In the previous section, we completed a discussion on kinetic energy. In this section, we will see potential energy.
We have seen some basics about potential energy in our high school classes. The following link will give those details
High school physics - Potential energy
It is recommended that the reader get a thorough understanding of those lessons before taking up this present discussion on potential energy
We have seen some basics about potential energy in our high school classes. The following link will give those details
High school physics - Potential energy
It is recommended that the reader get a thorough understanding of those lessons before taking up this present discussion on potential energy
We will write our present discussion on potential energy in steps:
1. Consider a block of mass m. It is resting on the ground
• The earth attracts it towards the center
• The force of this attraction is mg. This force acts downwards.
2. So if we want to raise the block to a height 'h' above the ground, we have to apply an upward force of mg.
• When it is raised to a height h, we can say this:
A force mg displaced the block through a distance h
■ So work done on the block = force × displacement = mg × h = mgh joules
3. This much work gets stored in the block as energy.
• The scientific method of describing this is by the following statement:
An amount of energy equal to mgh J has been added to the block-Earth system
4. We use the term 'system' because, we cannot describe the situation using the 'block' alone
• We have to use 'Earth' also
Let us see why:
♦ If there is no presence of earth, there is no gravitational force
♦ If there is no gravitational force, no force requires to be done to raise the block
♦ If there is no force, there is no work done
♦ If there is no work done, there is no energy to be stored.
• That is how the term 'system' comes in.
• We can write:
Energy stored in the block-Earth system = mgh
■ But for convenience, we usually write:
An energy of mgh joules is stored in the block
5. Also note that 'h' gives the separation between the two members of the system
• If there is no separation, that is., if h is zero, then there is no energy to be stored
• The level at which 'h = 0' is taken as the datum. So we take the 'surface of the earth' as the datum
6. The energy stored in this way is called gravitational potential energy
• We denote kinetic energy using the letter 'K'
• To denote potential energy, we use the letter 'P'
• Gravitational potential energy is denoted using an additional subscript 'g'
■ So we can write:
Pg stored in the block = mgh joules
Now we consider another important aspect. We will write it in steps:
1. When we push a block on an ordinary floor, the original force is the F that we apply on the block
• The frictional force f will be acting in the opposite direction
2. In the previous sections, we have seen the following two points related to this situation:
(i) The work done by F is stored as positive energy
(ii) The work done by f is negative energy
♦ That is., it takes away 'some energy' from the energy created by F
3. In our present case also, we encounter a similar situation
• The original force is the force of gravity
• With out that force, there is no potential energy at all
4. When we raise the block, we are applying a force against the original force
• So the work done in raising the block is negative energy
■ We can write: The energy stored in the block at a height h = -mgh
A mathematical explanation can be given in 4 steps as follows:
1. The magnitude of the force $\mathbf\small{\vec{F}}$ which does the work is mg
• The direction of this $\mathbf\small{\vec{F}}$ is downwards
2. The magnitude of the displacement $\mathbf\small{\vec{d}}$ is h
• The direction of this $\mathbf\small{\vec{d}}$ is upwards
3. Since the directions are opposite to each other, the angle θ between the two vectors is 180o
4. We have: Work done = $\mathbf\small{\vec{F}.\vec{d}=|\vec{F}|\times |\vec{d}|\times \cos \theta}$
• Substituting the known values, we get:
Work done = mg×h×cos 180 = mg×h×-1 = -mgh
• Level-0 is same as the surface level of earth
• We can do 3 different experiments
■ Experiment 1
• Consider a block of mass m on level-2
• With the block in level-2, we can think of 2 possible cases
Case 1
If the block is allowed to fall freely onto level-1, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mgh1 + (-mgh2) = -mgh1 - mgh2 = -mg(h1+h2)
2. Final potential energy Pgf = -mgh1
3. So energy released = Pgf - Pgi = -mgh1-[-mg(h1+h2)] = -mgh1+ mg(h1+h2) = mgh2
• Let us analyse the above result:
(i) The block was dropped from level-2 to level-1
(ii) The energy released in (3) is equal (in magnitude) to the work required to raise the block from level-1 to level-2
(iii)Also the energy released is positive. This is obvious because:
♦ Gravity pulls the block downwards. That means, gravity does some work on the block
♦ The displacement of the block is in the same direction as the direction of gravity
♦ So the work done by gravity (that is., the energy released) is positive
Case 2
If the block is allowed to fall freely onto level-0, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mgh1 + (-mgh2) = -mgh1 - mgh2 = -mg(h1+h2)
2. Final potential energy Pgf = 0
3. So energy released = Pgf - Pgi = 0-[-mg(h1+h2)] = mg(h1+h2)
• Clearly, mg(h1+h2) is greater in magnitude than mgh2
• That means, greater energy is released in case 2
■ Experiment 2
case 1:
• Let the block be on level-3
• If it is allowed to fall freely onto level-2, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mg(h1+h2+h3)
2. Final potential energy Pgf = -mg(h1+h2)
3. So energy released = Pgf - Pgi = mgh3
case 2:
• Let the block be on level-2
• If it is allowed to fall freely onto level-1, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mg(h1+h2)
2. Final potential energy Pgf = -mgh1
3. So energy released = Pgf - Pgi = -mgh1+mgh1+mgh2 = mgh2
• Clearly, more energy is released in case 2 because h2 > h3
• If h2 = h3, then the energy released is same in both cases
■ So if h2= h3, we see an interesting fact:
• The block has greater Pg when it is in the third floor.
