A smooth cube of mass m and edge length r slides with speed v on a horizontal su

Question

A smooth 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 / Answer

(a)   Applying the conservation of energy for cube,    1/2mv^2 = mgh h1 = v^2/2g -------------------- (1) and For cylender, 1/2 mv^2 + 1/2 I^2 = mgh here I = momentum of inetia of the cylender I = 1/2 m R^2   1/2 mv^2 + 1/2 (1/2mR^2)v^2/R^2 = mgh     1/2 mv^2 + 1/4mv^2 = mgh h2 = 3v^2/4g -------------- (2) From qeuations (1)&(2); therefore the cylinder will go the greater distance up the incline. (b)    h1-h2 = v^2/2g-3v^2/4g              = 0.25 v^2/g the difference between the maximum distances the objects travel up the incline is 0.25*v^2/g. (c) here we taken in to account of rotational energy.  

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