AP Physics C Mechanics 6.1: Rotational Kinetic Energy

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

[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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6.1.A—Describe the rotational kinetic energy of a rigid system in terms of the rotational inertia and angular velocity… question 1