Subsection 5.1.2 Energy
The capacity of doing work is known as energy. It is a scalar quantity and it’s unit in SI system is Joule. Energy exists in many different forms, such as thermal energy, sound energy, electrical energy, magnetic energy, gravitational energy, chemical energy, mechanical energy, etc. The energy associated with the motion and position of an object is termed as mechanical energy. We focus our understanding here in mechanical energy. Mechanical energy is of two types, kinetic energy and potential energy. The kinetic energy is associated with the motion of the object and potential energy is associated with the position of the object with respect to some other object. For example: elastic potential energy due to a spring force, gravitational potential energy due to gravity, etc. The sum of kinetic and potential energy gives the total mechanical energy of the object.
Subsubsection 5.1.2.1 Work-Energy Theorem
From equations of motion, we have
\begin{equation}
v^{2} =u^{2}+2as \tag{5.1.2}
\end{equation}
where \(v\text{,}\) \(u\text{,}\) \(a\) and \(s\) are final velocity, initial velocity, acceleration, and distance covered by the particle, respectively. From Newton’s second law of motion, we have -
\begin{equation}
F=ma \tag{5.1.3}
\end{equation}
\begin{equation*}
v^{2} =u^{2}+2\frac{F}{m}s
\end{equation*}
\begin{equation*}
\text{or,}\quad \frac{1}{2}mv^{2}-\frac{1}{2}mu^{2}=Fs
\end{equation*}
Here the term \(\frac{1}{2}mv^{2} \) is called final kinetic energy, \(KE_{f}\text{,}\) the term \(\frac{1}{2}mu^{2} \) is called initial kinetic energy, \(KE_{i}\text{,}\) and the term \(Fs\) is work done by a force. Hence,
\begin{equation*}
KE_{f}-KE_{i} = W\qquad \Rightarrow\quad \Delta k=W
\end{equation*}
Which is known as work-energy theorem. It states that work done by a net force acting on a body is equal to the chnage in its kinetic energy.
Subsubsection 5.1.2.2 Kinetic Energy:
It is the energy associated with a body during its motion. Suppose \(\,dW \) is the work done by a force \(F\) on a body to displace it to a distance \(\,ds\text{,}\) then
\begin{equation*}
\,dW=F\,ds = m\frac{\,dv}{\,dt} \,ds = m\,dv\frac{\,ds}{\,dt} =mv\,dv
\end{equation*}
\begin{equation*}
\therefore W = \int\,dW = \int\limits_{v_{i}}^{v_{f}}mv\,dv = \frac{1}{2}mv_{f}^{2}-\frac{1}{2}mv_{i}^{2}
\end{equation*}
The quantity \(\frac{1}{2}mv^{2} \) is called the kinetic energy, i.e.,
\begin{equation*}
KE=\frac{1}{2}mv^{2}.
\end{equation*}
Since this kinetic energy is associated with a body in translational motion, it is called a translational (linear) kinetic energy. The kinetic energy in rotational motion is called rotational kinetic energy which we will discuss in Subsection 7.3.2.
Subsubsection 5.1.2.3 Potential Energy:
The energy stored in a body due to its position is known as potential energy. If work is done by a force to change the position of a body, then this work is stored in the body at that position in the form of potential energy. The restoring force plays a crucial role to acquire potential energy. For example, a charged particle near or far from another charge; a squashed or stretched rubber ball, a body above the surface of the earth. The potential energy of the body at any position is the work done against the restoring force in moving the body from the zero point (reference point) to that position. The reference point (or, the zero point) is the point where potential energy is assumed to be zero. If work has to be done on a body to take it to a new location, then the potential energy increases in the process. If the external force does the work then the potential energy change is negative.
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The zero point is so chosen that it could be infinitely far away. Such as at infinity, at the center of the earth, at sea level, at the floor, at the table top, or at any other point. This is because only potential energy differences between two positions are physically measurable and cannot depend on the a rbitrary choice of the zero point.
- Gravitational Potential Energy: Suppose a body of mass \(m\) is lifted to a height \(h\) against gravity without acceleration, then work done by the applied force, \(F_{a}\) is given by\begin{equation*} W_{a}= F_{a}\cdot h= -F_{g}\cdot h =-mgh \end{equation*}This work is stored in the body in the form of gravitational potential energy. Hence when this body is released from the position the body drops down to the ground again. Therefore, the work done by the gravity to pull that object towards earth’s surface is given by\begin{equation*} W_{g}= F_{g}h = mgh = PE \end{equation*}This is gravitational potential energy in a uniform gravitational field, \(g\text{.}\) The gravitational potential energy due to two point masses is given by\begin{equation*} -G\frac{Mm}{r}. \end{equation*}Where \(M\) is the mass of an object at the origin, \(m\) is the mass of an object at a radial distance \(r\) from the origin and the zero point is chosen at infinity.
- Elastic Potential Energy: When the body (or, spring) is compressed or stretched external force has to do the work, this work is stored in a body (or, spring) in the form of elastic potential energy. It is the energy due to work done by a restoring force.\begin{equation*} W = \int\limits_{0}^{x}F_{s}\,dx =\int\limits_{0}^{x}(-kx)\,dx=-\frac{1}{2}kx^{2}=E_{s} \end{equation*}where \(F = -k x,\) is the restoring force, \(k\) is elastic (or, spring) constant (it defines the stiffness of the spring), and \(x\) is extension in the spring.
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[read Chapter 8 for more details]
Since work done by a force in principle depends on the path taken by the body, (as can be seen in \(W=fx\) where \(f\) is a frictional force), but if the force is conservative then work done is independent on the path taken by the body rather it depends on the initial and final position of the body [Subsection 5.1.3].
