Max. Height Insect Can Crawl in a Bowl: u & r

In summary, the problem requires finding the maximum height that an insect can crawl up to in a bowl with a given coefficient of friction and radius. The correct answer is given by r [ 1 - 1/root(1 + u2)]. The approach involves creating a diagram, finding the angle of inclination using the tangent, and using geometry and algebra to find the force components in terms of the height climbed. The maximum height is found by equating the frictional force and the restraining force.
  • #1
Mr Virtual
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4

Homework Statement



If the coefficient of friction between an insect and bowl surface is u (mu) and the radius of bowl is r, what is the max height upto which the insect can crawl in the bowl?

a) r/root(1 + u2)
b) r [ 1 - 1/root(1 + u2)]
c) r [root(1 + u2)]
d)r [ root(1 + u2) -1 ]

Correct Answer: b

Homework Equations



F=uR
tan theta = u

On a non-horizontal plane,

F=mg sin theta
R=mg cos theta

therefore, net f=mg(sin theta + u cos theta)


The Attempt at a Solution



Can't get the gist of how to approach this question.
 
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  • #2
With problems like this, first thing you do is go about making a diagram. For all the purposes of this question, we can take the bowl to be a hemisphere. Let's assume that there is a single meridian line of the sphere that the insect travels across. Let's start by taking the cross section of such a line:

http://img267.imageshack.us/img267/5157/insecthelpqi1.jpg [Broken]

Here, I've assumed a right-handed co-ordinate system, with the circle:

[tex]
x^2 + y^2 = r^2
[/tex]

Let the insect be at an arbitrary point on the circle (x, y). Now, a very small part of the circle can be taken to be as an inclined plane. The angle it makes can be found out using the angle the tangent on that point makes with the x-axis. It is given by:

[tex]
\frac{dy}{dx} = \tan{(\theta)} = -\frac{x}{y}
[/tex]

Once, you've done that, you can easily find out the angles, [itex]\sin(\theta)[/itex] and [itex]\cos(\theta)[/itex], using Pythagoras theorem wherever necessary. And hence, you can find the angle of the inclination as a function of the co-ordinates (x, y). But, you need it in the terms of height. With a little algebra and geometry, you can find the co-ordinates as:

[tex]
(x, y) \equiv (\sqrt{2hr - h^2}, r - h)
[/tex]

And hence, the inclination is given by:

[tex]
\frac{dy}{dx} = \tan{(\theta)} = -\frac{x}{y} = \frac{\sqrt{2hr - h^2}}{h - r}
[/tex]

Now, you have the force components in terms of the height climbed by the insect. Now, you need to find 'h' such that, the frictional force (caused by the normal force mg cos(θ)) becomes equal to the restraining force [mg sin(θ)] and beyond which it is smaller.

I think you can do the problem now.
 
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  • #3

As a scientist, it is important to understand the question and the given information before attempting to solve it. In this case, the question is asking for the maximum height that an insect can crawl in a bowl, given the coefficient of friction (u) and the radius of the bowl (r).

To solve this, we can use the equations F=uR and tan theta = u, where F is the force of friction, u is the coefficient of friction, and R is the normal force. Since the insect is crawling on a non-horizontal plane, we can use the equation F=mg(sin theta + u cos theta) where m is the mass of the insect, g is the acceleration due to gravity, and theta is the angle of the incline.

To find the maximum height, we need to find the angle of the incline at which the force of friction is equal to the weight of the insect (mg). This can be done by setting the two equations equal to each other and solving for theta.

mg(sin theta + u cos theta) = uR

mg(sin theta + u cos theta) = u(mg cos theta)

sin theta + u cos theta = u cos theta

sin theta = 0

theta = 0

Therefore, the maximum height that the insect can crawl in the bowl is when the angle of the incline is 0 degrees, which means the insect is crawling on a horizontal surface.

Substituting this into the given options, we can see that option b) r [ 1 - 1/root(1 + u2)] is the correct answer. This means that the maximum height is equal to the radius of the bowl multiplied by (1 - 1/root(1 + u2)).

In conclusion, as a scientist, it is important to understand the question and the given information, and to use relevant equations and logical reasoning to solve it.
 

1. What is the maximum height an insect can crawl in a bowl?

The maximum height an insect can crawl in a bowl depends on the size and type of insect. Smaller insects, such as ants, can crawl higher in a bowl compared to larger insects, such as beetles. On average, insects can crawl up to 2-3 times their body length in height.

2. How do insects crawl in a bowl?

Insects have specialized body structures, such as claws and adhesive pads, that allow them to grip and climb surfaces. They also use their legs and body movements to navigate and crawl in a bowl.

3. Can insects crawl in a bowl with smooth surfaces?

Some insects, such as cockroaches, can crawl on smooth surfaces due to the tiny hairs on their legs that provide grip. However, other insects may have difficulty crawling on completely smooth surfaces.

4. Do different types of bowls affect an insect's crawling ability?

Yes, different types of bowls can affect an insect's crawling ability. For example, a bowl with steep sides may be more difficult for an insect to crawl in compared to a shallower bowl with gradual sides.

5. Is there a limit to how high an insect can crawl in a bowl?

There is no definitive limit to how high an insect can crawl in a bowl. However, factors such as the size and type of insect, the surface of the bowl, and the height of the sides can all impact an insect's crawling ability.

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