It is the capacity of an object to do work by the virtue of its motion or configuration (position). Mechanical Energy is the sum of following two energy terms:
Here, V is the potential energy of the object in joules (J), m is the mass of the object in kilograms, g is the gravitational constant of the earth (9.8 m/s^{2}), and h is the height of the object from earth’s surface. Now, we know that the acceleration of an object under the influence of earth’s gravitational force will vary according to its distance from the earth’s centre of gravity.
But, the surface heights are so minuscule when compared to the earth’s radius, that, for all practical purposes, g is taken to be a constant.
Kinetic energy, a fundamental concept in physics, refers to the energy transferred to an object due to its motion. To put an object in motion, force is applied, requiring the performance of work. This work leads to energy transfer, resulting in the object moving at a new speed. The transferred energy is termed kinetic energy and is determined by the object's mass and the speed it attains.
Kinetic Energy
Understanding and quantifying kinetic energy require familiarity with its units. In the International System of Units (SI), the standard unit for kinetic energy is the Joule (J). One Joule is equivalent to 1 kilogrammeter squared per second squared (kg·m²/s²). This unit denotes the kinetic energy possessed by an object with a mass of 1 kilogram moving at a velocity of 1 meter per second.
In the CGS system (centimetergramsecond), the unit of kinetic energy is the erg. An erg is defined as the kinetic energy possessed by an object with a mass of 1 gram moving at a velocity of 1 centimeter per second.
Both the Joule and erg are derived from fundamental units of mass, length, and time, providing a standardized means to express and compare kinetic energy across different measurement systems.
Kinetic Energy Examples
However, the transformation continues. As the yoyo reaches the end of its descent, it starts climbing back up due to the tension in the string. The kinetic energy of the yoyo ball begins converting back into potential energy as it gains height. This process repeats as the yoyo moves up and down, with continuous conversions between potential and kinetic energy.
The formula of kinetic energy is an essential tool in understanding the energy associated with an object’s motion. The equation for kinetic energy is given by:
KE = 1/2 mv^{2}
The equation above represents the kinetic energy (KE) of an object, where "m" represents the mass and "v" represents the velocity.
According to this equation, kinetic energy is directly proportional to both the mass and the square of the velocity of the object. This means that an increase in either the mass or the velocity will result in an increase in the kinetic energy. It is important to note the significance of the relationship between kinetic energy and velocity, as the velocity term is squared in the equation. This implies that even small changes in velocity can have a substantial impact on the object's kinetic energy.To calculate the kinetic energy of an object using this equation, you need to know the mass and velocity of the object. First, square the velocity, and then multiply it by half of the object's mass. The resulting value will give you the kinetic energy of the object.
Kinetic energy is considered a scalar quantity because it does not have an associated direction. It represents the magnitude of energy possessed by an object. While velocity, which is a vector quantity, includes both magnitude and direction, the calculation of kinetic energy only takes into account the magnitude of velocity. The kinetic energy depends on the mass and the square of the magnitude of velocity. By squaring the velocity term, kinetic energy is always positive, regardless of the direction of motion. Therefore, even though velocity is a vector quantity, the calculation of kinetic energy focuses solely on magnitude, resulting in it being a scalar quantity.
The formula for calculating kinetic energy is an essential tool in understanding the energy associated with an object’s motion. The equation for kinetic energy is given by:
KE = 1/2 mv^{2}
To begin, let’s consider a particle of mass “m” moving with a velocity “v.” The kinetic energy of this particle can be defined as the work done to accelerate the particle from rest to its current velocity. According to the workenergy theorem, the work done on an object is equal to the change in its kinetic energy.
The work done on the particle is given by the equation:
Work = Force x distance
(i) In the case of constant force, the work done can be calculated as:
Work = Force x distance
Now, let’s express the force in terms of mass and acceleration. According to Newton’s second law of motion, force (F) is equal to mass (m) multiplied by acceleration (a). Since the particle is moving with a constant velocity, its acceleration is zero. Therefore, the force acting on the particle is zero.
Substituting the force into the work equation, we have:
Work = 0 x displacement
Work = 0
This means that no work is done on the particle as it moves with a constant velocity. Consequently, there is no change in its kinetic energy.
