Conservation of Mechanical Energy

Have you ever wondered how an automatic mechanical watch works? At the center of its complicated motion lies one of the most basic principles of classical physics: The law of conservation of mechanical energy. Let’s delve into the principle:

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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:

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.

Browse more Topics under Work Energy And Power

• Collisions
• Concepts of Potential Energy
• Potential Energy of a Spring
• Power
• The Scalar Product
• Work and Kinetic Energy
• Work-Energy Theorem
• Various Forms of Energy: The Law of Conservation of Energy

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.

Conservative Force

A conservative force has following characteristics:

• A conservative force is derived from a scalar quantity.  For example, the force causing displacement or reducing the rate of displacement in a single dimension without any friction involved in the motion.
• The work done by a conservative force depends on the end points of the motion. For example, if W is the work done, K_(f) is the kinetic energy of the object at final position and K_(i) is the kinetic energy of the object at the initial position:

• Work done by a conservative force in a closed path is zero. Here, W is the work done, F is the conservative force and d is the displacement vector.  In case of a closed loop, the displacement is zero.  Hence, the work done by the conservative force F is zero regardless of its magnitude.

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.

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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