AP Physics C: Mechanics 6.1 A Describe the Rotational Kinetic Energy of a Rigid System in Terms of the Rotational Inertia and Angular Velocity Questions

Relate rotational kinetic energy to rotational inertia and angular velocity, and compare rotational and translational energy descriptions.

Exam points

calculate rotational kinetic energy from one-half I omega squared for a rigid system
use gravitational-energy conservation to determine maximum angular speed during a swing
convert angular speed to a point's linear speed after solving the rotational energy balance
separate rotational kinetic energy from translational kinetic energy in a falling or rolling system
identify the energy dissipated while stopping a wheel as its initial rotational kinetic energy

[Maximum number: 4]

Object A is a long, thin, uniform rod of mass M and length 2 L that is free to rotate about a pivot of negligible friction at its left end, as shown above.

Derive an expression for the angular speed of object B when it is in the position shown above. Express your answer in terms of $M, L, I_{\mathrm{B}}$ , and physical constants, as appropriate.

Note: Figure not drawn to scale.

For using conservation of energy
$U_{g 1}=K_{2}$

1 point

For correctly relating the change in rotational kinetic energy to the change in gravitational potential energy
$m g h=\frac{1}{2} I \omega^{2}$

1 point

For correctly substituting for h into the equation above
$m g\left(\frac{L}{2}\right)=\frac{1}{2} I \omega^{2}$

1 point

For an expression for $\omega$ that uses only the allowed symbols and is algebraically consistent with the previous steps
$\begin{aligned} \omega=\sqrt{\frac{2 m g h}{I}}=\sqrt{\frac{2 M g(L / 2)}{I_{\mathrm{B}}}} \omega=\sqrt{\frac{M g L}{I_{\mathrm{B}}}} \end{aligned}$

1 point

Total for part (f)

for question 215 points

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AP Physics C: Mechanics 6.1 A Describe the Rotational Kinetic Energy of a Rigid System in Terms of the Rotational Inertia and Angular Velocity Questions question 1