Please explain your answer in words, in math, and check the NOTE in the PHOTO !!
ID: 1452827 • Letter: P
Question
Please explain your answer in words, in math, and check the NOTE in the PHOTO !! :
The Great Ramp Race! Suppose three objects of equal mass and radius are perched at the top of a large ramp as shown in the figure. One object is a uniform disk, one is a hoop or ring, and one is a wheel for which nearly all of the mass is concentrated at the center (only a very small amount is in the spokes and the rim). If all are released from the same height at the same time, in what order will they reach the bottom of the ramp? Please explain your answer and your reasoning. Yes, this is a pretty standard physics question, and you can search for explanations on the internet. Don't do that to answer this, but instead reason as far as you can with the tools we have so far. (If you want to look up the answers afterward, obviously, go for it! Also see Chapter 10.9.)Explanation / Answer
Moment of inertia of an object rolling down the ramp is I
Now, total energy at the bottom of ramp = total energy at the top of the ramp [ conservation of energy]
Total energy at teh top of ram p[assuming the objects start from rest] = potential energy = mgh
Total energy at teh bottom = rotational ke + translational ke = 0.5Iw^2 + 0.5mv^2
where w is the angular velocity and v is translational velocity, and the relation between them for no slip condition is w = v/r [where r is radius of rolling object]
so mgh = 0.5Iw^2 + 0.5mv^2 = 0.5Iv^2/r^2 + 0.5mv^2 = 0.5v^2[I/r^2 + m]
Now, for the three objects, mass is same and r is same
so, v is inversly proportional to I
hence, more the I is, more of the energy is used for rotational component of KE and hence less energy for translational ke
and lesser v means the object takes more time to come down
hence, time for hoop > time for spoked wheel > time for disc
because I hoop > I spoked wheel > I disc [ farther the mass is from centre of rotaion, more is the I]