Question

Figure 7-23 shows three arrangements of a block attached to identical springs that are in their relaxed state when the block is centered as shown. Rank the arrangements according to the magnitude of the net force on the block, largest first, when the block is displaced by distance d (a) to the right and (b) to the left. Rank the arrangements according to the work done on the block by the spring forces, greatest first, when the block is displaced by d (c) to the right and (d) to the left.

Short answer

1. Rank according to the net force acting on block it is displaced to the right is 3 > 2 > 1.
2. Rank according to the net force acting on block it is displaced to the left is 3 > 2 > 1.
3. Rank according to the work done on block when it is displaced to the right is 3 > 2 > 1.
4. Rank according to the work done on block when it is displaced to the left is 3 > 2 > 1.

Step-by-step solution

The given data

Figure shows the three arrangements of a block attached to identical springs is shown.

Understanding the concept of the force and the work done

Using the concept of the net force acting on a body attached to the spring, we can calculate the value of the ranks of the blocks' positions according to their forces for three identical springs.

Formulae:

The force acting on the block due to attached spring,

$\mathrm{F}=-\mathrm{k}.\mathrm{x}$ (i)

The work done due to an applied force,

$W=F.d$ (ii)

Calculation of the ranks of the situations due to force when block is displaced to the right

a)

When the spring is displaced by x in horizontal direction, then the spring force is given by equation (i).

For situation 1:

Here, two identical springs are attached in the system, so the net force is given using equation (i) as:

$$\begin{gathered} {\mathrm{F}}_{1\mathrm{net}}={\mathrm{F}}_{\mathrm{s}1}+{\mathrm{F}}_{\mathrm{s}1} \\ =2{\mathrm{F}}_{\mathrm{s}1} \\ =-2\mathrm{k}.\mathrm{x} \end{gathered}$$

For situation 2:

Here, three identical springs are attached in the system, so the net force is given using equation (i) as follows:

$$\begin{gathered} {\mathrm{F}}_{3\mathrm{net}}={\mathrm{F}}_{\mathrm{s}1}+{\mathrm{F}}_{\mathrm{s}1}+{\mathrm{F}}_{\mathrm{s}1} \\ =3{\mathrm{F}}_{\mathrm{s}1} \\ =-3\mathrm{k}.\mathrm{x} \end{gathered}$$

For situation 3:

Here, four identical springs are attached in the system, so the net force is given using equation (i) as:

$$\begin{gathered} {\mathrm{F}}_{1\mathrm{net}}={\mathrm{F}}_{\mathrm{s}1}+{\mathrm{F}}_{\mathrm{s}1}={\mathrm{F}}_{\mathrm{s}1} \\ =4{\mathrm{F}}_{\mathrm{s}1} \\ =-4\mathrm{k}.\mathrm{x} \end{gathered}$$

So, the rank for magnitude for force for both the directions is 3 > 2 > 1

Hence, the rank when block is displaced by the right is 3 > 2 > 1.

Calculation of the ranks of the situations due to force when block is displaced to the left

b)

From the calculations of part (a), we get that the rank is similar to both the directions.

Hence, the rank when block is displaced by the left is 3 . 2 > 1.

Calculation of the ranks of the situations due to work done when block is displaced to the right

c)

Work is done due to spring force. As displacement is the same for all cases, rank of the work is the same as the rank for spring force using equation (ii).

So, rank for the work done for both the directions is 3 > 2 > 1

Hence, the rank when block is displaced to the right is 3 > 2 > 1 .

Calculation of the ranks of the situations due to work done when block is displaced to the left

d)

From the above calculations of part (c), we get that the rank for both the directions is same.

Hence, the rank when block is displaced to the left is 3 > 2 > 1.