Q.A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s$^{-1}$. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

The key idea here is Rotational Dynamics, specifically calculating rotational kinetic energy and angular momentum using the moment of inertia.

1. First, calculate the moment of inertia ($I$) for a solid cylinder about its axis:

2. Next, determine the kinetic energy ($K$) associated with the rotation: $K = \frac{1}{2} I \omega^2 = \frac{1}{2} (0.625 \text{ kg m}^2) (100 \text{ rad s}^{-1})^2 = \frac{1}{2} (0.625) (10000) = 3125 \text{ J}$ …

When an object rotates, it possesses kinetic energy due to its rotation, and it also carries angular momentum, which is a measure of its rotational inertia in motion. These quantities are analogous to translational kinetic energy ($\frac{1}{2}mv^2$) and linear momentum ($mv$), but for rotational motion. The key difference is that instead of mass ($m$), we use the moment of inertia ($I$), and instead of linear velocity ($v$), we use angular velocity ($\omega$).

The moment of inertia ($I$) is a crucial concept here. It quantifies an object's resistance to changes in its rotational motion, much like mass quantifies resistance to changes in linear motion. Its value depends not only on the object's total mass but also on how that mass is distributed relative to the axis of rotation. For different shapes and different axes, the moment of inertia will vary.

For a solid cylinder rotating about its central axis, the mass is uniformly distributed, and its moment of inertia has a specific, well-known formula. Once we determine this value, calculating the rotational kinetic energy and angular momentum becomes a straightforward application of their respective formulas.

Let's break down the calculation:

1. Identify Given Information

   We are given the following parameters for the solid cylinder:

   • Mass, $M = 20 \text{ kg}$
   • Angular speed, $\omega = 100 \text{ rad s}^{-1}$
   • Radius, $R = 0.25 \text{ m}$

2. Determine the Moment of Inertia ($I$)

   The first step is to find the moment of inertia of the solid cylinder about its central axis. This is a standard result in rotational dynamics.

   For a solid cylinder of mass $M$ and radius $R$ rotating about its central axis, the moment of inertia is given by:

   $I = \frac{1}{2}MR^2$

   Now, we substitute the given values into this formula:

   $I = \frac{1}{2}(20 \text{ kg})(0.25 \text{ m})^2$

   $I = 10 \text{ kg} \times (0.0625 \text{ m}^2)$

   $I = 0.625 \text{ kg m}^2$

   Watch out

   Always ensure you use the correct moment of inertia formula for the specific object and axis of rotation. For example, a hollow cylinder or a sphere would have different formulas.

3. Calculate Rotational Kinetic Energy ($K_{rot}$)

   With the moment of inertia calculated, we can now find the kinetic energy associated with the rotation. This is the rotational analogue of $\frac{1}{2}mv^2$.

   The rotational kinetic energy of an object with moment of inertia $I$ and angular speed $\omega$ is: …

Concept: Rotational Kinetic Energy and Angular Momentum

Step 1: Moment of inertia of the solid cylinder

Step 2: Rotational kinetic energy

$K=\frac12I\omega^2=\frac12(0.625)(100)^2=\frac12(0.625)(10000)=3125\text{ J}$ …