Problem 48 - Master Resource Book in JEE Main Physics

Question

A plumb line is suspended from a ceiling of a car moving with horizontal acceleration of \(a\). What will be the angle of inclination with vertical? (a) \(\tan ^{-1}(a / g)\) (b) \(\tan ^{-1}(g / a)\) (c) \(\cos ^{-1}(a / g)\) (d) \(\cos ^{-1}(g / a)\)

Step-by-Step Solution

Answer

The angle of inclination with vertical is (a) \( \tan^{-1}\left(\frac{a}{g}\right) \).

Step 1: Understand the Problem

A plumb line is a weight suspended directly for measuring verticality. Due to the acceleration of the car, the plumb line inclines forming an angle with the vertical.

Step 2: Identify Forces Acting on the Plumb Line

The plumb line is subject to two forces: gravitational force acting downward and inertial force due to horizontal acceleration acting to the left.

Step 3: Setup the Problem

In the non-inertial frame of the car, the plumb line experiences a vertical gravitational force, \( mg \), and a horizontal pseudo-force, \( ma \), where \( g \) is the acceleration due to gravity.

Step 4: Apply Trigonometry to Find the Angle

The angle \( \theta \) of the plumb line with the vertical can be calculated using the tangent ratio: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{g} \).

Step 5: Solve for the Angle of Inclination

Using the tangent inverse function, the angle \( \theta \) is given by \( \theta = \tan^{-1}\left(\frac{a}{g}\right) \).

Step 6: Identify the Correct Answer Choice

Comparing the derived expression \( \tan^{-1}\left(\frac{a}{g}\right) \) with options provided, the correct answer is option (a).

Key Concepts

Inclined Angle Mechanics

Inclined angle mechanics comes into play when objects are subjected to forces that cause them to tilt or incline with respect to the vertical. In the context of a plumb line in a moving car, this concept is crucial to determine the angle of inclination that the line forms due to external forces.

• The plumb line tends to remain vertical under normal conditions.
• However, when the car accelerates, the plumb line inclines, forming an angle with the vertical axis.
• This inclination is caused due to the combined action of inertial force and gravitational force acting on the plumb line.

To find the angle, we analyze the forces involved and apply basic trigonometry. As the car accelerates horizontally, the line tilts until the forces reach equilibrium, resulting in a stable angle determined by both gravitational and inertial forces. By understanding the components of these forces, students can grasp how inclined angle mechanics apply to scenarios with moving vehicles.

Non-Inertial Reference Frame

A non-inertial reference frame is a viewpoint where observers are subject to apparent forces because the frame itself is accelerating. When the car moves with horizontal acceleration, it becomes a non-inertial reference frame for any object inside it, like the plumb line.

• This is different from an inertial reference frame, where Newton's laws apply without modification.
• In a non-inertial frame, objects appear to be acted upon by pseudo-forces due to the acceleration of the frame.
• For the plumb line, the pseudo-force acts horizontally, as though a force pushes it sideways.

Understanding non-inertial reference frames helps explain why the plumb line inclines as the car accelerates. Observers inside the moving car witness the plumb line tilted at an angle, indicative of a balance between real gravitational force and perceived inertial force presented in this accelerating frame.

Inertial and Gravitational Forces

Analyzing the plumb line's behavior requires comprehension of inertial and gravitational forces, which are pivotal in determining its angle of inclination.

• Gravitational force (\( mg \)) acts vertically downwards on the plumb line, influencing it to hang straight down when undisturbed.
• Inertial force, often called pseudo-force in a non-inertial frame, appears due to the car's acceleration and acts horizontally (\( ma \)).
• The combination of these forces results in the plumb line tilting with an angle calculated using trigonometric principles.

The interplay of these two fundamental forces determines the inclination angle. The tangent of the angle (\( \theta \)) formed by the plumb line with the vertical can be found using \( \tan(\theta) = \frac{a}{g} \), clearly showing the ratio of inertial to gravitational influences. This balanced setup allows students to solve for the precise inclination using inverse trigonometric functions, deepening their understanding of dynamics in non-stationary environments.