Problem 50 - Master Resource Book in JEE Main Physics

Question

If \(I\) is the moment of inertia and \(\omega\) the angular velocity, what is the dimensional formula of rotational kinetic energy (a) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]\) (b) \(\left[\mathrm{M}^{2} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]\) (c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (d) \(\left[\mathrm{M}^{2} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]\)

Step-by-Step Solution

Answer

The dimensional formula is \([ML^2T^{-2}]\), which corresponds to option (c).

Step 1: Recall the Formula for Rotational Kinetic Energy

The formula for rotational kinetic energy is \( KE = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

Step 2: Identify Dimensional Formula of Moment of Inertia

The moment of inertia \( I \) has the dimensional formula \( [ML^2] \), where \( M \) is mass and \( L \) is length.

Step 3: Identify Dimensional Formula of Angular Velocity

Angular velocity \( \omega \) has the dimensional formula \( [T^{-1}] \), as it is the rate of change of angular displacement with time.

Step 4: Calculate Dimensional Formula of \( \omega^2 \)

To find \( \omega^2 \), square the dimensional formula of \( \omega \): \([T^{-1}]^2 = [T^{-2}]\).

Step 5: Compute Dimensional Formula of Rotational Kinetic Energy

Substitute the dimensional formulas of \( I \) and \( \omega^2 \) into the kinetic energy formula. This results in \[ KE = \frac{1}{2} [ML^2] [T^{-2}] = [ML^2T^{-2}] \].

Key Concepts

Understanding Moment of Inertia

The moment of inertia is a fundamental concept when it comes to rotational dynamics. Think of it like the rotational equivalent of mass in linear motion. Just as mass reflects how much an object resists a change in its linear motion, the moment of inertia reflects how much an object resists rotational acceleration.

It depends on two main factors:

• The mass of the object
• The distribution of that mass relative to the axis of rotation

This means that even objects with the same mass can have different moments of inertia if their mass is distributed differently.

In terms of dimensional analysis, the formula for moment of inertia is represented as \( [ML^2] \). This indicates that moment of inertia is dependent on mass \( (M) \) and the square of length \( (L^2) \). Knowing this helps in the calculation of rotational kinetic energy, which is critical in scenarios involving spinning objects, like wheels or even planets.

Grasping Angular Velocity

Angular velocity describes how quickly an object rotates or revolves relative to another point. It's typically denoted by the symbol \( \omega \) and measured in radians per second.

But what does that mean in regular words? Imagine spinning in a chair: the rate at which you complete a circle is your angular velocity. It measures the speed of rotation.

• Higher angular velocity means the object spins faster.
• Lower angular velocity means it spins slower.

In terms of dimensional analysis, angular velocity has the dimensional formula \( [T^{-1}] \). This is because it's a measure of how much rotation occurs per unit of time, making it inversely proportional to time.

This concept becomes especially important when calculating rotational kinetic energy, as it greatly influences how much energy is being used or stored in a spinning object.

The Role of Dimensional Analysis in Physics

Dimensional analysis is a powerful tool in physics and engineering. It's a method to check the consistency of equations and derive potential formulas by comparing the dimensions of physical quantities.

In the context of rotational kinetic energy, dimensional analysis involves understanding the dimensions of each component, like the moment of inertia and angular velocity, and combining them to deduce the dimension of the resulting quantity.

• Helps in verifying equations
• Aids in reducing complex physical relationships to their simplest forms
• Facilitates the conversion and comparison of units

For example, when we plug in the dimensions of the moment of inertia \( [ML^2] \) and the squared angular velocity \( [T^{-2}] \) into the equation for rotational kinetic energy \( KE = \frac{1}{2} I \omega^2 \), we derive the dimensional formula \( [ML^2T^{-2}] \).

Dimensional analysis not only serves as a check for correctness but also offers insight into the relations and dependencies among various physical variables, making it indispensable for students and professionals alike.