But if h2 = h3, the following two energies are same:
(I) Energy released from level-3 to level-2
(ii) Energy released from level-2 to level-1 floor
• So an object being at a higher level does not guarantee that we can extract greater energy from it.
• The quantity of energy extracted will depend on initial and final positions
■ Experiment 3
• Let the block be on level-0
• In normal cases, we cannot make the block to fall freely from the level-0
• But in fig.6.21, there is a pit nearby
• If the block is allowed to fall from level-0 into the pit, how much energy will be released?
Solution:
1. Initial potential energy Pgi = 0
2. Final potential energy Pgf = mgh4
• Here the potential energy is positive because when we lower the block into the pit, the displacement is in the same direction as that of the gravitational force
3. So energy released = Pgf - Pgi = mgh4-0 = mgh4
• Note that, in this experiment also, 'energy released' is positive
(i) If the object is at a greater distance from the surface of the earth, it's Pg will be high
(ii) If the object is at a lower distance from the surface of the earth, it's Pg will be low
(iii) But the Pg released depends upon the initial and final positions of the object
Let us do one more experiment
■ Experiment 4:
• Let the block be on level-3
• It is brought down to level-1 by two paths
Path 1:
• Block is first taken from level-3 to level-0
• Then, from level-0, it is taken to the final position which is level-1
Path 2:
• Block is lowered directly from the third floor to the first floor
■ Let us analyze both the paths:
• Analysis of Path 1:
1. This path consists of two segments:
♦ Segment 1: Level-3 to Level-0
♦ Segment 2: Level-0 to Level-1
2. At the end of segment 1, the block would have lost a large quantity of energy
• If we take the earth surface as datum, the energy at the end of segment 1 is zero
3. At the end of segment 2, the block would have gained a potential energy of mgh1
• But 'end of segment 2' is the 'end of path 1'
• That means, the final energy of the block if it takes path 1 is mgh1
• Analysis of Path 2:
1. In this path, there is only one segment
• Block is lowered directly from level-3 to level-1
2. When the object is at level-1, we know that, it's energy is mgh1.
• So in this path also. the Pg at final position is mgh1.
■ So from this experiment 4, it is clear that, the final value of Pg does not depend on the path
■ Also note that, block could be moved straight up or straight down using pulleys and ropes
• It could also be moved 'by person' who takes long winding steps
■ What ever be the method, the 'final energy stored in the block' (or 'work done on the block') will depend on it's final level.
• It does not depend on the path taken
• An example for conservative force is the 'force of gravity'
• 'Spring force' is another example for conservative force. We will see this force in detail in a later section of this chapter
2. When it is released, it will be in free fall. Gravity will do work on it
• We know that, the amount of work that will be done by gravity is mgh joules
• This is positive because, work is done by gravity and so 'direction of displacement' is same as 'direction of force'
3. In fig.b, the same block is at the top of a smooth inclined side of a wedge ABC
• The height BC of the wedge is the same ‘h’ in fig.a
♦ That means the block in fig.b is at the same height as in fig.a
• The angle of inclination of the plane is θ
4. When released, the block will slide down along the inclined plane
■ What is the work done by gravity in this case?
