Problem 21 - Master Resource Book in JEE Main Physics

Question

The expression for centripetal force \((F)\) depends upon mass of body \((m)\), speed \((v)\) of the body and the radius ( \(r\) ) of circular path will be expression for centripetal force (a) \(F=\frac{m v^{2}}{2 r^{3}}\) (b) \(F=\frac{m v^{2}}{r}\) (c) \(F=\frac{m v^{2}}{r^{2}}\) (d) \(F=\frac{m^{2} v^{2}}{2 r}\)

Step-by-Step Solution

Answer

The correct expression is (b) \( F = \frac{mv^2}{r} \).

Step 1: Understand the Problem

We need to find the correct expression for centripetal force \( F \) in terms of mass \( m \), speed \( v \), and radius \( r \). We have four options to consider.

Step 2: Recall the Formula for Centripetal Force

The standard formula for centripetal force is given by:\[ F = \frac{mv^2}{r} \]This formula implies that centripetal force is directly proportional to the mass \( m \) and the square of the speed \( v \), and inversely proportional to the radius \( r \).

Step 3: Analyze the Given Options

We need to identify which of the provided options matches the standard formula for centripetal force. The options are:- (a) \( F = \frac{mv^2}{2r^3} \)- (b) \( F = \frac{mv^2}{r} \)- (c) \( F = \frac{mv^2}{r^2} \)- (d) \( F = \frac{m^2v^2}{2r} \)

Step 4: Compare Each Option to the Formula

- Option (a): Has extra powers and a factor of 2 in the denominator - not matching.- Option (b): Matches exactly with \( F = \frac{mv^2}{r} \).- Option (c): Incorrect denominator power of \( r \).- Option (d): Has an incorrect power on \( m \) and an extra factor of 2.

Step 5: Select the Correct Option

From the comparison, option (b) \( F = \frac{mv^2}{r} \) is identical to the known formula for centripetal force. Thus, option (b) is the correct choice for the expression.

Key Concepts

Circular Motion

When we think about circular motion, imagine a car moving along a racetrack or planets orbiting around the sun. Circular motion happens when an object moves along a circular path. An important feature of circular motion is that the object's direction is constantly changing, even if its speed remains the same.
This direction change means there's always an acceleration directed towards the center of the circle, known as centripetal acceleration. This center-seeking acceleration requires a force to sustain the circular motion, called centripetal force.
This force keeps the object going around the circle and prevents it from flying off in a straight line. Without centripetal force, no matter the object's speed, it would travel tangentially away from the circle due to its inertia. It’s the same reason why when you let go of a spinning stone tied to a string, it flies off straight away rather than continuing in a circle.

Physics Formulas

The world of physics is sophisticated yet fascinating, where things occurring around us get a formulaic representation. Among these, the formula for centripetal force is quite important when dealing with circular motions.
The formula for centripetal force is:

• \[ F = \frac{mv^2}{r} \]

This formula tells us how mass \( m \), speed \( v \), and radius \( r \) are related in circular motion.
In this formula:

• Mass \( m \) directly affects the force needed for circular motion. Heavier objects require more force.
• The speed \( v \) is squared, emphasizing that just a small increase in speed drastically increases the needed force.
• The radius \( r \) is in the denominator, meaning a larger circle needs less force to maintain the motion at the same speed.

This formula is essential to solving problems involving objects moving in a circle, from designing safe amusement park rides to calculating the forces in our galaxy.

Newton's Laws of Motion

Newton's Laws of Motion are the backbone of classical mechanics, which guide us in understanding how objects move.
These laws explain the relationship between an object's motion and the forces acting upon it. In the context of circular motion, the first and second laws are particularly relevant.

• The First Law, often called the law of inertia, tells us that an object will keep moving in a straight line unless a force acts upon it. For circular motion, this means centripetal force is necessary to continuously change the object's direction as it "wants" to move straight.
• The Second Law defines the link between force and acceleration: \( F = ma \). This is crucial in understanding centripetal force, as it represents the force needed to maintain the circular path by providing the necessary centripetal acceleration.
• Finally, the Third Law relates to actions and reactions, meaning if a force is applied in the circular motion context, there is an equal and opposite reaction force, accounting for the tension in a string or the friction on a track.

Newton's Laws help us comprehend how these concepts translate into the forces needed to sustain circular motion in everyday and cosmic scenarios.