A string is wrapped around the rim of a disk of radius \"r\" and mass \"m\". The

Question

A string is wrapped around the rim of a disk of radius "r" and mass "m". The free end of the string is tied to a block, also of mass 'm" as shown. The block starts from rest. a. Write out the force and torque equations on the objects. Given [m, r], calculate: b. The tension "T" in the lower string while the block descends. c. The tension "T"_2 in the upper string while the block descends. d. The acceleration of the block as it descends. e. The angular acceleration of the disk as it block descends. f. The time for the disk to make its first revolution.

Explanation / Answer

A) Force equation on the block:

ma = mg - T1

a = g - T1/m.......equation 1

Force equation on the disc:

T2 = mg + T1

Torque equation on disc:

I alpha = T1r

0.5 mr^2 alpha= T1r

0.5 ma = T1

a = 2T1/m...........equation 2

b) From equation 1 and 2,

g- T1/m =2T1/m

g = 3T1/m

T1 = mg/3

c) T2 = mg + T1

   = mg+ mg/3

   T2= 4mg/3

d) a = 2T1/m

= 2g/3

e) alpha = a/r

= 2g/3r

f) s = 0.5 at^2

2 pi r = 0.5*(2g/3) t^2

t^2 = 6 pi r /g

t = sqrt ( 6*pi* r /g)

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