Supermarket Ramp Design: Finding the Optimal Slope

[tman1]

here was the original question. in the design of a super market, there are to be several ramps connecting different parts of the store. cusomers will have to push grocery carts up the ramps. An engineer has done a survey and found that almost no one complains if the force require is no more that 50 N. Will a slope of 5 degrees be too steep, assuming a 30-kg grocery cart (full of groceries)? Assume friction can be accounted for by a coeffecient of 0.10.

we found that 5 degrees is too high. the new question requires us to find out the actual degree. How would i find it help!

i forgot to add the the formula that should be used is as follows
Fapp= (coefficient of friction x mass x gravity x cos thada) + (mass x gravity x sin thada)
i need help with the trig identities part more

PLZ solve full answer please

 

[Fermat]

Your formula is,

F_app = mg(sinø + μcosø)

There are two ways of solving this - analytically or using an approximate method, e.g. Newton's method.

Analytically
=========
Use the substitutions,

sinø = 2t/(1+t²)
cosø = (1-t²)/(1+t²)

where

t = tan(ø/2)

You will end up with a quadratic in t. Solve for t, then find ø = 2*arctan(t).

Or, if you have done Newton's method, you could do it that way.

 

[tman1]

ty

thatnks for ur help but i still don't understand what to do so i was wondering if u could coach me through it

 

[Fermat]

Have you made the substitutions for sinø and cosø in the formula F_app = mg(sinø + μcosø) ?

 

[tman1]

yes

yes i did but i can't solve for either cos thada or sin thada

 

[Fermat]

You have to substitute for
sinø = 2t/(1+t²)
and
cosø = (1-t²)/(1+t²)

In F_app = mg(sinø + μcosø), replace sinø by 2t/(1+t²) and replace cosø by (1-t²)/(1+t²).
This will give you a quadratic int t, which you then solve for, not for ø.
See my earlier post, #2.

 

[tman1]

im really slow

can u please write out step by step what to do. I am really slow sir please

 

[Fermat]

tman, this is a help forum and I have given you a lot of help and advice, but you have to something yourself.

Have you read the sticky at the top of this forum ? If not, will you please do so.

I have given you adivice and help, now it is up to you to do some work and show me what you have done.

All you have to do is make a substitution; show me your work and I will comment upon it.

 

[tman1]

thx

thanks for ur hellp and all and I am sorry i was a pest
i solved it differently from how u showed me tho