The way I would look at this is to consider the position of the fringes as a function of angle. That is, suppose we surround the slits by a cylindrical screen rather than a flat screen, then how are the fringes distributed on the cylindrical screen?
The reason for doing it this way is that on the bottom half of the diagram the screen subtends an angle of 90°, so it's easy to count the number of fringes.
At A the path difference is 5λ, and at O the path difference is zero, and we get a fringe whenever the path difference is an integral number of wavelengths, so including the central fringe, from A to O there will be 6 fringes Yes?
We get a bright fringe when nλ = d sinθ, and if we look at the range of angles θ can have from A to the maximum angle that hits the screen it is from θ = -90° to θ = +37°.
@PCMSE I used it because it made it obvious to me that we need only consider the angle i.e. the distance to the screen doesn't matter. But possibly this is the just the weird way my brain works :-)
All that matters is the angle that the screen subtends.
Tilting the screen makes it look as if it's going to be a hard calculation, but in fact it isn't as the angle the screen subtends is very simple to find.
It's an interesting question. I don't think I've seen a question like this before. The point is that the elastic cord is stretched by the weight of the bead, so the velocity of the bead relative to the cord is less than v.
Yes. The bead has a hole through it and the elastic cord passes through the hole. There is friction between the cord and the bead in this hole so the bead slides at constant velocity.
The frictional force is therefore equal to the weight of the bead.
I assume we can neglect the short period the bead takes to accelerate to the speed v, so we just assume the bead starts moving at speed v at the top of the cord.
Note that v is the speed relative to the ceiling not relative to the cord.
No, you will need to take into account the fact the cord stretches as the bead moves down it. This stretching of the cord is what makes the problem hard.
One thing I'm not sure about is exactly what the question means by the stiffness i.e. exactly how this is defined.
Have you been taught that stiffness has a precise definition?
We are told the bead is a distance x from the ceiling. Since the cord is being stretched by the bead the distance the bead has travelled down the cord is y, where y < x.
So we can consider this as a length y of the cord that has been stretched by the weight of the bead to a length x, i.e. the extension of the cord is x-y.
Now, if we take a length y of the cord then its spring constant is k𝓁/y i.e. if we apply a force F to the cord the force is related to the extension by F = k𝓁/y Δy
Is this step OK? i.e. you understand that the spring constant depends on the length of the cord?
Now, if we take a length y of the cord then its spring constant is k𝓁/y i.e. if we apply a force F to the cord the force is related to the extension by F = k𝓁/y Δy
Yes, and this looks reassuringly similar to the equations in the options so I think we are on the right track.
Yes $l$ is the natural length of the whole cord.
Given our result let's start looking at the options.
Option A is a bit bizarre. If the speed relative to the ceiling is v and and the distance relative to ceiling is x then the time taken is just t = x/v. Yes?
So B is false, but the questioner made it very close to the correct answer just to try and confuse us :-)
Now, D is simpler than it looks though again the algebra is a little messy.
Power is force times velocity, where here the force is the frictional force between the bead and the cord, and we know the frictional force is mg. OK so far?
Again it's not a question I've seen before, but my immediate reaction is that if we consider a height x above the base we can find the angle of the curve at that height from the derivative of the curve.
Pressure always exerts a force normal to the surface, so we can find the angle of the force and therefore it's horizontal and vertical components.
Though now I think about it there is an easier way to do this ...
In fact a very easy way
I'll draw a diagram
On the right of the dashed line we have the parabolic wall, and on the left we have the rest of the water above the flat base.
Consider the horizontal forces. The water to the left of the dashed line exerts a horizontal force on it that we can easily find by integrating the pressure ρgh. This is the force I've labelled F. OK so far?
And that means there must be the same horizontal force on the curved part of the wall. This way the net horizontal force is zero, which it has to be since the water isn't accelerating.
@JohnRennie To find weight of water in the curved part do we need to first complete rectangle and then find area through integration and then use unitary method?
Suppose a motorcycle starts at $v_i$ and then skids over a distance of$ d$, ending the skid at $v_f$. Find ${v_f}^2 - {v_i}^2 $ in terms of the distance $d$ and the deceleration a.
Answer assumed that acceleration is constant. I don't see that acceleration is constant is mentioned in the question. Does to skid mean that acceleration is constant?
"I don't see that acceleration is constant is mentioned in the question."
Why? As the motorcycle is deaccelerating, a force is acting (that is kinetic friction.) and kinetic friction is constant as the object's mass is not changing.
Problem Solving Strategies
Problem Solving Strategies