Number 66 , if wheel starts rrom rest, In the question what...

number 66 , if wheel starts rrom rest, In the question what is the kinetic energy of the wheel when $2 \mathrm{~m}$ of the cord is unwound?
(a) $20 \mathrm{~J}$
(b) $25 \mathrm{~J}$
(c) $45 \mathrm{~J}$
(d) $50 \mathrm{~J}$

Key Concepts

Relationship Between Linear and Angular Quantities

This concept highlights that in problems involving unwinding or rolling without slipping, the linear displacement of the cord is directly linked to the angular displacement of the wheel through the relation s = r?. This connection is fundamental in linking the energy provided by the linear movement of the cord to the rotational kinetic energy of the wheel.

Energy Conservation in Rotational Systems

In a system where motion is generated by an unwinding mechanism, the principle of conservation of energy applies. Energy transforms from one form (such as the potential energy due to the unwound cord) into kinetic energy (both rotational and possibly translational). Understanding how energy is conserved provides the basis for calculating the resulting kinetic energy when the system evolves from rest to motion.

Moment of Inertia

The moment of inertia is analogous to mass in linear motion but for rotation, representing a body's resistance to changes in its rotational state. It depends on how the object's mass is distributed relative to the axis of rotation, and it is crucial for determining the rotational kinetic energy when an object is set into rotational motion.

Rotational Kinetic Energy

This concept explains that a rotating body's kinetic energy depends on both its angular speed and its moment of inertia. The energy is calculated with the formula (1/2) I ?², which signifies that how mass is distributed (captured by the moment of inertia) and how quickly the body rotates (angular velocity) together determine its energy of rotation.

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