13.1 Work and energy  (Page 5/5)

This limited independence may appear to be disadvantageous. A complete independence for all forces would have allowed us to analyze motion without intermediate details in all situations. However, important point here is that major forces in nature are conservative forces - gravitational and electromagnetic forces. This allows us to devise techniques to calculate work by "non-conservative force" indirectly without details, using other concepts (work - kinetic energy theory) that we are going to develop in next module.

Details of motion

Let us check out on the details of the motion. Does work depend on whether a body is moved with acceleration or without acceleration? We can have a look at the illustration, in which we raise a block slowly against gravity through a certain height. Here, net force on the block is zero. Hence, work by net force is zero.

Now, let us modify the illustration a bit. We raise the block with certain constant velocity through the same height. As velocity is constant, it means that there is no acceleration and net force on the block is zero. Hence, work by net force is zero. However, if we raise the block with some acceleration, would it affect the amount of work by net force? The presence of acceleration means net force and as such, work by net force is not zero. Greater net force will mean greater work and acceleration.

In order to get the picture, we now look at the illustration from a different perspective. What about the work by component forces like gravity or normal force as applied by the hand? Work by individual force is multiplication of force and the displacement. Since gravity is constant near the surface, work by gravity is constant for the given displacement. As a matter of fact, work by gravity is only dependent on vertical displacement.

When the block is raised slowly or with constant velocity, the net force is zero. In these circumstances, the work by normal force is equal to work by gravity. This is an important deduction. It allows us to compute work by either force without any reference to velocity and acceleration.

We see that the work by conservative force is independent of the details of motion. Under certain situations, when work by net force is zero, we can determine work by other force(s) in terms of work by conservative force. Thus, the basic idea is to make use of the features of conservative force to simplify analysis such that our consideration is independent of the details of motion.

Role of newton's laws of motion

The discussion so far leads us to identify distinguishing features of laws of motion at one hand and work-energy concepts on the other - as far as analysis of motion is concerned.

The main distinguishing feature of the application of laws of motion is that we should know the details of motion to analyze the same. This feature is both a strength and a weakness. The strength in the sense that if we have to know the details like acceleration, then we are required to analyze motion in terms of laws of motion. "Work - energy" does not provide details.

On the other hand, if we have to know the broad parameters like energy or work, then "Work - energy" provides the most elegant solution. As a matter of fact, we would find that analysis by "Work - energy" is often supplemented with analysis by laws of motion to obtain detailed results.

Clearly, when both frameworks are used in tandem, we get the best of both worlds. The example here highlights this aspect. Carefully, note how two concepts are combined to achieve the result.

Example

Problem 2 : A block of 10 kg is being pulled by a force "F" applied at an angle 45° as shown in the figure. The coefficient of kinetic friction between the surfaces is 0.5. If the block moves with constant velocity, calculate the work done by the applied force in moving the block by 1.5 m.

Solution : We had earlier made the point that calculation of work does not require force analysis. This is indeed the case when we know the force. In this case, however, we do not know force and as such we can not do away with free body diagram and coordinate system.

In order to find work by applied force, "F", we are required to know the force. We can know force by analyzing force system on the block. Here, velocity is constant. This means that the block is moving with a constant speed along a straight line. As there is no acceleration involved, the forces on the blocks are balanced. Now the free body diagram for the balanced force system is shown here :

$$\begin{array}{l}\sum F_x=F\cos \theta -{\mu }_{k}N=0\\ \Rightarrow F\cos \theta ={\mu }_{k}N\end{array}$$

and

$$\begin{array}{l}\sum F_y=N+F\sin \theta -mg=0\end{array}$$

Combining two equations, we have :

$$\begin{array}{l}F\cos \theta ={\mu }_k(mg-F\sin \theta )\end{array}$$

$$\begin{array}{l}F=\frac{{\mu }_{k}mg}{\cos \theta +{\mu }_k\sin \theta }\end{array}$$

Work done by external force, F, :

$$\begin{array}{l}\Rightarrow W=\mathbf{F}\mathbf{.}\mathbf{r}=Fr\cos {45}^0=\frac{{\mu }_{k}mgr\cos {45}^0}{\cos {45}^0+{\mu }_k\sin {45}^0}\\ \Rightarrow W=\frac{0.5x10x10x1.5x\frac{1}{\surd 2}}{\frac{1}{\surd 2}+0.5x\frac{1}{\surd 2}}\\ \Rightarrow W=50J\end{array}$$