Question

In the figure, a cable supports a mass \(m,\) and a massless bar makes an angle \(\theta\) with the horizontal. The bar is hinged to the wall. Which one of the following statements is correct? a) As \(\theta\) approaches zero, the tension \(\vec{T}\) in the cable approaches zero. b) As \(\theta\) approaches \(90^{\circ}\) the tension \(\vec{T}\) in the cable approaches zero. c) As \(\theta\) approaches zero, the tension \(\vec{T}\) in the cable approaches \(m g\). d) As \(\theta\) approaches \(90^{\circ}\), the tension \(\vec{T}\) in the cable approaches \(m g\).

Short answer

Answer: As 𝜃 approaches 0° and 90°, the tension in the cable approaches mg in both cases.

Step-by-step solution

1. Identify forces on the mass

(1) Gravitational force: \(m g\) acts downwards. (2) Tension force in the cable: \(\vec{T}\) acts diagonally opposite to the hanging mass.

2. Resolve tension force into horizontal and vertical components

As per the given information, any one of the following statements would be correct. Hence, we'll use both the components of the tension force to analyze the situation. (1) Horizontal component: \(T\cos\theta\) (2) Vertical component: \(T\sin\theta\)

3. Analyze as \(\theta\) approaches zero

As the angle \(\theta\) approaches zero, the bar becomes horizontal and the tension force in the cable would oppose only gravitational force. In this case, the vertical component of the tension force must equal the gravitational force: \(T\sin \theta = m g\) Since \(\theta\) approaches zero, \(\sin \theta\) approaches zero, meaning the vertical component of tension force approaches zero. The horizontal component, \(T\cos \theta\), must be the main contributor to balance the gravitational force, which means the tension force must approach \(m g\). This corresponds to the statement (c) in the given options.

4. Analyze as \(\theta\) approaches \(90^\circ\)

As the angle \(\theta\) approaches \(90^{\circ}\), the bar becomes vertical and the tension force in the cable would act solely in the vertical direction to balance the gravitational force. In this case, the horizontal component of the tension force would be zero: \(T\cos \theta = 0\) The vertical component of the tension force now takes the full value of the tension force, which must balance the gravitational force: \(T\sin \theta = m g\) Since \(\theta\) approaches \(90^{\circ}\), \(\sin \theta\) approaches \(1\), meaning the vertical component of tension force approaches \(m g\). Then, the tension force \(\vec{T}\) itself approaches \(m g\). This corresponds to the statement (d) in the given options. By analyzing both the cases, we conclude that the correct options are: - As \(\theta\) approaches zero, the tension \(\vec{T}\) in the cable approaches \(m g\). (Option c) - As \(\theta\) approaches \(90^{\circ}\), the tension \(\vec{T}\) in the cable approaches \(m g\). (Option d)

Key concepts

Components of Tension Force

When it comes to tension forces in physics, especially in the context of cables or ropes, understanding the components is essential for solving problems. A tension force tends to act along the length of a cable or rope. This force, however, can be broken down into two components - horizontal and vertical.

The horizontal and vertical components of the tension force are identified using trigonometric functions based on the angle of inclination. Specifically, if a tension force \( \vec{T} \) acts along a cable at an angle \( \theta \) to the horizontal, then the horizontal component is \( T\cos\theta \) and the vertical component is \( T\sin\theta \).

It's these components that effectively contribute to balancing other forces in the system, such as gravitational force. When you're solving physics tension problems, always start by resolving the tension force into its components before proceeding with applying the equilibrium conditions. This method simplifies complex situations and allows you to focus on one direction at a time.

Gravitational Force

Gravitational force is a fundamental concept not just in physics tension problems but in understanding our universe at large. This force is the attractive pull that Earth exerts on any object with mass. In your textbook problem, the gravitational force acts directly downward, represented as \( m g \) - where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity, approximately 9.81 m/s^2 on the surface of Earth.

In constructing the solution to the problem, we must consider that the gravitational force is constant and acts as the counterforce that the tension in the cable must balance. Whether the object is stationary or in motion, gravitational force is always at play, impacting how the tension force will respond to different angles of inclination.

Equilibrium of Forces

The concepts of equilibrium are central when solving physics problems, serving as the bedrock principles that dictate when an object remains stationary or moves with constant velocity. In essence, equilibrium of forces implies that all the forces acting on an object are balanced, which means the net force is zero.

For the tension problem you've encountered, the tension force must adjust itself to counteract the gravitational force such that the sum of the forces in both the vertical and horizontal components equal zero. This condition helps us determine the magnitude of tension at different angles. Furthermore, by applying this principle, you can predict the behavior of the tension force as the angle of inclination varies, ultimately reaching the correct solution.

Angle of Inclination

The angle of inclination plays a pivotal role in physics tension problems by fundamentally altering the tension force's components. As the angle between the cable and the horizontal changes, so do the proportion of the tension force components responsible for balancing the gravitational pull.

At a zero-degree angle of inclination, a cable would be perfectly horizontal, making the vertical component of tension negligible; conversely, when the angle reaches 90 degrees, the cable aligns vertically, and the tension force's vertical component maximizes to counter gravity. Grasping the relationship between the angle of inclination and the components of tension force is crucial for resolving complex physics problems involving ropes, pulleys, or cables.