Figure 28-52 gives the orientation energy U of a magnetic...

Figure $28-52$ gives the orientation energy $U$ of a magnetic dipole in an external magnetic field $\vec{B}$, as a function of angle $\phi$ between the directions of $\vec{B}$ and the dipole moment. The vertical axis scale is set by $U_{s}=2.0 \times 10^{-4} \mathrm{~J} .$ The dipole can be rotated about an axle with negligible friction in order to change $\phi$. Counterclockwise rotation from $\phi=0$ yields positive values of $\phi$, and clockwise rotations yield negative values. The dipole is to be released at angle $\phi=0$ with a rotational kinetic energy of $6.7 \times$ $10^{-4} \mathrm{~J}$, so that it rotates counterclockwise. To what maximum value of $\phi$ will it rotate? (In the language of Module $8-3$, what value $\phi$ is the turning point in the potential well of Fig. $28-52 ?$ )

Figure $28-52$ gives the orientation energy $U$ of a magnetic dipole in an external magnetic field $\vec{B}$, as a function of angle $\phi$ between the directions of $\vec{B}$ and the dipole moment. The vertical axis scale is set by $U_{s}=2.0 \times 10^{-4} \mathrm{~J} .$ The dipole can be rotated about an axle with negligible friction in order to change $\phi$. Counterclockwise rotation from $\phi=0$ yields positive values of $\phi$,
and clockwise rotations yield negative values. The dipole is to be released at angle $\phi=0$ with a rotational kinetic energy of $6.7 \times$ $10^{-4} \mathrm{~J}$, so that it rotates counterclockwise. To what maximum value of $\phi$ will it rotate? (In the language of Module $8-3$, what value $\phi$ is the turning point in the potential well of Fig. $28-52 ?$ )

Key Concepts

Magnetic Dipole Potential Energy

This concept involves the energy associated with a magnetic dipole when placed in a magnetic field. The potential energy depends on the orientation of the dipole relative to the field and is commonly expressed as a function of the angle between the dipole moment and the magnetic field vector. It reaches a minimum (often the most stable state) when the dipole is aligned with the magnetic field and increases as the angle between them deviates from alignment.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation about an axis. In systems involving dipoles or other rotating bodies, this energy can be converted into potential energy and vice versa. Understanding rotational kinetic energy is crucial for analyzing the dynamics of rotating systems, especially when the object moves within a potential well and its speed and energy distribution change with position.

Energy Conservation in Rotational Systems

The principle of energy conservation is fundamental in analyzing mechanical systems. In rotational systems, the total energy is the sum of kinetic and potential energies. When a dipole rotates, its rotational kinetic energy can be progressively transformed into potential energy as it climbs the energy landscape defined by the magnetic field. The motion eventually stops at the turning point, where all the kinetic energy has been converted, and this point is critical for predicting the system's subsequent behavior.

Turning Point in a Potential Energy Well

A turning point in a potential energy well is the position at which an object in motion momentarily comes to rest before reversing direction. This occurs when the available kinetic energy is completely converted into potential energy. In the context of a rotating dipole within a magnetic field, the maximum angular displacement, or the turning point, is the angle where the rotational kinetic energy is zero and all energy is stored as potential energy in the system.

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