Question

A block and a hoop are both given an initial velocity of vo = 5 m/s at the bases of their identical inclines.

Diagram: https://imgur.com/a/xT0Wi

1. To what height, H, will the block slide up its incline if the incline is tilted at an angle of θ = 53º?

2. To what height, h, will the hoop roll up its incline if the incline is also tilted at an angle of θ = 53º?

3. What is the ratio of h/H?

Answer 1

(1) From an energy point of view the kinetic energy of the block has to balance the potential energy due to gravity. As the block is pushed along the slope, its potential energy increases. When KE=PE the block reaches its highest position x from the ground. So max=½mv², or 2gx=v² and x=v²/(2g)=25/19.62=1.274m. But this is the vertical height. The distance along the slope is H and x=Hsinθ so H=x/sinθ=v²/(2gsinθ)=25/(19.62×sin53)=1.60m approx.

(2) The hoop has translational and rotational kinetic energy: ½mv²+½I⍵² where I=mr² for a hoop and ⍵=v/r, so the rotational KE=½(mr²)(v²/r²)=½mv². Therefore we have total KE=mv²=mgx, so x=v²/g and h=v²/(gsinθ)=3.19m.

(3) So h/H=2.