Problem 26 - Master Resource Book in JEE Main Physics

Question

Read each of the following statements carefully and state with reasons, chose the correct statement (s) (i) The net acceleration of a particle in the circular motion is always along the radius of the circle towards the centre. (ii) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point. (iii) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector. (a) (i) and (iii) (b) (ii) and (iii) (c) (iii) Only (d) All the three

Step-by-Step Solution

Answer

The correct answer is (d) All the three.

Step 1: Understanding Statement (i)

For a particle in circular motion, the net acceleration includes centripetal acceleration, which is directed toward the center of the circular path. Thus, statement (i) is correct as it describes centripetal acceleration.

Step 2: Analyzing Statement (ii)

In circular motion, the velocity vector is tangent to the path at every point. This is because velocity is always directed in the direction of motion, which is along the tangent to the circle. Therefore, statement (ii) is correct.

Step 3: Evaluating Statement (iii)

In uniform circular motion, the acceleration is centripetal and constantly changing direction, averaging to zero over one full cycle, as the particles have no net displacement from starting point in terms of vector average. Hence, statement (iii) is correct as it describes the average acceleration vector being a null vector.

Step 4: Final Answer Selection

Since statements (i), (ii), and (iii) are all true, all of them are correct choices. The answer is (d) All the three.

Key Concepts

Centripetal Acceleration

In circular motion, centripetal acceleration is a crucial concept. It is the force that keeps an object moving in a circular path, as opposed to moving off in a straight line. This acceleration always points towards the center of the circle. It's important to remember that without this inward force, an object would continue in a straight line due to inertia.

Understanding centripetal acceleration can be simplified with the formula \(a_c = \frac{v^2}{r}\), where \(v\) is the object's speed and \(r\) is the radius of the circular path.

Key points to consider:

• Centripetal means "center seeking"; thus, it refers to the direction of the acceleration.
• Without centripetal acceleration, circular motion cannot happen, as suggested by statement (i). The particle would move tangentially away from the circular path.
• It is not a force in itself, but rather a requirement for maintaining a circular path through forces like tension, gravity, or friction acting towards the centre.

Uniform Circular Motion

Uniform circular motion describes the movement of a particle or object traveling at a constant speed along a circular path. Despite the constant speed, the direction of the velocity vector changes continuously, which is why there's a need for centripetal acceleration to maintain this motion.

Some important aspects of uniform circular motion include:

• The magnitude of the velocity remains constant.
• The direction of velocity changes continuously, leading to a change in velocity direction, thus requiring acceleration even though the speed is constant.
• Statement (iii) holds true because, over one complete cycle, the centripetal acceleration changes direction constantly, averaging out to zero. This means that there is no net change in the particle's position when considering the entire cycle.

This phenomenon can be observed in everyday occurrences, such as the motion of the moon around the Earth or a car navigating a circular road.

Velocity Vector

The velocity vector in circular motion is an essential part of understanding how objects move in a circle. It is always tangent to the circular path at any point. This means that at any given moment, the direction of the velocity vector represents the direction that the object would move if it were to suddenly stop undergoing circular motion.

Important aspects to remember:

• The velocity vector's magnitude in uniform circular motion is constant, as it represents speed.
• Its direction changes continuously to maintain the object's circular path.
• Statement (ii) acknowledges this by confirming that the velocity vector is tangent to the path, which is a defining characteristic of circular motion.

Thus, the velocity vector's tangential direction is crucial for understanding how circular motion can be sustained and illustrates why centripetal force is necessary in keeping the object from "flying off" the circle.