8.11 Unbalanced force system  (Page 2/3)

Let us work out a problem here to illustrate this point about the composite mass system.

Problem : Two blocks A and B of masses 10 kg and 20 kg respectively, connected by a string, are placed on a smooth surface. The blocks are pulled by a horizontal force of 300 N as shown in the figure. Find (i) acceleration of the blocks (ii) tension in the string and determine (iii) whether magnitudes of acceleration and tension change when force is applied on other mass? Consider, $g=10m/s^2$ .

Solution : Let us first investigate acceleration and tension for the given configuration in the figure above. Here, we see that two masses have same acceleration. We can use this opportunity to apply the concept of composite body system (comprising of both blocks). We neglect tension in the string as it constitutes internal force for the whole composite body system.

Let “a” be the acceleration in the direction of force. The mass of the composite body system, $m_c$ , is :

$$\begin{array}{l}{m}_c=10+20=30\mathrm{kg}\end{array}$$

$\text{Free body diagram of the composite body system}$

$$\begin{array}{l}\sum F_x=F=m{a}_x\\ \Rightarrow 300=30a\\ \Rightarrow a=10m/s^2\end{array}$$

Note that this force equation does not depend on whether we apply force on A or B. In either case, the force equation remains same – only direction of acceleration changes with the change in the direction of applied force. It means that magnitude of acceleration of the two body system will remain same in either case.

In order to find the tension in the string, however, we need to consider individual block. Here, we consider block "A" for the simple reason that it is acted upon by a single force (T) in x-direction and analysis of force on block "A" will be simpler than that of block "B", which is acted by two forces (T and 300 N force).

$\text{Free body diagram of the body A}$

$$\begin{array}{l}\sum F_x=T=m{a}_x\\ \Rightarrow T=10a\\ \Rightarrow T=10x10=100N\end{array}$$

Now, we answer the third part of the question : whether magnitudes of acceleration and tension change when force is applied on other mass? Let us consider the case when force is applied on body “A” as shown in the figure.

Now, we consider force and acceleration of block "B" as it is now acted by only one force i.e. tension in the string in x-direction.

$\text{Free body diagram of the body B}$

$$\begin{array}{l}\sum F_x=T=m{a}_x\\ \Rightarrow T=20a\\ \Rightarrow T=20x10=200N\end{array}$$

Thus, we see that acceleration (a) is independent, but tension (T) in the string is dependent on the point of application of the external force.

Time dependent force

The force as applied to a body may change with time. The change may occur in any combination of magnitude and direction of the force. The resulting acceleration will accordingly vary with time as given by force equation. In turn, the rate of change of velocity will change.

In a simplified scheme of thing, the velocity profile of a motion due to a particular variable force may look like the one shown here.

The velocity in first two seconds is increasing at constant rate, meaning that the body is under constant acceleration. The body is, thus, acted upon by a constant net force during this period. Subsequently, velocity is constant for time between 2s and 4s. Acceleration and hence force on the body , therefore, is zero in this interval. Finally, body is brought to rest with a constant force acting in opposite direction to that of velocity.