Chapter 12, Problem 1Q

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.

Figure 12-15 Question 1.

To determine

To rank:

a) The situations according to the magnitudes of the force on the rod from the cord.

b) The situations according to the magnitudes of the vertical force on the rod from the hinge.

c) The situations according to the magnitudes of the horizontal force on the rod form the hinge.

Answer to Problem 1Q

Solution:

a) Magnitudes of force on the rod from the cord in case (1) and (3) is same and greater than that in case (2).

b) Magnitudes of the vertical force on the rod from the hinge is same for all 3 cases.

c) Magnitudes of the horizontal force on the rod form the hinge is same in cases (1) and (3) and is zero in case (2)

Explanation of Solution

1) Concept:

We can use the concept of balancing of forces and torque at equilibrium to rank the situations according to the magnitude of the forces.

2) Formulae:

At equilibrium,

i. $\overrightarrow{{\tau }_{net}}=0$

ii. $\overrightarrow{F_{net}}=0$

3) Given:

i. The figure of rod-cable system.

ii. The angle made by the cord with the vertical direction in case 1 and 3 is 50⁰

4) Calculation:

a) We consider the hinge point as the point of rotation. The torques acting on the rod are due to tension in the string and the weight of the rod. In all the three cases, the rod is in static equilibrium, hence:

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

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

But the cord is making an angle with the vertical in cases (1) and (3). Hence we understand that the torque due to vertical component of the tension (T cos 50^o) is the same. 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).Thus, for cases (1) and (3) the tension in the string is same and it will be greater than this in case (2).

b) We consider the hinge point as the point of rotation. In all the 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.

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

c) The forces acting on the rod in the horizontal direction are the force from hinge and the horizontal component of tension in the cord. In cases (1) and (3), the tension in the string is the same. Hence their corresponding horizontal components are also the same.

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, hence the horizontal force from the hinge is also zero.

Conclusion:

The rod is in static equilibrium in all three cases. From the balancing conditions for torque and forces, we can determine the magnitudes of the forces acting on the rod.