As the disc descends calculat/e the tension in the string?

Question

A string is wound around a uniform disc of
radius 0.39 m and mass 2.1 kg . The disc is
released from rest with the string vertical and
its top end tied to a fixed support.
The acceleration of gravity is 9.8 m/s^2 .
As the disc descends, calculate the tension
in the string. Answer in units of N. And also Calculate the speed of the center of mass
when, after starting from rest, the center of mass has fallen 1.8 m. Answer in units of
m/s.

Answer 1

Consider a FBD of the disc
Summing force in the vertical with down as positive
m*g-T=m*a

where T is the tension and a is the acceleration
Now consider the sum of torques about the center of the disc
T*r=I*α
α=a/r
so
T*r=I*a/r

The moment of inertia of the disc is m*r^2/2
so
T*r=m*a*r/2
and
T=m*a/2
plugging in above
m*g-m*a/2=m*a
or
a=2*g/3
and
T=m*2*g/3
plugging in the numbers
T=2.1*2*9.8/3
T=13.72 N

since we know a to be
2*g/3
and
vy(t)=2*g*t/3
and
y(t)=g*t^2/3
When y=1.8
t=0.74 s
and
vy=4.85 m/s downward

j

Answer 2

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