• We know that, 'a component of the gravitational force' will be pulling the block down
• Also we know that this component is mg sinθ (Details here)
5. Since the plane is smooth, there is no friction to oppose the motion
• So mg sinθ will be doing work on the block for a distance of AC
6. Work done by mg sinθ = Force × Distance = mg sinθ × AC
• We have: $\mathbf\small{\sin \theta=\frac{BC}{AC}}$
• So $\mathbf\small{AC=\frac{BC}{\sin \theta}=\frac{h}{\sin \theta}}$
7. Thus work done by mg sinθ = $\mathbf\small{mg \sin \theta \times \frac{h}{\sin \theta}=mgh}$
■ So work done by gravity is the same in both figs. a and b
Another example:
• In fig.6.23 below, the x axis lies on a horizontal surface of the earth. The y axis is perpendicular to the surface of the earth
• That means, the x-y plane in fig.6.23 is a vertical plane
• A block of mass m is to be taken from point A to point B
• It can be taken by two paths
♦ Path 1 is APB shown in yellow colour
♦ Path 2 is AQRB shown in cyan colour
• Let us analyze each path
Analysis of path 1:
1. Work done by gravity on the block when it is taken from A to P is zero (Details here)
• Work done by gravity on the block when it is taken from P to B is mgh = -5mg
2. So total work done by gravity on the block when it is taken from A to B along the path 1 = 0+ (-5mg) = -5mg
Analysis of path 2:
1. Work done by gravity on the block when it is taken from A to Q is mgh = -7mg
• Work done by gravity on the block when it is taken from Q to R is zero
• Work done by gravity on the block when it is taken from R to B is mgh = 2mg
2. So total work done by gravity on the block when it is taken from A to B along the path 2 = (-7mg) + 0 + (2mg) = -5mg
■ So work done by gravity is same along both paths 1 and 2
1. Consider a block of mass m. It is resting on the ground
• The earth attracts it towards the center
• The force of this attraction is mg. This force acts downwards.
2. So if we want to raise the block to a height 'h' above the ground, we have to apply an upward force of mg.
• When it is raised to a height h, we can say this:
A force mg displaced the block through a distance h
■ So work done on the block = force × displacement = mg × h = mgh joules
3. This much work gets stored in the block as energy.
• The scientific method of describing this is by the following statement:
An amount of energy equal to mgh J has been added to the block-Earth system
4. We use the term 'system' because, we cannot describe the situation using the 'block' alone
• We have to use 'Earth' also
Let us see why:
♦ If there is no presence of earth, there is no gravitational force
♦ If there is no gravitational force, no force requires to be done to raise the block
♦ If there is no force, there is no work done
♦ If there is no work done, there is no energy to be stored.
• That is how the term 'system' comes in.
• We can write:
Energy stored in the block-Earth system = mgh
■ But for convenience, we usually write:
An energy of mgh joules is stored in the block
5. Also note that 'h' gives the separation between the two members of the system
• If there is no separation, that is., if h is zero, then there is no energy to be stored
• The level at which 'h = 0' is taken as the datum. So we take the 'surface of the earth' as the datum
6. The energy stored in this way is called gravitational potential energy
• We denote kinetic energy using the letter 'K'
• To denote potential energy, we use the letter 'P'
• Gravitational potential energy is denoted using an additional subscript 'g'
■ So we can write:
Pg stored in the block = mgh joules
Now we consider another important aspect. We will write it in steps:
1. When we push a block on an ordinary floor, the original force is the F that we apply on the block
• The frictional force f will be acting in the opposite direction
2. In the previous sections, we have seen the following two points related to this situation:
(i) The work done by F is stored as positive energy
(ii) The work done by f is negative energy
♦ That is., it takes away 'some energy' from the energy created by F
3. In our present case also, we encounter a similar situation
• The original force is the force of gravity
• With out that force, there is no potential energy at all
4. When we raise the block, we are applying a force against the original force
• So the work done in raising the block is negative energy
■ We can write: The energy stored in the block at a height h = -mgh
A mathematical explanation can be given in 4 steps as follows:
1. The magnitude of the force $\mathbf\small{\vec{F}}$ which does the work is mg
• The direction of this $\mathbf\small{\vec{F}}$ is downwards
2. The magnitude of the displacement $\mathbf\small{\vec{d}}$ is h
• The direction of this $\mathbf\small{\vec{d}}$ is upwards
