Skip to main content
Physics

Return to Answer

deleted 19 characters in body
Source Link
Puk
Puk
  • 13.9k
  • 1
  • 23
  • 42

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the elevation angle of the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_1 - \theta_2$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the elevation angle of the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_1 - \theta_2$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_1 - \theta_2$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.

edited body
Source Link
Puk
Puk
  • 13.9k
  • 1
  • 23
  • 42

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the elevation angle of the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_2 - \theta_1$$\theta_1 - \theta_2$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the elevation angle of the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_2 - \theta_1$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the elevation angle of the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_1 - \theta_2$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.

Source Link
Puk
Puk
  • 13.9k
  • 1
  • 23
  • 42

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the elevation angle of the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics: enter image description here

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_2 - \theta_1$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.