An Introduction
The concept of force is a crucial aspect for physicists to learn in detail. When studying it, one should not neglect its various types, including conservative and non-conservative forces. Consequently, a vital topic for students is to understand how to define conservative force. What is Conservative Force? Conservative force is any force necessary to move a particle from one point to another, independent of this particle’s path. Therefore, the work done by conservative force only relies on the start and end-point, and not on the path taken. The work done by a conservative force on a particle travelling between any two places does not rely on the path travelled by the particle as a direct result of the closed path test.
One of the most common conservative force examples is gravity. Other examples are magnetic, electrostatic and elastic forces. A force does work, and some forces, such as weight, have unique properties. A conservative force, such as gravity, is one in which the amount of work done by or against it is determined only by the starting and ending locations of a motion rather than the path travelled. Just like we did with the gravitational force, we can define potential energy (PE) for any conservative force.
Conservative force relies on the principles of conservation of energy. Conservation of energy primarily deals with the conservation of kinetic energy.
Example – 1
Suppose you drop a ball from the top of a ladder. As a conservative force, the gravity will be calculated while the ball is at the maximum height and again when it reaches the ground.
Gravitational force = m x g
Where m = ball’s mass
g = gravitational acceleration,
Therefore, we can say that work done on the body is
Wg = -mgh
Here, h refers to the difference between the final position of a body and its initial position.
Potential Energy And Conservative Forces
The energy a system have as a result of its position, form, or configuration is known as potential energy. It's a form of stored energy that can be fully recovered. A conservative force is one for which the amount of work done by or against it is determined only by the motion's beginning and ending positions, rather than the path travelled. For each conservative force, we may define potential energy (PE). The potential energy contributed is the effort done against a conservative force to achieve a final configuration, which is determined by the configuration rather than the path is taken. Noninertial frames of reference give rise to frictional forces in Newtonian mechanics. A particle moves according to Newton's equations of motion in an inertial frame. However, if we go to a non-inertial frame, we must incorporate extra forces to accurately characterise the particle's motion. The centrifugal and Coriolis forces, which arise in a rotating reference frame such as the Earth, are examples.
Determine the system, the potential energy function, and the external factors that affect it.
Calculate the amount of work done by external forces.
Calculate the total energy change.
Calculate the potential energy change.
Calculate the kinetic energy change.
Calculate the unknown amount using the extended increasing the effectiveness theorem.
Potential Energy
Gravitational Potential Energy
The amount of energy done in displacing a body of mass (m) into the force of gravity of a source mass (M) is typically composed of potential energy when it is transported from infinity to a location inside the pull of gravity of a source surface area (M) without accelerating it. Gravitational potential energy is the term for this. The letter Ug is used to signify it.
The work done by the gravitational force is independent of the path used for a change in position, making it a conservative force. Besides, all of these forces possess some potential.
At infinity, the gravitational force on a body is zero, hence potential energy is zero, which is referred to as a reference point.
The gravitational potential energy equation is:
GPE = m.g.h
The mass in kilogrammes is denoted by m.
The letter g represents the acceleration due to gravity. (9.8 on Earth)
h is the height in metres above the ground.
Elastic Potential Energy
Elastic potential energy is the vitality produced as a result of applying a force to deform an elastic object. The energy is retained until the force is withdrawn, at which point the item returns to its original shape and does work. The item might be compressed, stretched, or twisted as a result of the deformation. Many products, for example, are designed to store elastic potential energy in a certain way:
A bent diving board just before a diver leaps.
A wind-up clock's coil spring
An archer's weapon is stretched outward.
A toy aeroplane is powered by a twisted rubber band.
A bouncing ball that has been squeezed as it bounces off a brick wall.
An item designed to hold elastic potential energy has a high elastic limit, yet all elastic items have a load limit. If an object is deformed far beyond the elastic limit, it will not return to its original shape. Wind-up mechanical timepieces powered by coil springs were prominent in prior eras.
Hooke's Law
A spring is a deformable item that may be distorted by a force and then maintain its shape after the force has been removed.
Springs occur in a wide range of shapes and sizes, but the most common is the plain metal coil spring. From ballpoint pens to race car engines, springs are an integral feature of practically all somewhat sophisticated mechanical equipment.
The form of a coil spring isn't very miraculous in terms of making it act like a spring. The elasticity, or springiness,' of the wire that the spring is formed of is a key feature of the wire. Following a stretching or twisting movement, a long straight metal wire can also 'spring back.' We can only use the qualities of a long piece of wire in a tiny space by looping it into a spring. It's a lot easier to manufacture rotary gadgets this way.
When a force is applied to a material, the substance responds by stretching or compressing.
We're all familiar with materials that stretch readily, such as rubber. What matters in mechanics is the force exerted per unit area, which is known as stress (symbol ϵ ) Engineers must pick materials for their constructions and equipment that perform reliably under predicted loads, and the intricacies of the response are vital to them.
When a little stress is applied to most materials, the strain experienced is determined by the tightness of the chemical bonds inside the substance. The chemical structure of the material and the sort of chemical bonds present are closely connected to its stiffness. When the tension is eliminated, what occurs depends on how far the atoms have been shifted. There are two types of deformation:
Deformation due to elastic forces When the tension is released, the material reverts to its original dimensions before the load was applied. The distortion is non-permanent and reversible.
When a material is subjected to a substantial amount of stress, plastic deformation happens. The tension is so great that the material does not bounce back to its original dimensions when it is removed. A permanent and irreparable deformity has occurred. The elastic limit of a material is the smallest amount of stress that causes plastic deformation.