3. Since the directions are opposite to each other, the angle θ between the two vectors is 180o
4. We have: Work done = $\mathbf\small{\vec{F}.\vec{d}=|\vec{F}|\times |\vec{d}|\times \cos \theta}$
• Substituting the known values, we get:
Work done = mg×h×cos 180 = mg×h×-1 = -mgh
Differences in Gravitational Potential Energy
• Consider different levels in a hill shown in fig.6.21 below: |
| Fig.6.21 |
• We can do 3 different experiments
■ Experiment 1
• Consider a block of mass m on level-2
• With the block in level-2, we can think of 2 possible cases
Case 1
If the block is allowed to fall freely onto level-1, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mgh1 + (-mgh2) = -mgh1 - mgh2 = -mg(h1+h2)
2. Final potential energy Pgf = -mgh1
3. So energy released = Pgf - Pgi = -mgh1-[-mg(h1+h2)] = -mgh1+ mg(h1+h2) = mgh2
• Let us analyse the above result:
(i) The block was dropped from level-2 to level-1
(ii) The energy released in (3) is equal (in magnitude) to the work required to raise the block from level-1 to level-2
(iii)Also the energy released is positive. This is obvious because:
♦ Gravity pulls the block downwards. That means, gravity does some work on the block
♦ The displacement of the block is in the same direction as the direction of gravity
♦ So the work done by gravity (that is., the energy released) is positive
Case 2
If the block is allowed to fall freely onto level-0, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mgh1 + (-mgh2) = -mgh1 - mgh2 = -mg(h1+h2)
2. Final potential energy Pgf = 0
3. So energy released = Pgf - Pgi = 0-[-mg(h1+h2)] = mg(h1+h2)
• Clearly, mg(h1+h2) is greater in magnitude than mgh2
• That means, greater energy is released in case 2
■ Experiment 2
case 1:
• Let the block be on level-3
• If it is allowed to fall freely onto level-2, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mg(h1+h2+h3)
2. Final potential energy Pgf = -mg(h1+h2)
3. So energy released = Pgf - Pgi = mgh3
case 2:
• Let the block be on level-2
• If it is allowed to fall freely onto level-1, how much energy will be released?
Solution:
1. Initial potential energy Pgi = -mg(h1+h2)
2. Final potential energy Pgf = -mgh1
3. So energy released = Pgf - Pgi = -mgh1+mgh1+mgh2 = mgh2
• Clearly, more energy is released in case 2 because h2 > h3
• If h2 = h3, then the energy released is same in both cases
■ So if h2= h3, we see an interesting fact:
• The block has greater Pg when it is in the third floor.
But if h2 = h3, the following two energies are same:
(I) Energy released from level-3 to level-2
(ii) Energy released from level-2 to level-1 floor
• So an object being at a higher level does not guarantee that we can extract greater energy from it.
• The quantity of energy extracted will depend on initial and final positions
■ Experiment 3
• Let the block be on level-0
• In normal cases, we cannot make the block to fall freely from the level-0
• But in fig.6.21, there is a pit nearby
• If the block is allowed to fall from level-0 into the pit, how much energy will be released?
Solution:
1. Initial potential energy Pgi = 0
2. Final potential energy Pgf = mgh4
• Here the potential energy is positive because when we lower the block into the pit, the displacement is in the same direction as that of the gravitational force
3. So energy released = Pgf - Pgi = mgh4-0 = mgh4
• Note that, in this experiment also, 'energy released' is positive
■ So from the above 3 experiments, the following 3 facts can be written:
(ii) If the object is at a lower distance from the surface of the earth, it's Pg will be low
(iii) But the Pg released depends upon the initial and final positions of the object
Let us do one more experiment
■ Experiment 4:
• Let the block be on level-3
• It is brought down to level-1 by two paths
Path 1:
• Block is first taken from level-3 to level-0
• Then, from level-0, it is taken to the final position which is level-1
Path 2:
• Block is lowered directly from the third floor to the first floor
■ Let us analyze both the paths:
• Analysis of Path 1:
1. This path consists of two segments:
♦ Segment 1: Level-3 to Level-0
♦ Segment 2: Level-0 to Level-1
2. At the end of segment 1, the block would have lost a large quantity of energy
• If we take the earth surface as datum, the energy at the end of segment 1 is zero
3. At the end of segment 2, the block would have gained a potential energy of mgh1
• But 'end of segment 2' is the 'end of path 1'
• That means, the final energy of the block if it takes path 1 is mgh1
• Analysis of Path 2:
1. In this path, there is only one segment
• Block is lowered directly from level-3 to level-1
2. When the object is at level-1, we know that, it's energy is mgh1.
• So in this path also. the Pg at final position is mgh1.
