10 - Angular Momentum and Rolling Motion Free Online FlashCards

What determines a particle’s angular momentum about an axis?

For a particle, angular momentum magnitude is $L=r_\perp p=r_\perp mv$, or equivalently $L=rp\sin\phi$. It depends on the chosen axis.

What is the angular momentum of a rigid object?

For a rigid object rotating about a fixed axis, angular momentum is $L=I\omega$, where $I$ is the moment of inertia and $\omega$ is angular velocity.

How does mass distribution affect moment of inertia?

The moment of inertia is $I=\sum_i m_i r_i^2$. Mass farther from the axis contributes more strongly because distance is squared.

How is net external torque related to angular momentum?

Net external torque changes angular momentum according to $\sum\tau=\frac{dL}{dt}$. For constant $I$, this becomes $\sum\tau=I\alpha$.

When is angular momentum conserved?

Total angular momentum is conserved about a selected axis when the net external torque about that axis is zero or negligible: $\sum\tau_{\text{ext}}=0\Rightarrow L_i=L_f$.

Why does a spinning student speed up when pulling in their arms?

With negligible external torque, $I_i\omega_i=I_f\omega_f$. If the moment of inertia decreases, the magnitude of angular velocity increases.

What constraint defines rolling without slipping?

For rolling without slipping, the center-of-mass speed and angular speed satisfy $v_{\text{CM}}=R\omega$. The corresponding acceleration constraint is $a_{\text{CM}}=R\alpha$.

What are the ground speeds at the contact point and top of a rolling wheel?

For a wheel rolling without slipping, the contact point is instantaneously at rest relative to the ground, while the top of the wheel moves at speed $2v_{\text{CM}}$.

What changes when a rolling object slips?

During slipping, $v_{\text{CM}}\ne R\omega$. Kinetic friction acts at the contact surface and can transform mechanical energy into thermal energy.

What role does static friction play in ideal rolling?

Ideal rolling without slipping generally involves static friction. It can provide torque and change angular speed while doing no net work when the contact point has no displacement.

What is the rotational kinetic-energy formula?

The rotational kinetic energy of a rigid object is $K_{\text{rot}}=\frac12 I\omega^2$. Moment of inertia plays the rotational role analogous to mass in translation.

What kinetic energies does a rolling object possess?

A rolling object has both forms: $K=\frac12 Mv_{\text{CM}}^2+\frac12 I_{\text{CM}}\omega^2$. The translational and rotational terms must both be included.