Problems about torque and image charge

[dilloncyh]

http://farm9.staticflickr.com/8513/8460991004_d774e743cc.jpg

1. As shown in the figure, a gyroscope consists of a uniform disk of radius r and an axle of length R through its center and along its axis. The other end of the axle is hinged on a table but its otherwise free to rotate in any direction. The gyroscope is spinning with angular velocity w with the axle inclined to the vertical direction. Find its angular velocity of precession.

My attempt:
What I thought is that the torque due to the weight of the gyroscope equals to its rate of change of angular momentum. mgRsin(theta) = d(Iw)/dt, and I get
angular velocity = 2gR/(rw^2). Is it right?2. A point charge q is at x=3R/2 on the x-axis in front of a grounded conductor hemisphere of radius on a large conductor plate perpendicular to the x-axis and in the y-z plane. The center of the hemisphere is at (0,0,0). Find the potential energy energy of the point charge.

My attempt:
Can it be done using image charge? If so, how to set up the image charge configuration? Have no idea in how to deal with that hemisphere. If I put image charges to make the hemisphere zero potential, then the plane is not and I have to put another image charge to make the plane zero potential, but that will affect the potential of the hemisphere. So do I have to keep putting image charges (and end up with an infinite series)? Or are there other important things that I have omitted?

thanks

 

[rude man]

http://farm9.staticflickr.com/8513/8460991004_d774e743cc.jpg

1. As shown in the figure, a gyroscope consists of a uniform disk of radius r and an axle of length R through its center and along its axis. The other end of the axle is hinged on a table but its otherwise free to rotate in any direction. The gyroscope is spinning with angular velocity w with the axle inclined to the vertical direction. Find its angular velocity of precession.

My attempt:
What I thought is that the torque due to the weight of the gyroscope equals to its rate of change of angular momentum. mgRsin(theta) = d(Iw)/dt, and I get
angular velocity = 2gR/(rw^2). Is it right?

Nope.

Look at your dimensions - your angular velocity has dimensions of length ? So you know that can't be right. Always check dimensions term-by-term. It's the most powerful checking tool there is.

But you started right - τ = r x F with τ = dL/dt
and |τ| = mgRsin(θ). τ = torque, L = angular momentum.

Where you went wrong is understanding how angular momentum changes. You need to realize we are dealing with vectors here, and angular momentum is a vector that here changes not magnitrude but direction. In small interval of time Δt, ΔL = τΔt. You should draw this out & realize that ΔL is perpendicular to L. Just as dv/dt for circular motion is perpendicular to v - |v| stays constant but dv/dt changes direction and stays perpendicular to v.

The last steps involve coming up with an expression for the precession angle Δψ which is the amount of precession in time Δt, in terms of τ, which you know, ΔL, L and θ (all scalars). You need to draw these vectors to understand how Δψ relates to L, ΔL, τ and θ. If you do it right you will be left with an expression for the precession frequency ω_p = dψ/dt as a function of g, r, R, and w only. It might surprise you that it is not a function of θ.

2. A point charge q is at x=3R/2 on the x-axis in front of a grounded conductor hemisphere of radius on a large conductor plate perpendicular to the x-axis and in the y-z plane. The center of the hemisphere is at (0,0,0). Find the potential energy energy of the point charge.

My attempt:
Can it be done using image charge? If so, how to set up the image charge configuration? Have no idea in how to deal with that hemisphere. If I put image charges to make the hemisphere zero potential, then the plane is not and I have to put another image charge to make the plane zero potential, but that will affect the potential of the hemisphere. So do I have to keep putting image charges (and end up with an infinite series)? Or are there other important things that I have omitted?

thanks

Get back to you on this one I hope.

 

[dilloncyh]

opps, just a typo, my answer to q1 is angular velocity = 2gR/(wr^2), is it correct (at least it has the dimension of 1/time)?

 

[rude man]

opps, just a typo, my answer to q1 is angular velocity = 2gR/(wr^2), is it correct (at least it has the dimension of 1/time)?

Yes, that is correct. You must have been more closely on the right track than I gave you credit for! My apologies for my lengthy rambling ...

 

[paranormal]

1. Your attempt at finding the angular velocity of precession for the gyroscope is correct. This equation is known as the precession equation and is commonly used in analyzing the motion of gyroscopes. It is important to note that the angle theta in this equation is the angle between the axle and the vertical direction.

2. Yes, you can use the method of image charges to solve this problem. The key is to choose the right location and magnitude for the image charges. In this case, the grounded hemisphere can be replaced by a point charge located at the center of the hemisphere (0,0,0) with a magnitude equal to the charge on the hemisphere. This will ensure that the potential on the conductor plate is zero. Then, you can add another image charge at x=-3R/2 with the same magnitude as the original point charge q, to ensure that the potential at x=3R/2 is also zero. This process can be repeated for an infinite series of image charges, but the potential energy of the point charge can be approximated by using only a few image charges.