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.  

the body's situation won't change the forces are equal opposit and on the same body

increasing transnational kinetic energy. because the transnational kinetic energy formula consists of only two variables, mass and velocity, increasing one of those properties is the only way to increase an object's transnational kinetic energy. increases to mass and velocity, however, do not have the same impact.