Problem 21 - Master Resource Book in JEE Main Physics

Question

The speed of revolution of a particle moving round a circle is doubled and its angular speed is halved. What happens to the centripetal acceleration? (a) Unchanged (b) Halved (c) Doubled (d) 4 times

Step-by-Step Solution

Answer

The centripetal acceleration becomes 4 times greater.

Step 1: Understanding the Problem

We need to find out how the centripetal acceleration of a particle changes when its speed of revolution is doubled and its angular speed is halved.

Step 2: Formula for Centripetal Acceleration

The centripetal acceleration formula is given by:\[ a_c = \frac{v^2}{r} \]Where \( a_c \) is the centripetal acceleration, \( v \) is the linear speed, and \( r \) is the radius of the circle.

Step 3: Expressing Angular Speed

Angular speed \( \omega \) is related to linear speed \( v \) by:\[ \omega = \frac{v}{r} \]From this equation, we can express \( v \) in terms of \( \omega \):\[ v = \omega \cdot r \]

Step 4: New Expressions for Speed and Acceleration

If the original speed is \( v \) and this speed is doubled, the new speed becomes \( 2v \). If the original angular speed is \( \omega \) and is halved, the new angular speed becomes \( \frac{\omega}{2} \).The new centripetal acceleration \( a_c' \) with speed \( 2v \) is:\[ a_c' = \frac{(2v)^2}{r} = \frac{4v^2}{r} \]

Step 5: Analyzing Angular Speed Influence

The relation between the new speed and new angular speed confirms that the change in centripetal acceleration resulting from doubling the speed overrides the angular speed influence since centripetal acceleration depends only on linear speed in the circular motion context provided by \[ a_c = \frac{v^2}{r} \].

Step 6: Compare Acceleration Changes

Centripetal acceleration initially was \( \frac{v^2}{r} \) and is now \( \frac{4v^2}{r} \). Therefore, the centripetal acceleration becomes 4 times greater.

Key Concepts

Circular Motion

Circular motion occurs when an object moves along a circular path at a constant distance from a fixed point. This type of motion is common in our everyday lives, from carousels at a fairground to the Earth's orbit around the Sun.

• The key characteristic here is that the motion is along a circular path.
• The direction of motion is continuously changing, even if the speed is constant.
• This requires a constant force directed towards the center of the circle, known as the centripetal force.

Understanding circular motion involves recognizing that even though the speed might be constant, the velocity is not, because velocity is a vector quantity that includes direction. This change in direction causes what is known as centripetal acceleration, acting perpendicularly to the motion but towards the center of the circle.

Angular Speed

Angular speed, often denoted as \( \omega \), is a measure of how fast an object rotates or revolves relative to another point. It's a concept that translates well from our intuitive understanding of things spinning around or moving in a circle.

• The angular speed is defined as the change in the angle over time.
• It is distinct from linear speed, which measures how fast something moves along a path.

The relation between angular speed \( \omega \) and linear speed \( v \) is given by the equation \( \omega = \frac{v}{r} \), where \( r \) is the radius of the circle. Thus, angular speed is fundamentally a measure of an object's rotational speed rather than its path speed. In this problem, understanding angular speed helps in determining how changes in speed affect centripetal acceleration.

Linear Speed

Linear speed is the distance traveled per unit of time without regard to direction. In circular motion scenarios, linear speed is crucial as it determines the kinetic energy of the moving object.

• For any object moving in a circle, linear speed is given by \( v = \omega \cdot r \).
• This shows a direct relationship between angular speed and linear speed.

In a situation where the speed of revolution doubles, it means the linear speed is also doubled. The centripetal acceleration, which is based on linear speed, thus responds directly to these changes. Specifically, since it depends on \( v^2 \), doubling the speed quadruples the centripetal acceleration, as shown by the formula \( a_c = \frac{v^2}{r} \).

Physics Problem Solving

Physics problem-solving involves applying concepts to understand how systems behave under various conditions. In this exercise, we need to determine how changes in speed affect centripetal acceleration, using the relationship between linear and angular speed.

• First, analyze the problem and identify what is changing (linear and angular speeds).
• Next, apply relevant formulas, such as \( a_c = \frac{v^2}{r} \) for centripetal acceleration.
• Consider how these changes influence the key variables within these formulas.

Checking your work involves ensuring that all steps logically follow each other and correctly lead to the solution. In this problem, doubling the linear speed results in a quadruple increase in centripetal acceleration due to the squared variable in the formula. Even if angular speed is halved, it does not directly influence the final outcome as much in this specific context. Hence, the solution becomes clear by systematically solving through each step.