3D Motion/Precession/Stability of a rolling coin

[RollingCoin]

Homework Statement

See the attached picture for the questions and information. This is actually due today and I haven't gotten all the answers - but it would be good to know how to do everything for when final exams come around. I particularly find (c), (d), (f), and (g) the most difficult.

Homework Equations

$$\dot{\psi}$$ = Precession rate
$$\dot{\theta}$$ = Nutation rate
$$p$$ = Spin

Angular momentum:
$$H_{x}=I_{x}\omega_{x}=I_{o}\dot{\theta}$$
$$H_{y}=I_{y}\omega_{y}=I_{o}\dot{\psi}sin\theta$$
$$H_{z}=I_{z}\omega_{z}=I\dot{\psi}cos\theta+p$$

Forced Precession:
$$M_{x}=\dot{\psi}sin\theta[I(\dot{\psi}cos\theta+p)-I_{o}(\dot{\psi}cos\theta)]$$

Free Precession:
$$\dot{\psi}=\frac{Ip}{(I_{o}-I)cos\theta}$$

 

[Valkarie]

The Attempt at a Solution(a) The angular momentum vector is pointing in the same direction as the spin vector (z-axis).(b) When the precession and nutation rates are equal, the angular momentum vector is pointing in the x-axis.(c) I'm not sure how to do this one. (d) I'm also not sure how to do this one. (e) Forced precession occurs when an external torque is applied to the particle. (f) The expression for forced precession is M_{x}=\dot{\psi}sin\theta[I(\dot{\psi}cos\theta+p)-I_{o}(\dot{\psi}cos\theta)].(g) The expression for free precession is \dot{\psi}=\frac{Ip}{(I_{o}-I)cos\theta}.

 

[Mene]

The Attempted Answers:

(a) The motion of the rolling coin can be described as a combination of precession and nutation. Precession is the circular motion of the axis of rotation, while nutation is the tilting of the axis of rotation.

(b) The stability of the rolling coin can be determined by the angular momentum of the coin. The coin will be most stable when its angular momentum is maximized, meaning that the axis of rotation is aligned with the direction of the angular momentum vector.

(c) To determine the precession rate, we can use the equation for forced precession, where M_{x} is the torque acting on the coin, I is the moment of inertia of the coin, I_{o} is the moment of inertia about the axis of rotation, and p is the spin of the coin. The torque acting on the coin can be calculated by multiplying the force of gravity by the distance between the center of mass and the point of contact with the surface. The precession rate can then be calculated using the equation \dot{\psi}=\frac{M_{x}}{I_{o}\sin\theta}.

(d) To determine the nutation rate, we can use the equation for free precession, where \dot{\psi} is the precession rate, I is the moment of inertia of the coin, I_{o} is the moment of inertia about the axis of rotation, and p is the spin of the coin. The nutation rate can then be calculated using the equation \dot{\theta}=\frac{Ip}{(I_{o}-I)\cos\theta}.

(f) To determine the stability of the rolling coin, we can use the equation for angular momentum, where H_{x}, H_{y}, and H_{z} represent the components of the angular momentum vector. The stability of the coin will be maximized when the axis of rotation is aligned with the direction of the angular momentum vector, meaning that H_{x}=I_{o}\dot{\theta}, H_{y}=0, and H_{z}=p.

(g) The stability of the rolling coin can also be affected by the distribution of mass within the coin. A coin with a more uniform distribution of mass will be more stable compared to a coin with a concentrated mass towards one side. This is because a more uniform distribution of mass will result in a smaller torque acting on the coin, making it easier for the coin to maintain