Imagine a downhill race between four objects each of mass M. One object such as a skier (P) is best modeled as a particle (shown as a small flat object) translating down the hill. The other objects are modeled as a solid cylinder (C), hoop (H) and a solid sphere (S) each of radius R rolling down the hill without slipping. All four objects start from rest a t a height h. Assuming mechanical energy losses due to friction may be ignored, determine the order in which the objects cross the finish line. Use their letter designation: P, C, H and S and arrange them from first to last to arrive at the finish line.

Answer 1

Answer: From the energy conservation

1/2mv^2+1/2Iw^2 = mgh

here I = moment of inertia = cmr^2

w = v/r

1/2mv^2+1/2cmr^2*w^2 = mgh

v = sqrt(2gh/1+c)

c is a constant

for particle c = 1

vp = sqrt(2gh/1+1) = sqrt(2gh/2)

for cylinder

vc = sqrt(2gh/1+1/2) = sqrt(4gh/3)

for hoop

vh = sqrt(2gh/1+1) = sqrt(2gh/2)

for solid sphere

vs = sqrt(2gh/1+2/5) = sqrt(10gh/7)

vs>vc>vh=vp