A small wheeled cart and a solid uniform disk are both rolled (without slipping)

Question

A small wheeled cart and a solid uniform disk are both rolled (without slipping) along a flat surface towards a gently sloping hill that makes an angle of 5 degrees with the horizontal. The wheels of the cart have frictionless bearings and are small enough that their rotational kinetic energy can be neglected. The cart and the disk each have a mass of M=2.5kg. The radius of the disk is 0.15m, and it is 0.05m thick. Both objects are given an initial speed of V = 4.8 m/s towards the hill. Each of them rolls without slipping up the hill, reaches a maximum height, and then rolls back down. Air resistance can be neglected.

a) Do the two objects reach the same maximum height? If not, which one reaches the higher maximum height? Explain your answer.
b) Find the maximum height reached by the two objects, or, if they reach different heights, by the one that reaches the higher maximum height.

Explanation / Answer

a) no. disk reaches more height compared to cart.


cart
==========
0.5*m*v^2 = m*g*h

h = v^2/(2*g)

= 1.1755 m

disk
======

0.5*m*v^2 + 0.5*I*w^2 = m*g*h

0.5*m*v^2 + 0.5*0.5*m*r^2*w^2 = m*g*h

0.5*v^2 + 0.25*v^2 = g*h

0.75*v^2 = g*h

h = 0.75*v^2/g

= 0.75*4.8^2/9.8

= 1.7633 m

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