Problem 41 - Master Resource Book in JEE Main Physics

Question

Centripetal acceleration is (a) a constant vector (b) a constant scalar (c) a magnitude changing vector (d) not a constant vector

Step-by-Step Solution

Answer

The correct answer is (d) not a constant vector.

Step 1: Understanding Centripetal Acceleration

Centripetal acceleration occurs in a body moving in a circular path at a constant speed. It is directed towards the center of the circle.

Step 2: Analyzing Vector and Magnitude

Centripetal acceleration is a vector; it has both magnitude and direction. While the magnitude can be constant, the direction continuously changes as the body moves along the circular path.

Step 3: Determining Constancy

Since the direction of the vector changes continuously, centripetal acceleration is not a constant vector, even if its magnitude remains constant during uniform circular motion.

Step 4: Conclusion on Options

Given that centripetal acceleration has a changing direction, it does not qualify as (a) a constant vector. Similarly, it isn't (b) a constant scalar because it is a vector, and it remains at a constant magnitude during uniform motion, ruling out (c). Thus, the best fit is (d) not a constant vector.

Key Concepts

Vector Analysis

Vector analysis is an essential part of understanding physics concepts like centripetal acceleration. Vectors are quantities that have both magnitude (size) and direction, setting them apart from scalars, which only have magnitude. In the context of centripetal acceleration, vector analysis helps us understand how the acceleration behaves in circular motion.
When analyzing vectors, it’s crucial to remember:

• Magnitude determines how large the vector is.
• Direction shows where the vector points.

Applying this to centripetal acceleration, we see that while the speed may be constant, the direction of the acceleration vector changes continuously to point towards the center of the circle. This consistent change of direction is why centripetal acceleration is not a constant vector.

Circular Motion

Circular motion occurs when an object moves in a path that forms a circle. In circular motion, the object may travel at constant speed. However, even if the speed is unchanging, the direction of the object continuously alters to stay on its circular path.
Circular motion can be:

• Uniform, where the speed is constant.
• Non-uniform, where both speed and direction change.

For uniform circular motion, like that of a car going around a circular track at a steady speed, the direction of movement constantly changes although the velocity's magnitude remains steady. The essential force that keeps this motion directed towards the circle’s center is centripetal force, resulting in centripetal acceleration directed inward.

Acceleration in Physics

Acceleration in physics refers to the rate of change of velocity over time. It can involve a change in speed or direction, or both. Acceleration is a vector, meaning it has both magnitude and direction, making it crucial to use vector analysis when studying it.
In circular motion:

• Even if an object moves at a constant speed, it can still be accelerating due to the continuous change in direction.
• Centripetal acceleration is the specific type of acceleration for objects moving in a circular path, always pointed towards the circle’s center.

This acceleration is crucial in keeping the object on its circular path. While the speed remains constant, the changing direction of movement embodies acceleration in physics, challenging us to think beyond speed changes alone.