Mechanical Energy: Types & Conservation of Mechanical Energy | Physics Class 11 - NEET PDF Download

Mechanical Energy

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:

• Kinetic Energy. It is the ability of an object to do work by the virtue of its motion. For example, the kinetic energy of Wind has the capacity to rotate the blades of a windmill and hence produce electricity. Kinetic energy is expressed as,where, K is the kinetic energy of the object in joules (J), m is the mass of the object in kilograms and v is the velocity of the object:
  
• Potential energy. It is the ability of an object to do work by the virtue of its configuration or position. For example, a compressed spring can do work when released. For the purpose of this article, we will focus on the potential energy of an object by the virtue of its position with respect to the earth’s gravity. Potential energy can be expressed as:
  V = mg h

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/s²), 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

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.

• In the realm of physics, the kinetic energy of an object signifies the work it can accomplish through its motion. It is crucial to note that kinetic energy is a scalar quantity, representing solely its magnitude without any directional component.
• In summary, kinetic energy is the energy associated with the motion of an object, acquired by applying work to it. It relies on both the object's mass and the square of its speed, playing a crucial role in understanding various physical phenomena.

Kinetic Energy

Units of 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 kilogram-meter 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 (centimeter-gram-second), 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

• The kinetic energy of a river is evident as it flows at a specific speed, possessing both velocity and mass.
• As an asteroid descends towards Earth, it exhibits an immense amount of kinetic energy due to its high velocity and substantial mass.
• During flight, an airplane attains higher kinetic energy owing to its considerable mass and rapid velocity.
• When a person walks, their legs and body in motion result in the acquisition of kinetic energy.
• As a thrown ball moves through the air, it gains kinetic energy, the amount of which is determined by its throwing speed.
• Objects falling towards the ground amass kinetic energy as a result of the force of gravity, with the impact's kinetic energy directly proportional to the object's mass and velocity.
• A truck traveling at the same speed as a car possesses greater kinetic energy due to its significantly higher mass.

Kinetic Energy Examples

Transformation of Kinetic Energy

• Elastic Collision: During collisions between objects like billiard balls, kinetic energy can be transferred. When the balls collide and rebound, some of the kinetic energy transforms into other forms, such as heat and sound energy.
• Pendulum Swing: As a pendulum swings back and forth, it continually converts between potential and kinetic energy. At the highest point of the swing, potential energy is dominant, while descending converts potential energy into kinetic energy.
• Car Braking: When a car applies its brakes, the kinetic energy of the moving vehicle transforms into other forms, primarily heat energy. Friction between the brake pads and wheels slows down the car by converting kinetic energy into thermal energy.
• Wind Turbine: In a wind turbine, the kinetic energy of moving air is captured and converted into electrical energy. As the wind blows, it rotates the turbine blades, transforming the wind's kinetic energy into mechanical energy, which then drives a generator to produce electricity.
• Waterfall: As water falls from a higher elevation in a waterfall, its potential energy gradually converts into kinetic energy. Gravity causes the flowing water to gain speed and kinetic energy, which can be harnessed to generate hydroelectric power.
• Yo-Yo Transformation: A yo-yo demonstrates the transformation of kinetic energy. Initially at rest in a person's hand, the yo-yo stores energy as potential energy. When released, gravity pulls the yo-yo downward, converting potential energy into kinetic energy. The yo-yo ball accelerates, gaining speed and kinetic energy. At the bottom of its path, the energy is entirely in the form of kinetic energy.

However, the transformation continues. As the yo-yo reaches the end of its descent, it starts climbing back up due to the tension in the string. The kinetic energy of the yo-yo ball begins converting back into potential energy as it gains height. This process repeats as the yo-yo moves up and down, with continuous conversions between potential and kinetic energy.

Kinetic Energy Formula

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 mv²
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.

Why is Kinetic Energy a Scalar Quantity?

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.

Deriving Kinetic Energy Formula

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 mv²

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 work-energy 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.

Various Forms of Kinetic Energy

Kinetic energy manifests in different forms depending on the context. Here are some common types:

• Translational Kinetic Energy: This form of kinetic energy pertains to the energy associated with the linear motion of an object. It is most commonly observed when objects move along a straight path.

• Rotational Kinetic Energy: Objects that rotate or spin around an axis exhibit rotational kinetic energy. Examples include spinning tops, wheels, and rotating machinery.

• Vibrational Kinetic Energy: Vibrational kinetic energy arises from the oscillatory motion of particles within a system. It is observed in vibrating strings, molecules, or atoms undergoing vibrational movement.

