_f}_{\vec{x}_i} \vec{F} \cdot d \vec{x}$$

Where \( \vec{x}_i \) and \( \vec{x}_f \) are the initial and final positions of the body, respectively. You can prove that this is an equivalent definition to the previous definitions by using the relation \( \vec{v} = d \vec{x} / dt \), and the chain rule of calculus.