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Define total kinetic energy.
The sum of translational and rotational kinetic energies of an object or system.
Define translational kinetic energy.
The kinetic energy due to the linear motion of an object, calculated as $\frac{1}{2}mv^2$.
Define rotational kinetic energy.
The kinetic energy due to the rotation of an object, calculated as $\frac{1}{2}I\omega^2$.
Define moment of inertia.
A measure of an object's resistance to rotational acceleration about a given axis.
Define angular velocity.
The rate at which an object rotates or revolves relative to another point, i.e. how many radians per second the object rotates.
Define 'rolling without slipping'.
A condition where an object rolls without any sliding at the point of contact with the surface. Static friction acts to prevent slipping, and mechanical energy is conserved.
Define 'rolling with slipping'.
A condition where an object slides while rotating. Kinetic friction is present, and mechanical energy is not conserved.
Compare rolling without slipping and rolling with slipping in terms of energy conservation.
Rolling without slipping: Mechanical energy is conserved. | Rolling with slipping: Mechanical energy is not conserved; it is dissipated by kinetic friction.
Compare the friction involved in rolling without slipping vs. rolling with slipping.
Rolling without slipping: Static friction acts as a constraint, preventing slipping. | Rolling with slipping: Kinetic friction dissipates energy as heat.
Compare the applicability of $v_{cm} = r\omega$ in rolling with and without slipping.
Rolling without slipping: $v_{cm} = r\omega$ applies. | Rolling with slipping: $v_{cm} = r\omega$ does not apply; linear and rotational motion must be analyzed separately.
What is the effect of kinetic friction during rolling with slipping?
Kinetic friction dissipates mechanical energy, converting it into thermal energy (heat).
What is the effect of static friction during rolling without slipping?
Static friction acts as a constraint, preventing slipping, and allowing mechanical energy to be conserved.
What happens to the linear speed of a cylinder at the bottom of an incline if it slips instead of rolls?
The linear speed will be greater because some energy is lost to thermal energy due to kinetic friction in slipping. In rolling, some of the initial potential energy is converted to rotational kinetic energy.