Consider a device consisting of two uniform disks of radii r1= 5cm and mass m1=1

Question

Consider a device consisting of two uniform disks of radii r1= 5cm and mass m1=1kg and r2=10cm
and m2= 4kg that are welded together and mounted on a frictionless axis about their common center.
String of negligible mass is wound about both disks and allowed to hang freely.

a) Assume a 1 kg mass is attached to the rope hanging from the smaller disk and released from
rest. What is the angular acceleration of the disk system and the linear acceleration of the
mass? (Assume the rope unwinds from the wheel without slipping and that air resistance can be
neglected).
b) Now assume the same 1 kg mass is removed and attached instead to the rope hanging from the
larger disk. What is the angular acceleration of the disk system and the linear acceleration of
the mass? (Assume the rope unwinds from the wheel without slipping and that air resistance
can be neglected).
c) In which case is the acceleration greater? Does this make sense?

Consider a device consisting of two uniform disks of radii r1= 5cm and mass m1=1kg and r2=10cm and m2= 4kg that are welded together and mounted on a frictionless axis about their common center. String of negligible mass is wound about both disks and allowed to hang freely. Assume a 1 kg mass is attached to the rope hanging from the smaller disk and released from rest. What is the angular acceleration of the disk system and the linear acceleration of the mass? (Assume the rope unwinds from the wheel without slipping and that air resistance can be neglected). Now assume the same 1 kg mass is removed and attached instead to the rope hanging from the larger disk. What is the angular acceleration of the disk system and the linear acceleration of the mass? (Assume the rope unwinds from the wheel without slipping and that air resistance can be neglected). In which case is the acceleration greater? Does this make sense?

Explanation / Answer

r1 = 0.05 m
m1 = 1 kg

r2 = 0.1 m
m2 = 4 kg


I = 0.5*m1*r1^2 + 0.5*m2*r2^2

= 0.02125 kg.m^2

a)

mass of hanging body, m = 1 kg

let T is the tencion in the rope

net force acting on the hanging body, Fnet = m*g - T

m*a = m*g - T

a = g - T/m

here m = 1 kg

a = g - T --(1)

net torque acting on the pulley syste = r1*T

I*alfa = r1*T

I*a/r1 = r1*T

a = r1^2*T/I

g - T = r1^2*T/I

g = T(1 + r1^2/I)

T = g/(1 + r1^2/I)

= 9.8/(1 + 0.05^2/0.02125)

= 9.686 N

a = 9.8 - 9.686

= 0.114 m/s^2

alfa = a/r1

= 0.114/0.05

= 2.28 rad/s^2

b)

mass of hanging body, m = 1 kg

let T is the tencion in the rope

net force acting on the hanging body, Fnet = m*g - T

m*a = m*g - T

a = g - T/m

here m = 1 kg

a = g - T --(1)

net torque acting on the pulley syste = r2*T

I*alfa = r2*T

I*a/r1 = r2*T

a = r2^2*T/I

g - T = r2^2*T/I

g = T(1 + r2^2/I)

T = g/(1 + r2^2/I)

= 9.8/(1 + 0.1^2/0.02125)

= 6.664 N

a = 9.8 - 6.664

= 3.316 m/s^2

alfa = a/r2

= 3.316/0.1

= 33.16 rad/s^2

c)

in the second case angular acceleration is more. Because the net torque in the second case is more.

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