91Ó°ÊÓ

The moment of inertia is a crucial concept in understanding angular motion, especially in rotating objects like disks. Think of it as the rotational analogue of mass in linear motion. It measures how much torque is needed for a desired angular acceleration around an axis.
In simpler terms, it tells us how hard it is to change the rotational speed of an object. The formula to calculate the moment of inertia for a solid disk is given by:

• \( I = \frac{1}{2} m r^2 \)

In the problem of Disks A and B, we calculated the moment of inertia as follows:

• For Disk A: \( I_A = \frac{1}{2} \times 4 \times (0.3)^2 = 0.18 \, \text{kg}\cdot\text{m}^2 \)
• For Disk B: \( I_B = \frac{1}{2} \times 1.6 \times (0.18)^2 = 0.02592 \, \text{kg}\cdot\text{m}^2 \)

These calculations show how the moment of inertia depends on both mass and radius, illustrating why larger and heavier objects have higher moments of inertia.

Kinetic friction plays a key role when two surfaces are in motion relative to each other, such as the friction between Disks A and B when they come into contact. This type of friction opposes the relative motion and is calculated using the formula:

• \( F_k = \mu_k \times N \)

Here, \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force, which in this problem is the weight of Disk B (\( mg \)). So, for Disk B:

• Normal force, \( N = mg = 1.6 \times 9.81 = 15.696 \, \text{N} \)
• Kinetic friction force, \( F_k = 0.35 \times 15.696 = 5.4936 \, \text{N} \)

This frictional force is crucial as it serves as the torque that changes the angular speeds of the disks, illustrating the connection between friction and rotational dynamics.

Angular acceleration indicates how quickly an object's angular velocity is changing with time. It is analogous to linear acceleration in rectilinear motion. To calculate angular acceleration, we need to know the torque applied and the moment of inertia, using the formula:

• \( \tau = I \alpha \)

This can be rewritten to solve for angular acceleration (\( \alpha \)):
• \( \alpha = \frac{\tau}{I} \)
In the case of Disk A, the frictional torque \( \tau_f \) is:
• \( \tau_{f,A} = r F_k = 0.3 \times 5.4936 = 1.64808 \, \text{N}\cdot\text{m} \)
• \( \alpha_A = \frac{1.64808}{0.18} = 9.156 \, \text{rad/s}^2 \)
Disk B experiences the same torque but is less resistant to acceleration due to its lower moment of inertia:
• \( \alpha_B = \frac{1.64808}{0.02592} = 63.58 \, \text{rad/s}^2 \)
These calculations highlight how different masses and sizes influence the angular acceleration of objects.

Support reactions are forces that are exerted by a support to keep a structure in equilibrium. In the context of Disk A, the support at point C must counteract the forces acting on the disk to ensure it remains in a steady state.
When Disk B is spun up due to kinetic friction, Disk A experiences an equal and opposite reaction at its support due to Newton's Third Law – which states that every action has an equal and opposite reaction. Here, the reactive force at support C is calculated as:

• \( R_C = F_k = 5.4936 \, \text{N} \)

This reaction force is equal to the frictional force exerted by Disk B. Understanding support reactions illuminates how different forces in a mechanical system interact to maintain equilibrium and prevent unwanted motion. This showcases the interrelationship between dynamic forces and structural responses.