■ So from this experiment 4, it is clear that, the final value of Pg does not depend on the path
■ Also note that, block could be moved straight up or straight down using pulleys and ropes
• It could also be moved 'by person' who takes long winding steps
■ What ever be the method, the 'final energy stored in the block' (or 'work done on the block') will depend on it's final level.
• It does not depend on the path taken
■ If the work done by a force on an object does not depend on the path taken by the object, then that force is called a conservative force
• 'Spring force' is another example for conservative force. We will see this force in detail in a later section of this chapter
Let us prove that 'force of gravity' is a conservative force:
1. In fig.6.22(a) below, a block of mass m is at a height h.
1. In fig.6.22(a) below, a block of mass m is at a height h.
 |
| Fig.6.22 |
• We know that, the amount of work that will be done by gravity is mgh joules
• This is positive because, work is done by gravity and so 'direction of displacement' is same as 'direction of force'
3. In fig.b, the same block is at the top of a smooth inclined side of a wedge ABC
• The height BC of the wedge is the same ‘h’ in fig.a
♦ That means the block in fig.b is at the same height as in fig.a
• The angle of inclination of the plane is θ
4. When released, the block will slide down along the inclined plane
■ What is the work done by gravity in this case?
• We know that, 'a component of the gravitational force' will be pulling the block down
• Also we know that this component is mg sinθ (Details here)
5. Since the plane is smooth, there is no friction to oppose the motion
• So mg sinθ will be doing work on the block for a distance of AC
6. Work done by mg sinθ = Force × Distance = mg sinθ × AC
• We have: $\mathbf\small{\sin \theta=\frac{BC}{AC}}$
• So $\mathbf\small{AC=\frac{BC}{\sin \theta}=\frac{h}{\sin \theta}}$
7. Thus work done by mg sinθ = $\mathbf\small{mg \sin \theta \times \frac{h}{\sin \theta}=mgh}$
■ So work done by gravity is the same in both figs. a and b
Another example:
• In fig.6.23 below, the x axis lies on a horizontal surface of the earth. The y axis is perpendicular to the surface of the earth
 |
| Fig.6.23 |
• A block of mass m is to be taken from point A to point B
• It can be taken by two paths
♦ Path 1 is APB shown in yellow colour
♦ Path 2 is AQRB shown in cyan colour
• Let us analyze each path
Analysis of path 1:
1. Work done by gravity on the block when it is taken from A to P is zero (Details here)
• Work done by gravity on the block when it is taken from P to B is mgh = -5mg
2. So total work done by gravity on the block when it is taken from A to B along the path 1 = 0+ (-5mg) = -5mg
Analysis of path 2:
1. Work done by gravity on the block when it is taken from A to Q is mgh = -7mg
• Work done by gravity on the block when it is taken from Q to R is zero
• Work done by gravity on the block when it is taken from R to B is mgh = 2mg
2. So total work done by gravity on the block when it is taken from A to B along the path 2 = (-7mg) + 0 + (2mg) = -5mg
■ So work done by gravity is same along both paths 1 and 2
■ A force is conservative if the net work that it does on an object when that object moves in a closed pathway is zero
• Let us see whether this definition is applicable to gravity. We will write the steps:
1. We can use the same fig.6.23
• This time, assume that there is only one path
2. The block is taken from A. It moves along the closed path APBRQA
• That means, the block returns to the original position
• In other words, the initial and final positions of the block, are the same
3. We want to find the work done by gravity on the block when it moves along APBRQA
• Work done by gravity on the block when it is taken from A to P is zero
• Work done by gravity on the block when it is taken from P to B is mgh = -5mg
• Work done by gravity on the block when it is taken from B to R is mgh = -2mg
• Work done by gravity on the block when it is taken from R to Q is zero
• Work done by gravity on the block when it is taken from Q to A is mgh = 7mg
4. So total work done by gravity on the block when it is taken along the path APBRQA
= 0 + (-5mg) + (-2mg) + 0 +7mg = 0
■ So gravity is indeed a conservative force
1. In fig.6.24 below, the x axis lies on a horizontal surface of the earth. The y axis also lies on the same horizontal surface
• That means, the x-y plane in fig.b is a horizontal plane
• It lies on the surface of the earth
2. This surface is an ordinary surface. It is not very smooth. So there will be friction
3. A block of mass m is to be pushed on the ground from point M to point N
