Problem 35 - Master Resource Book in JEE Main Physics

Question

An aircraft executes a horizontal loop of radius \(1 \mathrm{~km}\) with a speed of \(900 \mathrm{~km} / \mathrm{h}\). Compare its centripetal acceleration with the acceleration due to gravity. (a) 6 (b) 7 (c) \(\underline{8}\) (d) 5

Step-by-Step Solution

Answer

The closest approximation is (c) 8.

Step 1: Understand the formula for centripetal acceleration

The formula for centripetal acceleration is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the speed of the object and \( r \) is the radius of the circular path. Our goal is to calculate \( a_c \) and compare it to gravitational acceleration, \( g \), which is approximately \( 9.8 \text{ m/s}^2 \).

Step 2: Convert units

The speed of the aircraft is given in km/h, so we must convert it to m/s before using it in the centripetal acceleration formula. Use the conversion factor: \( 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \). Therefore, \( 900 \text{ km/h} = 900 \times \frac{1}{3.6} = 250 \text{ m/s} \).

Step 3: Calculate centripetal acceleration

Substitute \( v = 250 \text{ m/s} \) and \( r = 1000 \text{ m} \) into the centripetal acceleration formula: \[a_c = \frac{v^2}{r} = \frac{(250)^2}{1000} = \frac{62500}{1000} = 62.5 \text{ m/s}^2 \].

Step 4: Compare centripetal acceleration with gravitational acceleration

Compute the ratio of the centripetal acceleration to gravitational acceleration: \[\frac{a_c}{g} = \frac{62.5}{9.8} \approx 6.38\].Since this value is approximately equal to 6.38, check the multiple choice options for the closest value.

Step 5: Identify the correct answer

From the given options, the one that is closest to 6.38 is option (c) 8, which suggests an acceptance of rounding in choices. However, evaluate your steps if a more precise option isn't available.

Key Concepts

Horizontal Loop Motion Concept

Imagine an aircraft soaring through the skies, tracing a circular path on a flat, horizontal plane. This type of motion is precisely what we mean by *Horizontal Loop Motion*. In this scenario, the aircraft maintains a constant altitude and speed but continuously changes direction to follow the loop's circle. The continuously changing direction provides the needed centripetal force to keep the aircraft on this path.

Key aspects to understand about horizontal loop motion include:

• The **radius** of the path, which is the distance from the loop's center to the path itself.
• The **speed** or velocity, which must remain consistent for true horizontal loop motion.
• The **centripetal force**, which acts towards the circle's center to change the direction of motion effectively without changing the speed.

The interesting part here is that an object, like our aircraft, moving in a circle at a constant speed is continuously accelerating. This acceleration, known as *centripetal acceleration*, keeps it on a curved path.

Unit Conversion Simplification

In understanding physics problems like the one at hand, converting units is crucial for accurate calculations. When dealing with speed in physics, it's often necessary to convert from kilometers per hour (km/h) to meters per second (m/s) because the standard scientific calculations mainly use the SI units.

The conversion factor to remember is:

• For speed: 1 km/h equals approximately 0.27778 m/s, or more simply, divide the number of km/h by 3.6 to get m/s.

In our example, the aircraft's speed of 900 km/h is converted to 250 m/s by multiplying 900 by the reciprocal of 3.6. This step ensures all units are standardized, making calculations, such as those for centripetal acceleration, straightforward and reliable.
Maintaining this consistency across calculations helps prevent errors and simplifies the comparison with gravitational acceleration, directly measured in m/s².

Understanding Acceleration due to Gravity

On Earth, gravity gives objects an acceleration towards the planet's center. This is known as *acceleration due to gravity*, symbolized by the letter 'g'. Its average value is approximately 9.8 m/s², though it can vary slightly depending on where you are on the planet due to Earth's shape and rotation.

Gravity's pull is what makes a dropped object speed up as it falls. It is also the standard measure against which other accelerations are compared in many physics problems.

• **Why compare with 'g'?** Understanding how other forces, like centripetal forces in loop motion, compare to gravitational forces gives context. It lets us grasp how much greater or smaller these forces are, relative to what we already know.
• **Practical implications:** In our aircraft example, comparing centripetal acceleration to 'g' tells us how intense the forces experienced by the aircraft are relative to gravity. A ratio greater than 1 means the centripetal force is stronger than the gravitational pull.

In the exercise example, the aircraft's centripetal acceleration is calculated and then compared to 'g' to determine the relative strength of forces involved, using a simple division operation.