The stress versus strain curve for many materials contains a linear area, according to 17th-century scientist Robert Hooke, who was investigating springs and elasticity. The force needed to stretch an elastic item like a metal spring is precisely proportional to the spring's extension within specified limitations. Hooke's law is typically worded as follows:
f= -kx
Despite the fact that the force's direction has not been expressly determined, the small difference is usually supplied. The restoring force of the spring is the polar opposite of the force that caused the displacement. When you pull down on a spring, it extends downward, causing the spring to exert bullish force.
In one crucial element, dier is forced. They die in the manner in which they transmit energy from one item to another. When it comes to transmitting energy "from" a moving object, all forces are constant and identical. When motion is reversed, they really lose their capacity to return energy "to" the object. When motion is reversed, one type of force conserves energy in the form of potential energy and transmits it.
Furthermore, these forces restore the energy to the item in the same proportion as it was removed from it. This indicates that when a conservative force(s) interact with an object's motion, the item's motion is reversed and the object regains its original motion. "Conservative" forces are a type of force that conserves energy for re-use. Forces that do not store energy for reuse are referred to as "nonconservative" forces.
Understanding the Properties of Conservative Force
Now that you know the conservative force definition, let us proceed to some of its known properties of it.
When you consider a loop, the work completed by a conservative force will always be zero. This is because there is no defined starting or ending point inside a loop.
You can easily reverse the work completed when it comes to a force which is conservative. Therefore, work done by a gravitational force can be easily reversed.
Lastly, the conservative force is only dependent on the final and initial position of the object. All other positions in between do not affect the final outcome.
Now that you know what is conservative force, you must also assess what non-conservative force means.
What is a Non-conservative Force?
Non-conservative force refers to any force where the path taken by the object defines the work done using that force. You can consider friction as an example of non-conservative force.
Example – 2
Suppose a body moves from point A to point B. Another body moves from point A to Point B, after visiting point C. Therefore, the second body will require greater force to complete its journey to point B.
As you can see, here, the path of a body greatly determines the force acting on it and the resulting work done.
Refer to the table below to understand the distinction between a conservative force field and a non-conservative one.
Differences Between Conservative Force and Non-conservative Force
Friction
The most prevalent nonconservative factor is a restriction, and we shall show why it is not conservative. Consider a W-pound crate on a bumpy floor. The box is pushed h meters from one end of the floor to the other, then returned to its original location. What is the status of the crate's network? Friction resists the motion of the container at all times, producing a force of kW at all times. As a result, the total work done on the journey is (- 2)(kW)(h) = - 2hwk, which is clearly not zero. Friction's network over a closed path isn't zero, and it's nonconservative.
FAQs on Conservative Force
1. What defines a force as a 'conservative force' in Class 11 Physics?
A force is defined as a conservative force if the work done by it in moving an object from one point to another depends only on the object's initial and final positions, not on the path taken between them. This means the work done is path-independent. For any conservative force, a potential energy function can be defined.
2. What are the key examples of conservative and non-conservative forces?
Understanding the examples helps clarify the concept:
- Conservative Forces: These include the Gravitational Force, Electrostatic Force, and the Elastic Force in a spring. For these forces, energy can be stored as potential energy and fully recovered.
- Non-Conservative Forces: These include Frictional Force, Air Resistance, and Viscous Force. The work done by these forces depends on the path taken, and energy is typically dissipated as heat or sound.
3. What is the fundamental difference between a conservative and a non-conservative force?
The fundamental difference lies in two key properties:
- Path Dependence: Work done by a conservative force is path-independent, whereas work done by a non-conservative force is path-dependent.
- Energy Conservation: Conservative forces conserve mechanical energy and are associated with potential energy. Non-conservative forces dissipate mechanical energy from the system, usually as heat.
4. How is the concept of potential energy exclusively linked to conservative forces?
Potential energy can only be defined for conservative forces. This is because the work done by a conservative force is stored within the system and can be fully recovered. This stored energy is what we call potential energy. The change in potential energy (ΔU) is defined as the negative of the work done (Wc) by the conservative force: ΔU = -Wc. For non-conservative forces, energy is lost from the system (e.g., as heat due to friction), so a potential energy function cannot be defined.
5. Why is the total work done by a conservative force over a closed path always zero?
The work done by a conservative force depends only on the start and end points. In a closed path, the object returns to its starting position, meaning the initial and final points are identical. Since the displacement between the identical start and end points is zero, the net work done by the conservative force must also be zero. This property is a definitive test for a conservative force.
6. Explain with an example why friction is a classic non-conservative force.
Friction is a non-conservative force because the work it does depends on the total distance travelled. Imagine pushing a book across a table from point A to B and then back to A.
- Friction always opposes the direction of motion.
- On the path from A to B, friction does negative work.
- On the return path from B to A, friction again does negative work.
The total work done is not zero; it is a negative value proportional to the total path length. This energy is lost as heat, demonstrating the dissipative and path-dependent nature of friction.
7. In a real-world system like a swinging pendulum, which forces are conservative and which are non-conservative?
In a real-world pendulum system, both types of forces are at play:
- Conservative Force: The gravitational force is the primary conservative force acting on the pendulum bob. It is responsible for the conversion between kinetic and gravitational potential energy that keeps the pendulum swinging.
- Non-Conservative Forces: Air resistance and friction at the pivot point are non-conservative forces. They do negative work on the pendulum, gradually dissipating its mechanical energy and causing the swings to decrease in amplitude until it eventually stops.
