Q. Hoop, Disk, Cylinder and Sphere
A disk, a hoop a cylinder and a sphere have the same mass and the same diameter. Every is rolled toward a ramp with the same initial velocity. Which one will arrive at a higher point on the ramp?
This problem merely involves the conversion of kinetic energy to potential energy. The total kinetic energy when every object is released consists of its forward kinetic energy and its rotational kinetic energy. Thus, for each object
P.E. = K.E. =1/2(mv2) +1/2(Iω2)
Where I be the rotational inertia
The forward kinetic energy for every object is the same except the rotational kinetic energy depends on the distribution of mass around the centre.
The following table illustrate the values of I and KER for several simple shapes.
|
Shape
|
Inertia
|
KER
|
|
hoop
|
mr2
|
mv2/2
|
|
hollow cylinder
|
mr2
|
mv2/2
|
|
disk
|
mr2/2
|
mv2/4
|
|
solid cylinder
|
mr2/2
|
mv2/4
|
|
hollow sphere
|
2mr2/3
|
mv2/3
|
|
solid sphere
|
2mr2/5
|
mv2/5
|
A little thought contras that I be the same for a hoop and hollow cylinder having equal masses and diameters. Likewise I be the same for a disk and solid cylinder.
Obviously the object with the largest rotational inertia will reach the greatest height on the ramp. Specified the values for I from the table the hoop will reach the highest point.