Question

(a) An 11.0 kgSalami is supported by a cord that runs to a spring scale, which is supported by a cord hung from the ceiling (Figure-a). What is the reading on the scale, which is marked in weight units? (b) In Figure-bthe salami is supported by a cord that runs around a pulley and to a scale. The opposite end of the scale is attached by a cord to a wall. What is the reading on the scale? (c) In Figure-c the wall has been replaced with second11.0 kg salami, and the assembly is stationary. What is the reading on the scale?

Short answer

1. Reading on Spring Scale, in the situation (a) is 108 N.
2. Reading on Spring Scale, in the situation (b) is 108 N.
3. Reading on Spring Scale, in the situation (c) is 108 N.

Step-by-step solution

The given data

Mass of salami, m = 11 kg.

Understanding the concept of the free-body diagram

The free body diagram helps us to understand the different forces acting on the body along with their direction. It is very useful to resolve the forces and find the net force acting on the body.

By drawing the free body diagram for each situation, we can find the reading on the spring scale. All forces acting on the body are to be balanced for the body to be at equilibrium. So, the total force in a given direction is zero.

Formula:

Force acting on a body according to Newton’s second law, F = ma (i)

Here, F is the force, mis the mass and ais the acceleration of the body.

(a) Calculation of reading on a spring scale in the situation (a)

Free body diagram on salami in the situation (a):

As there is no acceleration for the section (a), using equation (i), we get the net force to be:

$$\begin{gathered} {\mathrm{F}}_{\mathrm{Net}}=\mathrm{ma} \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)-\left(\begin{array}{c}\mathrm{Weight}\mathrm{of}\\ \mathrm{salami}\end{array}\right)=0 \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)=\left(\begin{array}{c}\mathrm{Weight}\mathrm{of}\\ \mathrm{salami}\end{array}\right) \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)=11.0\mathrm{kg}\times 9.8\mathrm{m}/{\mathrm{s}}^2 \\ =107.8\mathrm{N} \\ \approx 108\mathrm{N} \end{gathered}$$

Hence, the reading for situation (a) is 108 N

(b) Calculation of reading on scale for situation (b)

Free body diagram of the spring scale

Free body diagram of salami on salami in the situation (b):

From F.B.D. of Spring Scale, we get the net force is zero for equilibrium. Hence,

$$\begin{gathered} {\mathrm{F}}_{\mathrm{Net}}=0 \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)-\left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{String}\end{array}\right)=0 \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)=\left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{String}\end{array}\right)............................\left(\mathrm{a}\right) \end{gathered}$$

From F.B.D. of Salami, the net force at equilibrium is zero. Hence,

$$\begin{gathered} {\mathrm{F}}_{\mathrm{Net}}=0 \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)-\left(\mathrm{Weight}\mathrm{of}\mathrm{the}\mathrm{salami}\right)=0 \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)=\left(\mathrm{Weight}\mathrm{of}\mathrm{the}\mathrm{salami}\right) \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)=11.0\mathrm{N}\times 9.8\mathrm{m}/{\mathrm{s}}^2 \\ =107.8\mathrm{N} \\ =108\mathrm{N} \end{gathered}$$

As we know from equation (a), we get the reading value as:

$\left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{Spring}\mathrm{Scale}\end{array}\right)=108\mathrm{N}$

Hence, the value of reading is 108 N

(c) Calculation of reading on scale in situation (c)

From F.B.D. of spring scale

Free body diagram of salami in the situation (c)

For both the salamis, the free body diagram will be the same.

As the same force is acting on both sides, we consider only the left side for calculation

$$\begin{gathered} {\mathrm{F}}_{\mathrm{Net}}=0 \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)-\left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{String}\end{array}\right)=0 \\ \left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{spring}\mathrm{scale}\end{array}\right)=\left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{String}\end{array}\right)............................\left(b\right) \end{gathered}$$

From F.B.D. of Salami, we get the net forces on it is zero. Hence,

$$\begin{gathered} {\mathrm{F}}_{\mathrm{Net}}=0 \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)-\left(\mathrm{Weight}\mathrm{of}\mathrm{the}\mathrm{salami}\right)=0 \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)=\left(\mathrm{Weight}\mathrm{of}\mathrm{the}\mathrm{salami}\right) \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)=11.0\mathrm{kg}\times 9.8\mathrm{m}/{\mathrm{s}}^2 \\ \left(\begin{array}{c}\mathrm{Tension}\mathrm{in}\mathrm{the}\\ \mathrm{Spring}\end{array}\right)=107.8\mathrm{N} \\ =108\mathrm{N} \end{gathered}$$

As we know from equation (b), we get the reading as:

$\left(\begin{array}{c}\mathrm{Reading}\mathrm{on}\\ \mathrm{Spring}\mathrm{Scale}\end{array}\right)=108\mathrm{N}$

Hence, the reading value is 108 N.