91Ó°ÊÓ

Torque is a fundamental concept in physics that describes how a force causes an object to rotate. Imagine trying to open a door – pushing at the handle makes it swing open, whereas pushing near the hinges requires much more effort. That's torque in action! It depends on three factors:

• The magnitude of the force applied
• The distance from the pivot point (also known as the lever arm)
• The angle at which the force is applied

Mathematically, torque \( (\tau) \) is expressed as \( \tau = F \times r \times \sin \theta \), where \( F \) is the force applied, \( r \) is the length of the lever arm, and \( \theta \) is the angle between the force vector and the lever arm. In our problem, the pivot is at the center of the rod, and the forces are due to the springs pulling at either end, creating a rotational effect or torque. For equilibrium, the torques from the weights and the springs must balance each other, ensuring that the net torque is zero.

The spring constant, denoted as \( k \), is characteristic of a spring's stiffness. It tells us how much force is needed to stretch (or compress) the spring by a unit distance. Think of a spring as a like a bouncy rubber band; the stiffer it is, the higher the spring constant.In the exercise, there are two springs with different spring constants, 59 N/m and 33 N/m. This difference is crucial because it causes an imbalance in the forces, influencing the tilt of the rod. Hooke's Law relates the spring constant to force and extension: \( F = k \times x \), where \( F \) is the force exerted by the spring, \( x \) is the extension, and \( k \) is the spring constant. When equilibrium is achieved, the forces from the springs match the gravitational force on the rod, playing a key role in determining the system's behavior.

Rotational equilibrium occurs when an object is not experiencing any net torque, meaning it is either at rest or rotating at a constant angular velocity. For the rod in our problem, being in rotational equilibrium implies that the torque forces generated by the springs and the weight of the rod are balanced.To find the condition of rotational equilibrium, we use the principle that the sum of clockwise torques must equal the sum of counterclockwise torques. This can be represented equation-wise as:\[ \tau_\text{clockwise} = \tau_\text{counterclockwise} \]For the rod, this took into account the weights pulling down at its center and the forces exerting upward torque at each end. Achieving rotational equilibrium is essential for determining the tilt angle of the rod because any unbalance would cause it to rotate until equilibrium is restored.

In equilibrium scenarios, understanding force balance helps us ensure that the sum of forces acting in any direction is zero. The rod hangs at rest because all forces acting vertically are perfectly balanced.Considering forces on the rod, the upward forces from the two springs (\( F_1 \) and \( F_2 \)) must counterbalance the downward gravitational force (\( m\cdot g \)). Mathematically, this is represented as:\[ m\cdot g = F_1 + F_2 \]Breaking it down:- \( m \) is the mass of the rod (1.4 kg)- \( g \) is the acceleration due to gravity (9.8 m/s²)- \( F_1 = 59x_1 \) for the spring with 59 N/m- \( F_2 = 33x_2 \) for the spring with 33 N/mBalancing these forces ensures the rod remains in place without any vertical motion, setting the stage to address rotational aspects, which ultimately define the angle of tilt.