• Thermal Kinetic Energy: Thermal kinetic energy is linked to the random motion of particles within a substance. It determines temperature and facilitates heat transfer. Higher thermal kinetic energy corresponds to faster particle movement.

• Electrical Kinetic Energy: Electrical kinetic energy is associated with the movement of charged particles, such as electrons, in an electric current. It is employed in various electrical devices and serves as a means of energy transfer.

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.

Potential Energy

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.

Potential Energy Formula

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:

• m represents the mass of the object in kilograms
• g denotes the acceleration due to gravity
• h signifies the height of the object in meters

Potential Energy Unit

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).

Types of Potential Energy

Potential energy manifests in various forms. The two main types are gravitational potential energy and elastic potential energy.

• Gravitational Potential Energy: Gravitational potential energy refers to the energy possessed by an object that has been raised to a certain height against the force of gravity. Let's illustrate this with an example: Imagine an object with mass (m) placed at height (h) above the ground. When the object is raised against gravity, work (W) is done on it. According to the law of conservation of energy, the energy gained by the object, in this case, is its potential energy (E). Thus, the gravitational potential energy (E) of an object raised to a height (h) above the ground is given by the formula: E = m * g * h. Notably, the distance traveled by the object is not considered; only the displacement between the initial and final heights matters.

Potential Energy due to height

• Elastic Potential Energy: Elastic potential energy resides in objects that can be compressed or stretched, such as rubber bands, trampolines, or bungee cords. The greater an object's capacity for deformation, the more elastic potential energy it possesses. Some examples of objects designed to store elastic potential energy include a twisted rubber band propelling a toy plane, a stretched bow in archery, or a coiled spring in a wind-up clock. Elastic potential energy can be calculated using the formula: U = 0.5 * k * x², where U represents elastic potential energy, k symbolizes the spring force constant, and x denotes the stretch length in meters.

Potential Energy Surfaces

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.

Potential Energy Curves: Understanding 1-D Surfaces

• The potential energy curve (PEC) represents the energy of a molecule as a function of the positions of its nuclei, denoted by r. 
• In a system with two atoms, the energy is proportional to their separation. At longer distances, no contact exists, resulting in zero energy. 
• As the distance decreases, attractive forces dominate until the atoms reach the minimum point of the curve, where attractive and repulsive forces balance. 
• This minimum point determines the bond length or equilibrium bond length, which reflects the average distance at which the atoms oscillate due to thermal motion. Shorter bonds typically indicate stronger bonding.

Analysis of Potential Energy Curves

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.

• Mechanical Properties: Mechanical characteristics define how materials deform or break under stress, time, and temperature. Understanding potential energy curves helps elucidate the behavior of materials during deformation, elongation, compression, or fracture.
• Thermal Properties: Heat capacity, thermal expansion, and thermal conductivity describe a material's response to heat. Potential energy curves provide insights into how thermal energy is absorbed and transmitted through atom vibrations in solids.
• Electrical Properties: Electrical conductivity and resistivity characterize a material's ability to conduct electricity. The energy levels and electron availability in potential energy curves influence the electrical properties of solid materials.

Conclusions

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.

Conservation of Mechanical Energy

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.

Proof of Conservation of Mechanical energy

Let us consider the following illustration:

Here, Δx is the displacement of the object under the conservative force F. By applying the work-energy 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.

Case Study: Simple Pendulum

The pendulum is a very good example of conservation of mechanical energy.  Following illustration will help us understand the pendulum motion:

• At position A, Potential energy is zero and the kinetic energy is at maximum.
• When the object travels from position A to B, it’s kinetic energy reduces and potential energy increases.
• At position B, the object stops momentarily.  At this position, the object’s kinetic energy becomes zero and its potential energy reaches the maximum.  The law of conservation of mechanical energy comes into play here.  The object’s entire kinetic energy at position A has been converted to potential energy at position B.
• Now, the object retraces its path, this time from position B to position A.  Back at position A, the object’s kinetic energy has been restored to its initial level.  Object’s Potential energy is zero.
• Now, the object travels the exact same path as AB, but in reverse direction of AC.
• This process repeats itself infinitely because the mechanical energy of the object remains constant.

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!

Solved Examples For You

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) = (mv²)/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: (mv²)/2 = 2304
(2 × v²)/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 = x² 

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 = x² so now

(a) when y = x

Law of Conservation of Energy

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.