2 concentric cylinders with radius r and 2r having a common axis are present . if the bottom part is given a twist @ then the twist at the middle part is ?

1. @
2. @/2
3. 16@/17
4. 17@/16

To solve the problem of determining the twist at the middle part of two concentric cylinders when the bottom part is twisted, we need to consider the relationship between the angles of twist and the geometry of the cylinders involved. Let's break this down step by step.

Understanding the Setup

We have two cylinders: the inner cylinder with radius \( r \) and the outer cylinder with radius \( 2r \). Both cylinders share a common axis. When the bottom part of the inner cylinder is twisted by an angle \( \theta \), we want to find out how much the middle part of the outer cylinder twists as a result.

Twist Distribution in Cylinders

The twist in a cylindrical structure is not uniform; it varies with the radius. The angle of twist is proportional to the distance from the axis of rotation. This means that the outer cylinder will experience a different amount of twist compared to the inner cylinder.

Using the Concept of Torque and Angle of Twist

The angle of twist \( \theta \) can be expressed in terms of the shear modulus \( G \), the polar moment of inertia \( J \), and the length \( L \) of the cylinder:

• For the inner cylinder: \( \theta_1 = \frac{T L}{G J_1} \)
• For the outer cylinder: \( \theta_2 = \frac{T L}{G J_2} \)

Here, \( T \) is the torque applied, \( J_1 \) and \( J_2 \) are the polar moments of inertia for the inner and outer cylinders, respectively. The polar moment of inertia for a solid cylinder is given by:

• For the inner cylinder: \( J_1 = \frac{\pi r^4}{2} \)
• For the outer cylinder: \( J_2 = \frac{\pi (2r)^4}{2} = 8 \frac{\pi r^4}{2} = 4 \pi r^4 \)

Calculating the Twist Ratio

The ratio of the twists can be derived from the relationship of the polar moments of inertia:

Since the torque \( T \) and length \( L \) are the same for both cylinders, we can express the ratio of the angles of twist as:

\( \frac{\theta_1}{\theta_2} = \frac{J_2}{J_1} = \frac{4 \pi r^4}{\frac{\pi r^4}{2}} = 8 \)

This means that for every unit of twist in the inner cylinder, the outer cylinder twists \( \frac{1}{8} \) of that amount. If the inner cylinder is twisted by an angle \( \theta \), the outer cylinder will twist by:

\( \theta_2 = \frac{\theta_1}{8} \)

Finding the Middle Part Twist

Now, if we consider the middle part of the outer cylinder, which is at a radius of \( r \) (the average of \( r \) and \( 2r \)), we can apply the same principle. The twist at the middle part will be proportional to its radius relative to the outer cylinder:

Since the middle part is at radius \( r \), we can find the twist at this point as:

\( \theta_{middle} = \frac{\theta}{2} \)

Final Answer

Thus, if the bottom part of the inner cylinder is twisted by an angle \( \theta \), the twist at the middle part of the outer cylinder will be \( \frac{\theta}{2} \). Therefore, the correct answer is \( \frac{\theta}{2} \).