A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane?

Question

A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane

(a) Calculate the speed of the disk at the bottom of the inclined plane if the height of the incline is 0.84 m.

(b) Calculate the speed of the hoop at the bottom of the inclined plane if the height of the incline is 0.84 m.

Answer 1

The objects will each have two types of kinetic energy at the bottom of the inclide:
Translatinal
.5*m*v^2
and rotational
.5*I*w^2
since w=v/R
.5*I*v^2/R^2

the energy of the system will be equal to the potential energy lost
m*g*h

For the disk
I=.5*m*R^2
plugging in
m*g*h=
.25*m*v^2+.5*m*v^2
v=sqrt(g*h*4/3)
or
v=sqrt(9.81*0.84*4/3)
=3.315 m/s

for the hoop
I=m*R^2
plugging in
m*g*h=
.5*m*v^2+.5*m*v^2
v=sqrt(g*h)
v=sqrt(9.81*0.84)
v=2.87 m/s

j

Answer 2

The speed at the bottom of the inclined plane depends on the amount of friction between the plane and the objects rolling down it. You can't just assume no friction, because if there is not enough friction, the objects will slide down the plane, not roll.