IBDP Physics HL - Advanced Dynamics & Fields

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Moment of Inertia ($I$)

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A scalar measure of a rigid body's resistance to rotational acceleration about a specific axis. Defined as the sum of the products of mass ($m$) and the square of the perpendicular distance ($r$) from the axis of rotation for each particle in the body: $I = \sum m_i r_i^2$. It depends on mass distribution relative to the axis.

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Rotational Kinetic Energy

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The kinetic energy due to the rotation of a body. It is given by $K_{rot} = \frac{1}{2} I \omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. This is analogous to translational kinetic energy $K = \frac{1}{2} mv^2$.

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Calculating Angular Momentum ($L$)

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A vector quantity defined as the cross product of the position vector $\vec{r}$ and the linear momentum $\vec{p}$ (where $\vec{p} = m\vec{v}$). The equation is $\vec{L} = \vec{r} \times \vec{p}$. For a rigid body rotating about a fixed axis, it simplifies to $L = I\omega$. Its conservation law explains why a spinning skater pulls in their arms to spin faster.

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Torque ($\tau$) and Rotational Newton's Second Law

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Torque is the rotational equivalent of force, causing angular acceleration. The rotational equivalent of Newton's Second Law is $\tau_{net} = I \alpha$, where $\tau_{net}$ is the net external torque, $I$ is the moment of inertia, and $\alpha$ is the angular acceleration.

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Lorentz Transformations (Time Dilation & Length Contraction)

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Mathematical formulas relating space and time coordinates between two inertial reference frames moving at relative velocity $v$. - **Time Dilation:** $\Delta t = \gamma \Delta t_0$, where $\Delta t_0$ is the proper time and $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$. - **Length Contraction:** $L = \frac{L_0}{\gamma}$, where $L_0$ is the proper length. Length contracts only in the direction of motion.