4. It is taken by two paths
♦ Path 1 is MPN
♦ Path 2 is MQRN
[Assume that the frictional force acting on the block is 3 N]
5. Let us analyze each path
Analysis of path 1:
• Work done by friction on the block when it is taken from M to P
= Force × Distance = -5 × 6 = -30 N
• Work done by friction on the block when it is taken from P to N
= Force × Distance = -5 × 4 = -20 N
• All the above works are negative because, the frictional force is always opposite to the direction of displacement, no matter whether the block is pushed leftward/rightward or forward/backward
Analysis of path 2:
• Work done by friction on the block when it is taken from M to Q
= Force × Distance = -5 × 8 = -40 N
• Work done by friction on the block when it is taken from Q to R
= Force × Distance = -5 × 6 = -30 N
• Work done by friction on the block when it is taken from R to N
= Force × Distance = -5 × 4 = -20 N
• All the above works are negative because, the frictional force is always opposite to the direction of displacement, no matter whether the block is pushed leftward/rightward or forward/backward
■ We see that, the works done are not equal for the two paths. So friction is not a conservative force
■ Also it is clear that, if the block is pushed through the complete circuit MPNRQM, the total work done will not be zero. It will be: -50 + (-90) = -140 joules
■ The work done by friction, air resistance etc., are not available for later use. They are dissipated in the form of heat, sound etc.,
• Let us see whether this definition is applicable to gravity. We will write the steps:
1. We can use the same fig.6.23
• This time, assume that there is only one path
2. The block is taken from A. It moves along the closed path APBRQA
• That means, the block returns to the original position
• In other words, the initial and final positions of the block, are the same
3. We want to find the work done by gravity on the block when it moves along APBRQA
• Work done by gravity on the block when it is taken from A to P is zero
• Work done by gravity on the block when it is taken from P to B is mgh = -5mg
• Work done by gravity on the block when it is taken from B to R is mgh = -2mg
• Work done by gravity on the block when it is taken from R to Q is zero
• Work done by gravity on the block when it is taken from Q to A is mgh = 7mg
4. So total work done by gravity on the block when it is taken along the path APBRQA
= 0 + (-5mg) + (-2mg) + 0 +7mg = 0
■ So gravity is indeed a conservative force
Let us apply this definition to friction force and see whether it is conservative or not. We will write the steps:
 |
| Fig.6.24 |
• It lies on the surface of the earth
2. This surface is an ordinary surface. It is not very smooth. So there will be friction
3. A block of mass m is to be pushed on the ground from point M to point N
4. It is taken by two paths
♦ Path 1 is MPN
♦ Path 2 is MQRN
[Assume that the frictional force acting on the block is 3 N]
5. Let us analyze each path
Analysis of path 1:
• Work done by friction on the block when it is taken from M to P
= Force × Distance = -5 × 6 = -30 N
• Work done by friction on the block when it is taken from P to N
= Force × Distance = -5 × 4 = -20 N
• All the above works are negative because, the frictional force is always opposite to the direction of displacement, no matter whether the block is pushed leftward/rightward or forward/backward
• So total work done by friction on the block when it is taken along the path MPN
= -30+(-20) = -50 NAnalysis of path 2:
• Work done by friction on the block when it is taken from M to Q
= Force × Distance = -5 × 8 = -40 N
• Work done by friction on the block when it is taken from Q to R
= Force × Distance = -5 × 6 = -30 N
• Work done by friction on the block when it is taken from R to N
= Force × Distance = -5 × 4 = -20 N
• All the above works are negative because, the frictional force is always opposite to the direction of displacement, no matter whether the block is pushed leftward/rightward or forward/backward
• So total work done by friction on the block when it is taken along the path MPN
= -40+(-30)+(-20) = -90 N■ We see that, the works done are not equal for the two paths. So friction is not a conservative force
■ Also it is clear that, if the block is pushed through the complete circuit MPNRQM, the total work done will not be zero. It will be: -50 + (-90) = -140 joules
■ The word 'conservative' is very appropriate because, the work done by forces like gravity, spring force etc., are conserved. That is., the work is stored as energy and is available for later use
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