Asmooth cube of mass m and edge length r slides with speed v on a horizontal surface with negligible friction. the cube then moves up a smooth incline that makes an angle ? with the horizontal. a cylinder of mass m and radius r rolls without slipping with its center of mass moving with speed v and encounters an incline of the same angle of inclination but with sufficient friction that the cylinder continues to roll without slipping.
(a) which object will go the greater distance up the incline?
(b) find the difference between the maximum distances the objects travel up the incline. (use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)
(c) explain what accounts for this difference in distances traveled.

Explanation:

(a)

 Applying the conservation of energy for cube,

i.e the initial kinetic energy of the body = its final potential energy

  1/2mv² = mgh

h1 = v²/2g (1)

and For cylinder,

the initial kinetic and rotational energy equals the final potential energy

1/2 mv² + 1/2 Iω² = mgh

here I = momentum of inertia of the cylinder

I = 1/2 m R²

∴  1/2 mv² + 1/2 (1/2mR²)v²/R² = mgh

   1/2 mv² + 1/4mv² = mgh

h2 = 3v²/4g (2)

From equations (1)&(2);

therefore the cylinder will go the greater distance up the incline.

(b)

  h1-h2 = v²/2g - 3v²/4g

            = 0.25 v²/g

the difference between the maximum distances the objects travel up the incline is 0.25*v²/g.

(c)

here we are taking into account of rotational energy.  

elly's long hair:

a. if the salon only takes cash, then this isn't the problem. it doesn't matter how much she has in her checking account.   she can't get a haircut until she has the cash.

b. if she has overdraft protection, then there's no problem.   she can go ahead and write a check that her account can't cover, then sometime later, pay the bank some outrageous usury for loaning her a few bucks for a few days, but with beautiful hair.

c. if elly has some principles, and doesn't want to write a bad check, then now she has a problem.   her desire for the new hairstyle is in conflict with her honesty and integrity.

d.   if the bank will not process a check she writes, then even if she had a million $$ in her account, she couldn't use any of it.   the amount she has in her account is not the problem.   and if i may suggest, she might as well close her checking account, and put the money where it'll work for her.   it's not doing her a bit of good in a checking account, since she can't write any checks.

justin can't pay the full balance:

a. he can't even handle the credit limit he has now.   under no circumstances should he ask for a higher limit.

b.   he can ask for the grace period to be extended.   but that's no solution, the bank will almost certainly refuse anyway, and there are much better options.   he knew the rules of his credit card when he started using it.

c.   he definitely should pay the interest on the balance, plus as much of the balance as he can cover at the same time.

d.   he definitely should pay the minimum otherwise they could shut down his card.   he should also pay as much more as he possibly can.   he should also remember the awkward situation he's in, stop buying stuff on credit that he can't afford, and avoid adding any more debt.

i'd say that justin's best best option isn't on this list of choices.   what he has to do:   pay as much as he can at least the minimum payment plus as much more as he can and then avoid adding more debt.