In the figure below, two pulleys are mounted on fixed axles that have negligible
ID: 1342436 • Letter: I
Question
In the figure below, two pulleys are mounted on fixed axles that have negligible friction. The small pulley has a moment of inertia of 9.0 kg · m2; it is made up of two cylinders welded together, one of radius 7.0 cm and the other of radius 15.0 cm. The large pulley has a radius of 42.0 cm and a moment of inertia of 88 kg · m2; the pulleys are coupled using a light belt. A 6.50 kg mass hangs from the smaller pulley by a rope that is wound around the smaller cylinder. The system is initially at rest. The mass is let go and starts to fall.
(a) Find the acceleration of the mass.
m/s2
(b) Find the tension in the rope.
N
(c) Is the tension in the belt the same everywhere? Explain.
Yes, but only once the mass is let go.No, the calculated tension in the rope is too small to balance the tension in the belt. Yes, the rope is exerting a force on the smaller pulley which balances the tension in the belt.No, the belt is exerting a force on the larger pulley, so the tension in the upper part of the belt is not the same as the tension in the lower part.
Explanation / Answer
from the energy conservation principle :
J1*1^2 = J2*2^2
J2 = J1*1^2/2^2
Je = J1*(15/42)^2
= 88*0.127
= 11.176 kgm^2
Jt = Je+J2
= 11.176+9.0
= 20.176 kgm^2
the formula for accelerating torque is
Ta = m*g*r
= 6.50*9.8*0.07
= 4.459 Nm
from equation Ta = Jt*d/dt
a) d/dt = Ta/Jt
= 4.459/20.176
= 0.221 rad/sec^2
.corresponding to 0.221*0.07 = 0.015 m/s^2
the acceleration of the mass. is 0.015 m/s2
Tension T evaluation is
Ta1 = Ta*Je/Jt
= 4.459*11.176/20.176
= 2.46 Nm
b) T = Ta1/r
= 2.46 Nm/0.07 m
= 35.15 N
(c) Yes, but only once the mass is let go.No, the calculated tension in the rope is too small to balance the tension in the belt. Yes, the rope is exerting a force on the smaller pulley which balances the tension in the belt.No, the belt is exerting a force on the larger pulley, so the tension in the upper part of the belt is not the same as the tension in the lower part.