(ii) In the case when the particle is initially at rest and then accelerated to a final velocity “v” by an external force.
In this case, work is done on the particle to change its velocity.
The work done on the particle is equal to the change in its kinetic energy. So, we can write:
Work = Change in kinetic Energy
Using the work formula:
Work = Force x displacement
The force acting on the particle is equal to the mass of the particle (m) multiplied by its acceleration (a). Since the initial velocity is zero, the final velocity is “v,” and the displacement is “d,” we have:
Work = (m x α) x d
Now, recall that the formula for acceleration (α) is given by:
Since the initial velocity is zero, the change in velocity is “v” and the time taken is “t,” we can rewrite the equation as:
α = v/t
Substituting this back into the work equation, we have:
The quantity (d / t) represents the average velocity (v_{avg}) of the particle. So, we can rewrite the equation as:
Now, the average velocity (vavg) is equal to half of the final velocity (v), as the particle starts from rest and reaches a final velocity “v.” Therefore, we can further simplify the equation as:
Finally, we equate the work done to the change in kinetic energy:
Hence, we have derived the equation for kinetic energy:
This equation relates the kinetic energy (K.E) of an object to its mass (m) and velocity (v). It demonstrates that the kinetic energy is directly proportional to the square of the velocity and the mass of the object.
Kinetic energy manifests in different forms depending on the context. Here are some common types:
These diverse forms of kinetic energy illustrate the broad range of phenomena where energy is connected to the motion of objects or particles. Each type of kinetic energy possesses distinct characteristics and contributes to different areas of physics and everyday life.
At its core, potential energy stems from an object's ability to store energy due to its position. For instance, when a bow is drawn, it accumulates potential energy, which transforms into kinetic energy upon release. Similarly, a stretched spring gains potential energy, evident in the tension we feel when stretching it. Therefore, potential energy can be defined as a form of energy resulting from positional or state alterations.
The formula for potential energy relies on the specific force acting on the objects. In the case of gravitational force, the formula is as follows:
W = m×g×h = mgh
Where:
Gravitational potential energy shares the same units as kinetic energy: kg m²/s². It's important to note that all forms of energy possess the same units and are measured in joules (J).
Potential energy manifests in various forms. The two main types are gravitational potential energy and elastic potential energy.
Potential Energy due to height
A potential energy surface (PES) represents the potential energy of a collection of atoms, typically in terms of their spatial coordinates. The PES can describe energy as a function of one or more coordinates, with a potential energy curve or energy profile representing a PES with a single coordinate. The analogy of a landscape is often used, where energy values correspond to different bond lengths or other relevant variables. Each geometry of atoms in a chemical reaction corresponds to a unique potential energy, resulting in a smooth energy "landscape" that allows the study of chemistry from a topological perspective.
Crystal Structure Packing and Bonding Energies: Different atoms arrange themselves in various crystalline formations based on their nature, leading to diverse potential energy curves. Random and dense ordered packing of atoms exhibit contrasting potential energy curves, highlighting the role of crystal structure in determining bonding energies.
The potential energy surface and curve are valuable conceptual tools for analyzing molecular geometry and chemical reaction kinetics. The characteristics of bonding energy and the shape of potential energy curves vary from one material to another. A deep and narrow trough in the curve indicates significant bond energy, high melting temperature, large elastic modulus, and a small coefficient of thermal expansion. The diameter and asymmetry of the potential energy curve reveal distinct material properties. Different materials exhibit varying potential energy curves based on their bonding types, such as metallic bond for metals and covalent and secondary bonding for polymers.
The sum total of an object’s kinetic and potential energy at any given point in time is its total mechanical energy. The law of conservation of energy says “Energy can neither be created nor be destroyed.”
So, it means, that, under a conservative force, the sum total of an object’s kinetic and potential energies remains constant. Before we dwell on this subject further, let us concentrate on the nature of a conservative force.
Let us consider the following illustration:
Here, Δx is the displacement of the object under the conservative force F. By applying the workenergy theorem, we have: ΔK = F(x) Δx. Since the force is conservative, the change in potential Energy can be defined as ΔV = – F(x) Δx. Hence,
ΔK + ΔV = 0 or Δ(K + V) = 0
Therefore for every displacement of Δx, the difference between the sums of an object’s kinetic and potential energy is zero. In other words, the sum of an object’s kinetic and potential energies is constant under a conservative force. Hence, the conservation of mechanical energy is proved.
