Question

Figure 12-15 shows three situations in which the same horizontal rod is supported by a hinge on a wall at one end and a cord at its other end. Without written calculation, rank the situations according to the magnitudes of (a) the force on the rod from the cord,

(b) the vertical force on the rod from the hinge, and

(c) the horizontal force on the rod from the hinge, greatest first.

Short answer

a) The magnitude of the force on the rod from the cord in cases (1) and (3) is the same and greater than that in case (2); thus, the rank is 1 = 3 > 2.

b) The magnitude of the vertical force on the rod from the hinge is the same for all 3 cases; thus, the rank is 1 = 2 = 3 .

c) The magnitude of the horizontal force on the rod from the hinge is the same in cases (1) and (3) and is zero in case (2); thus, the rank is 1 = 3 > 2 .

Step-by-step solution

The given data

1. The figureof the rod-cablesystem.
2. The angle made by the cord with the vertical direction in cases 1 and 3 is$\theta =50°$ .

Understanding the concept of the net force

Using the concept of balancing the forces and the torque at equilibrium, we can get the rank of the situations according to the magnitude of the forces.

Formulae:

At equilibrium, the net torque of the system is given by, ${\overrightarrow{\tau }}_{net}=0$ (i)

At equilibrium, the net force of the system is given by, ${\overrightarrow{F}}_{net}=0$ (ii)

a) Calculation of the magnitudes of the force on the rod from the cord to rank the situations

We consider the hinge point as the point of rotation. All the torques acting on the rod is due to tension in the string and the weight of the rod. In all three cases, the rod is in static equilibrium; thus, using equations (i) and (ii), we can say that,

$\tau \left(tension\right)=\tau \left(weight\right)$

The weight of the rod is acting at its center and is the same in magnitude. Thus, the torque equation tells us that torque due to tension is the same in all the cases.

But the cord makes an angle with the vertical in cases (1) and (3). Hence we understand that the torque due to the vertical component of the tension is the same that is given by, T cos$50°$ .

Since it is a component of the total tension, we know that the total tension is greater than the components in cases (1) and (3).

Hence, for cases (1) and (3), the tension in the string is the same, and it will be greater than this in case (2), and the rank can be given as 1 = 3 > 2 .

b) Calculation of the magnitudes of the vertical force on the rod from the hinge to rank the situations

We consider the hinge point as the point of rotation. In all three cases, the rod is in static equilibrium. So the torques acting on the rod due to the tension in the string and the weight of the rod are balanced, and the forces are also balanced considering equations (i) and (ii).

Thus, the vertical force from the hinge on the rod is the same in all three cases.

c) Calculation of the magnitude of the horizontal force on the rod from the hinge to rank the situations

The forces acting on the rod in the horizontal direction are the forcefrom the hingeand the horizontal component of tension in the cord. In cases (1) and (3), the tension in the string is the same. Thus, their corresponding horizontal components are also the same and can be stated using equation (i).

Thus, the horizontal force on the rod from the hinge is the same in cases (1) and (3). In case (2), there is no horizontal component of tension.

So, the horizontal force from the hinge is also zero. Hence, the rank is 1 = 3 > 2 .