Optimum Angle for Object Pulling: A Plot Analysis

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In summary, the conversation discusses the topic of finding the most efficient angle for pulling an object, dependent on the coefficient of friction (mu). The equation F=\frac{\mu mg}{cos(\alpha)(\mu tan(\alpha)-1)} or F=\frac{\mu mg}{\mu sin(\alpha)+cos(\alpha)} is used to calculate the optimum angle. The conversation also mentions using calculus to find the maximum or minimum of a curve and a geometrical approach for those who are not familiar with calculus.
  • #1
Joans
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I found interesting to me this topic and tryed to analyze.
http://img505.imageshack.us/img505/6655/49209931.jpg [Broken]

At which angle it is eseaiest to pull an object?
I got equation
[tex]F=\frac{\mu mg}{cos(\alpha)(\mu tan(\alpha)-1)}[/tex] or [tex]F=\frac{\mu mg}{\mu sin(\alpha)+cos(\alpha)}[/tex]


I don't know this math very well unfortunately, but I would be interested to see a plot:
how mostly optimum angle is dependent from mu, and for example then mu is 1 alpha is 45, and so on...

heh sorry for the paint and spelling...
 
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  • #2


It depends on mu actually. Think of 2 extreme cases. First, mu is 1 (friction is extremely high). In which case X = Y, the answer is 45 degree as you said. But if mu is 0 (frictionless), the answer is clearly 0 degree (the force required to move the object is close to zero).

I'm not sure though; ha haaa

Mr Peetiya
 
  • #3


You're looking for a maximum. Use your calculus brain. How does one find the maximum or minimum of a curve?
 
  • #4


Brin said:
You're looking for a maximum. Use your calculus brain. How does one find the maximum or minimum of a curve?

I would use calculus if i know how to use it, in school i do not have lessons with it, unfortunately, since I am 11grader. But infact i know quite a lot about it. But still I don't know how to found derirative of the bottom. [tex]f(\alpha)=\mu sin(\alpha)+cos(\alpha) f'(\alpha)=\mu cos(\alpha)-sin(\alpha)[/tex] ?? When to make it to zero and solve it? How to solve what equation when? It's homogenic .. divide by cos alpha and whola? :)
And does best angle depends from [tex]\mu[/tex] lineraly? In fact this topic is quite clear, just math's is not very clear.
 
  • #5


Well, you seemed to have gleaned the important part of the derivation anyways.

[tex] f(\alpha) = \frac{\mu mg}{\mu sin(\alpha) + cos (\alpha)}[/tex]

[tex] f'(\alpha) = \frac{\mu mg (\mu cos(\alpha) - sin(\alpha))} {(\mu sin(\alpha) + cos(\alpha))^2} = 0 [/tex]

because [tex]\mu mg[/tex] is constant, and the denominator can't be zero, we can simplify this problem a bit by focusing on the only part that can be zero.

So, you see why I thought you did well on calculating the f' you did:
[tex]\mu cos(\alpha) - sin(\alpha) = 0 [/tex]

Then

[tex]
\mu cos(\alpha) = sin (\alpha)
[/tex]

So,
[tex]
tan(\alpha) = \mu
[/tex]

Then arc tan both sides to get an explicit value for alpha. By analyzing this function, e.g. you can see that if there is no friction mu = 0, the best pull is the directly horizontal pull (i.e. alpha = 0). If you have mu = 1.0 the best pull is at alpha = 45 degrees.

If you don't know calculus, this problem probably seems a little out of your league. But I am fairly confident there is a geometrical approach as well, that is within your limits if you're an intelligent high school student, or a bored undergrad. If you have the time, and are still curious, I'd recommend seeking out that way.
 
  • #6
Great, thanks!
 

What is the easiest way to pull an object?

The easiest way to pull an object is by using a rope or a chain. This allows you to have a strong grip on the object and pull it with more force.

Do I need any special equipment to pull an object?

It depends on the weight and size of the object you are trying to pull. For smaller objects, a rope or chain may be sufficient. However, for larger or heavier objects, you may need specialized equipment such as a winch or a pulley system.

What is the best technique for pulling an object?

The best technique for pulling an object is to use your body weight and strength to your advantage. This means using your legs and core muscles to generate the force needed to pull the object, rather than relying solely on your arms.

Can I damage the object by pulling it?

Yes, it is possible to damage the object if you pull it with too much force or in the wrong direction. It is important to assess the weight and fragility of the object before attempting to pull it and to use caution and proper technique to avoid causing damage.

Are there any safety precautions I should take when pulling an object?

Yes, it is important to wear appropriate protective gear, such as gloves and sturdy shoes, when pulling an object. It is also important to have a clear path and to communicate with any other individuals helping with the task to avoid accidents or injuries.

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