A hoop and a solid disc are released from rest at the top of an incline and allo
ID: 1353265 • Letter: A
Question
A hoop and a solid disc are released from rest at the top of an incline and allowed to roll down the incline without slipping. You want to figure out (carefully) which one gets to the bottom first. Set up the equations symbolically and put in as much details as you can before you choose an answer.
a. The hoop gets to the bottom first and I can show why.
b. The hoop gets to the bottom first, but this is just a guess.
c. The disk gets to the bottom first and I can show why.
d. The disk gets to the bottom first, but this is just a guess.
e. The two objects tie, and I can show why.
f. The two objects tie, but this is just a guess.
A hoop and a solid disc are released from rest at the top of an incline and allowed to roll down the incline without slipping. You want to figure out (carefully) which one gets to the bottom first. Set up the equations symbolically and put in as much details as you can before you choose an answer.
a. The hoop gets to the bottom first and I can show why.
b. The hoop gets to the bottom first, but this is just a guess.
c. The disk gets to the bottom first and I can show why.
d. The disk gets to the bottom first, but this is just a guess.
e. The two objects tie, and I can show why.
f. The two objects tie, but this is just a guess.
Explanation / Answer
C.The disk gets there first because it has the smaller moment of inertia, so less of the energy finishes up as rotational energy, instead becoming translational energy and getting the disc moving faster
EXPLANATION:
When the mass center of the hoop moves with velocity V, then the hoop rotates with the angular frequency =V/R. The kinetic energy of the hoop is the sum of its translational and rotational energies:
E_{kin} = mV^2/2 + I ^2 /2=
mV^2/2 + mV^2 /2 = mV^2.
Since the sum of kinetic and potential energies is conserved, E_{kin} = mgh at the bottom of the incline. From this condition we have
V_{bottom} = (gh).
Disc also rotates with the angular frequency =V/R. However it has smaller moment of inertia, I = mR^2/2. Repeating above calculations we find the kinetic energy of the disc and its speed at the bottom of the incline:
E_{kin} = mV^2/2 + mV^2 /4 =3 mV^2/4,
V_{bottom} = (4gh/3)
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.