91Ó°ÊÓ

When a body rolls without sliding, like the object in this exercise, it means there is a perfect balance between the linear and rotational movements. Frictional force plays a crucial role in this balance. It's the force that prevents the object from sliding and helps it roll.

In the case of the rolling body, the frictional force acts at the point of contact with the surface. This force provides the necessary torque that enables the rotation of the body. It's important to note that the frictional force does not do work because the point of contact is momentarily at rest as the body rolls.

The frictional force can be calculated using the known values of mass and acceleration. In this example, using the formula derived from Newton's second law for rotation, the frictional force is found to be:\[f = \frac{ma}{3}\]

Substituting in the given values of mass and acceleration, the frictional force is calculated to be \(\frac{4}{3} \text{ N}\). This value shows how much force is needed to keep the body rolling without slipping, perfectly balancing forces and torques.

Moment of inertia is a concept that quantifies how much torque is needed for a body to achieve a certain angular acceleration. It's analogous to mass in linear motion but for rotational motion.

The moment of inertia depends on how the mass of the object is distributed relative to the axis of rotation. For our object in the exercise, the radius of gyration is provided as \(k = \frac{R}{\sqrt{3}}\). This enables the calculation of the moment of inertia \(I\):

\[I = mk^2 = m\left(\frac{R}{\sqrt{3}}\right)^2 = \frac{mR^2}{3}\]

The moment of inertia is a crucial factor in determining the angular acceleration of the body when subjected to a torque. Here it is used to establish the relationship between frictional force and angular acceleration, which ensures the body rolls without slipping.

Newton's second law for rotation is similar to its linear counterpart, detailing how a rotational motion occurs. It states that the net torque acting on a body is equal to the product of its moment of inertia and angular acceleration.

Mathematically, this is expressed as \( \tau = I \alpha \), where \(\tau\) is the net torque, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration. This principle helps in understanding how forces cause changes in the rotational state of a body.

In the context of this problem, the frictional force works as the torque to produce angular acceleration. The equation \(fR = \frac{mR^2}{3} \left(\frac{a}{R}\right)\) is a direct application of this law, connecting the linear acceleration to the angular one. It elegantly ties together mass, acceleration, and radius, showing how they interplay to achieve rolling motion without slipping. The frictional force is vital for maintaining this rotational dynamic, as it provides the needed torque for the body to rotate.