The pendulum is a very good example of conservation of mechanical energy. Following illustration will help us understand the pendulum motion:
This property of mechanical energy has been harnessed by watchmakers for centuries. Of course, in the real world, one has to account the other forces like friction and electromagnetic fields. Hence, no mechanical watch can run perpetually. But, if you get a precise mechanical watch like Rolex, you can expect long power reserves!
Q: A mass of 2kg is suspended by a light string of length 10m. It is imparted a horizontal velocity of 50m/s. Calculate the speed of the said mass at point B.
Solution: Potential energy at point A, V(A) = mgh(A)
Kinetic energy at point A, K(A) = (mv²)/2 = (2 × 2500)/2 = 2500J
Hence, total mechanical energy at point A, K(A) + V(A) = [2500 + V(A)]J
Potential energy at point B, V(B) = mg h(B) = mgh (A+10) = mg h(A) + 2 × 9.8 × 10
= [V(A) + 196]J
Kinetic energy at point B, K(B) = (mv^{2})/2
Hence, total mechanical energy at point B, K(B) + V(B) = [K(B) + V(A) + 196]J
By applying the law of conservation of energy,
V(A) + K(A) = V(B) + K(B)
Therefore, V(A) + 2500 = K(B) + V(A) + 196
or K(B) = 2500 – 196
Which gives: (mv^{2})/2 = 2304
(2 × v^{2})/2 = 2304
v = [2304]½
Therefore, velocity of the mass at point B = 48m/s
Work done by conservative forces
I^{st} format : (When constant force is given)
Ex.8 Calculate the work done to displace the particle from (1, 2) to (4, 5). if
Sol.
w = (16  4) + (15  6) = w = 12 + 9 = 21 Joule
II format : (When F is given as a function of x, y, z)
then
Ex.9 An object is displaced from position vector under a force Find the work done by this force.
Sol.
III^{rd} format (perfect differential format)
then find out the work done in moving the particle from position (2, 3) to (5, 6)
Sol.
Now ydx + xdy = d(xy) (perfect differential equation)
⇒ dw = d(xy)
for total work done we integrate both side
⇒ w = (30  6) = 24 Joule
Internal Work
Suppose that a man sets himself in motion backward by pushing against a wall. The forces acting on the man are his weight 'W' the upward force N exerted by the ground and the horizontal force N exerted by the wall. The works of 'W' and of N are zero because they are perpendicular to the motion. The force N' is the unbalanced horizontal force that imparts to the system a horizontal acceleration. The work of N', however, is zero because there is no motion of its point of application. We are therefore confronted with a curious situation in which a force is responsible for acceleration, but its work, being zero, is not equal to the increase in kinetic energy of the system.
The new feature in this situation is that the man is a composite system with several parts that can move in relation to each other and thus can do work on each other, even in the absence of any interaction with externally applied forces. Such work is called internal work. Although internal forces play no role in acceleration of the composite system, their points of application can move so that work is done; thus the man's kinetic energy can change even though the external forces do no work.
"Basic concept of work lies in following lines
Draw the force at proper point where it acts that give proper importance of the point of application of force.
Think independently for displacement of point of application of force, Instead of relation the displacement of applicant point with force relate it with the observer or reference frame in which work is calculated.
Ex.11 Calculate the work done by the force to move the particle from (0, 0) to (1, 1) in the following condition
(a) y = x (b) y = x^{2}
Sol. We know that
dw = ydx ...(1)
In equation (1) we can calculate work done only when we know the path taken by the particle.
either
y = x or y = x^{2} so now
(a) when y = x
Energy can be classified into two distinct categories : one based on motion, namely kinetic energy; the other on configuration (position), namely potential energy. Energy comes in many a forms which transform into one another in ways which may not often be clear to us.
The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
102 videos411 docs121 tests

1. What is the law of conservation of energy? 
2. What are the different forms of energy? 
3. How does conservation of mechanical energy apply to reallife situations? 
4. Can mechanical energy be completely converted into other forms of energy? 
5. How is conservation of mechanical energy related to the concept of work? 

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