Consider a sphere of non-uniform density with radius R=0.5m and mass M=1 kg. The
ID: 1313998 • Letter: C
Question
Consider a sphere of non-uniform density with radius R=0.5m and mass M=1 kg. The rotational inertia of this sphere can be taken to be I=BMR^2 where B is unknown. The sphere is initially rotating with angular velocity w0=4.00rad/s and is placed on a rough horizontal surface. The velocity of the center of mass of the sphere as it is placed on the surface is 0. The sphere at first rolls and slips. Just as it begins to roll without slipping, the velocity of the center of mass is found to be 1.5 m/s. Calculate B.
*Please show work, thanks!
Explanation / Answer
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Given Data
mass of the sphere m= 1kg
Radius R = 0.5 m
moment of inertia I = BmR^2
intial angular velocity w0 = 4 rad/s
intially the sphere has only rotational kinetic energy
K1 =0.5IW0^2 =0.5*BmR^2*4^2
K1 = 8BMR^2 = 8*B*1*0.5^2 = 2B ------------(1)
and the condition for rolling without slipping is
V =RW
final angular speed W =V/R = 1.5/0.5 = 3 rad/s
final kinetic energy when it is rolling is =trans KE +Rotational KE
K2 =0.5mv^2+0.5IW^2 = 0.5*1*1.5^2 + 0.5*BmR^2*3^2 = 2.25/2 + 0.5*B*1*0.5^2*9
K2 = 1.125 + 9B/8 ------------(2)
from the principle of conservation of energy
K1=K2
2B = 1.125 +9B/8
(16B-9B)/8 = 1.125
7B/8 = 1.125
7B = 9
B = 9/7 = 1.286
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