Problem 655 - JEE Physics

Question

A cylinder of mass \(\mathrm{M}\) has length \(\mathrm{L}\) that is 3 times its radius what is the ratio of its moment of inertia about its own axis and that about an axis passing through its centre and perpendicular to its axis? \(\\{\mathrm{A}\\} 1\) \(\\{\mathrm{B}\\}(1 / \sqrt{3})\) \(\\{\mathrm{C}\\} \sqrt{3}\) \(\\{\mathrm{D}\\}(\sqrt{3} / 2)\)

Step-by-Step Solution

Answer

The short answer based on the step-by-step solution is: None of the given options match the correct ratio, which is \(\frac{3}{7}\).

Step 1: Moment of Inertia about its own axis

The moment of inertia, \(I_A\), of a cylinder about its own axis, assuming it is a solid cylinder, can be found using the following formula: \[ I_A = \frac{1}{2}MR^2, \] where \(M\) is the mass of the cylinder and \(R\) is its radius. In this case, \(R = r\). So, the moment of inertia about its own axis can be calculated as: \[ I_A = \frac{1}{2}Mr^2. \]

Step 2: Moment of Inertia about the perpendicular axis

The moment of inertia of a cylinder about an axis passing through its center and perpendicular to its axis, \(I_B\), can be found using this formula: \[ I_B = \frac{1}{12}M(3d^2 + h^2), \] where d is the diameter of the cylinder which is \(2r\) (2 times the radius) and h is the height of the cylinder which is \(3r\), given the length is 3 times its radius. So, the moment of inertia about the perpendicular axis can be calculated as: \[ I_B = \frac{1}{12}M(3(2r)^2 + (3r)^2). \] Now, we have both moments of inertia \(I_A\) and \(I_B\). To find the ratio of these moments of inertia, we just need to divide them.

Step 3: Finding the ratio

We will now find the ratio as follows: \[ \text{ratio} = \frac{I_A}{I_B} = \frac{\frac{1}{2}Mr^2}{\frac{1}{12}M(3(2r)^2 + (3r)^2)}. \] We can simplify the fraction, canceling out \(M\) and \(r^2\) from both the numerator and denominator and simplifying the expression: \[ \text{ratio}=\frac{\frac{1}{2}}{\frac{1}{12}(3(4) + 9)} = \frac{\frac{1}{2}}{\frac{1}{12}(12+9)} = \frac{\frac{1}{2}}{\frac{1}{12}(21)} = \frac{1}{2}\times\frac{12}{21}. \] Further simplifying the expression: \[ \text{ratio} = \frac{1}{2}\times\frac{6}{7} = \frac{6}{14} = \frac{3}{7}. \] Hence, none of the given options matches the correct ratio.

Key Concepts

Cylinder

A cylinder is a three-dimensional shape that has a circular base and a specific height. Unlike a sphere or a cube, a cylinder has two parallel bases connected by a curved surface. In this particular exercise, it is a solid cylinder. That means it's filled throughout without any hollow spaces.
A solid cylinder can be imagined as a can with liquid inside it. Here, the length of the cylinder is given to be three times its radius. This means if you were to unroll the curved surface, the length would be three times the distance from the center of the circle to its edge.
The mass of the cylinder plays an important role since it affects its moment of inertia - a measure of how much force is needed to rotate the cylinder around an axis.

Axis of Rotation

An axis of rotation is an imaginary line that an object rotates around. Think of it like the axle of a wheel or the Earth's axis on which it spins. For our cylinder, we consider two different axes of rotation to determine the moments of inertia: one is its own central axis along the length, and the other is perpendicular to it, passing through the center.

• Its own axis: This is a straight line running from one circular base to the other. Imagine placing a skewer through the center of the two circular ends.
• Perpendicular axis: This is a line that runs through the middle of the cylinder at a right angle to its length. Think about an axis that pierces through the cylinder from one side to the other, like a flagpole through a flag sideways.

Considering these different axes helps us understand how different forces affect the cylinder's rotation.

Perpendicular Axis Theorem

The perpendicular axis theorem is a handy tool in physics that comes into play when calculating the moment of inertia. It states that for a planar object, the moment of inertia about an axis perpendicular to its plane can be calculated by adding the moments of inertia around two perpendicular axes that lie in its plane. In our case, the cylindrical symmetry means we often use well-known formulas. However, this theorem helps illustrate that the cylinder's inertia for axes within and through its geometry is interconnected. The moment of inertia around the perpendicular central axis (which lies through its center and is perpendicular to its length) incorporates both its height and width.
Understanding this principle aids in visualizing how every small piece of mass within the cylinder contributes to the total rotational inertia differently, depending on its orientation.

Mass Distribution

The way mass is spread out within a body affects its moment of inertia - similar to how easier it is to rotate a baseball bat by holding it near the center rather than at one end.For a cylinder, the mass distribution is directly associated with its radius and length. All the mass is uniformly distributed within its volume, impacting how resistant the cylinder is to any changes in its state of rotation.

• Uniform Mass Distribution: In simple terms, this means every part of the cylinder contains an equal amount of mass, like a solid log.
• Impact on Inertia: Because the entire structure contributes equally to rotation around its central axis, calculating inertia based on uniform mass allows using shorter formulas like \(\frac{1}{2}MR^2\) for its own axis.

Grasping how mass distribution affects rotational movement helps in calculating and understanding different real-world scenarios, such as designing engines or evaluating structural stability.