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.  

causes

fossil fuels burn

cities become more industrialized

human population grows worldwide