No, a system cannot have rotational kinetic energy without having linear kinetic energy. Rotational kinetic energy and linear kinetic energy are interconnected and depend on each other.
To understand this, let's consider a rigid body that can rotate about a fixed axis. The rotational kinetic energy (KE_rot) of the body is given by the formula:
KE_rot = (1/2) * I * ω^2
where I is the moment of inertia of the body about the axis of rotation, and ω is the angular velocity of the body.
Now, for a rigid body to have rotational kinetic energy, it must have an angular velocity (ω). Angular velocity represents the rate of change of angular displacement with respect to time. In order for the angular velocity to be nonzero, the body must have linear motion, which means it must have linear velocity (v).
The linear velocity (v) and angular velocity (ω) are related by the equation:
v = ω * r
where r is the distance of a point on the body from the axis of rotation. This equation shows that there is a direct connection between linear velocity and angular velocity.
Since linear velocity is a necessary component of rotational motion, a system cannot have rotational kinetic energy without having linear kinetic energy. If there is no linear motion, the angular velocity would be zero, and thus the rotational kinetic energy would also be zero.
Therefore, the limits are such that if a system does not possess any linear kinetic energy (i.e., all linear velocities are zero), it would not have any rotational kinetic energy either. The presence of linear kinetic energy is a prerequisite for the existence of rotational kinetic energy in a system.