One implication of external force on the system is that we are dealing with special “closed” system, which permits exchange of energy with surrounding via “work by external force” only. The form of conservation law for the mechanical process is :

However, we pointed out that we shall limit change in thermal energy to “heat energy”, which is equal to the negative of work by friction only. Hence, we can rewrite the equation as :

In words, we can put the conservation of energy for mechanical process as :

“Work by external force on the system and work by friction within the system is equal to the change in potential and kinetic energy of the system.”

**Example**

Problem 1: Two blocks of 1 kg and 2 kg are connected by a spring as shown in the figure. If spring constant is 500 N/m and coefficient of static and kinetic friction between surfaces are each 0.5, then what minimum constant horizontal force, F, (in Newton) is required to just initiate block on the left ? (Consider g = 10

Block spring system |
---|

Solution : The system consists of two blocks and one spring. The system is subjected to an external force. Thus, system is “closed” system, which allows energy exchange with system via work by external force. As friction is also involved, the corresponding energy statement is :

We are required to find minimum force. We need to understand the implication of this phrase. It is clear that we can apply external force in such a manner that block of “2 kg” acquires kinetic energy by the time block of “1 kg” is initiated in motion. Alternatively as a base case, we can apply external force gradually and slowly in increasing magnitude till the block of “1 kg” is initiated in motion. In this case, the block of “2 kg” does not acquire kinetic energy. This mode of application of external force represents the situation when minimum external force will be required to initiate block of “1 kg”. Hence,

Motion of block is constrained in horizontal direction. There is no change of vertical elevation. Hence, there is no change in gravitational potential energy. However, spring is stretched from its neutral state. As a consequence, there is change in elastic potential energy of the spring. Let “x” be the extension in the spring by the time block of “1 kg” is initiated in motion. Then, the total change in potential energy is :

The work by friction is done only on the right block of “2 kg” :

On the other hand, the work by external force is :

Putting all these values in the equation of conservation of energy :

Clearly, we can not solve this equation as there are two unknowns, “F” and “x”. We, therefore, make use of the fact that spring force on the block of mass “1 kg” is equal to maximum static friction,

Forces on the block |
---|

Combining two equations, we have :