Heavy Pulleys. Pulleys are classically considered to have zero mass and zero ine

Question

Heavy Pulleys. Pulleys are classically considered to have zero mass and zero inertia. This is however not the case and these mass properties do affect the solution. Consider first the single heavy pulley shown in Fig. (a) below and determine a relation between the tensions (in and out of the pulley) and the angle theta (and/or its derivatives). Proceeding as in (a), analyze the system shown in Fig. (b) which is composed of two identical, heavy pulleys through which is passing an inextensible cable connected to the mass m resting on a spring and dashpot foundation. Note that the left pulley moves up and down in addition to rotating. Determine all necessary equations to obtain the equation of motion of the system (i.e. a differential equation for x).

Explanation / Answer

Answer (a)

FORCE RELATION FOR MASS:-

mg-T = ma

Mg-T =ma

Put it together, T = a/R2

mg = ma(1+I/MR2)

TORQUE RELATION FOR PULLEY:-

€=Ia

€=Tr

a=a/R

TR = 1a=1a/R

Note Postive down

a=g(I/1+1/MR2)

Answer (b)

-The mechanical energy of the system is conserved.

-Theblocks undergo changes in transtational kinetic energy and gravitational potential energy.

- The pulleys undergoes a change inpotential kinetic energy.

(A) M1>M2

(B)M1=M2

(C)M1<M3

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