Consider the pulley system shown below. A block with mass m_1 is lifted by a sec
ID: 2077965 • Letter: C
Question
Consider the pulley system shown below. A block with mass m_1 is lifted by a second block with equal mass m_2 (m_1 = m_2 = m). The spool mass moment of inertia is t_0, and the spool has internal and external radiuses equal to R_1 and R_2, respectively. Assume that: The ropes connecting the blocks to the spool are unstretchable and have negligible masses. All forces are conservative. (a) Write the kinematic equations that relate the translational motion of the blocks and the rotational motion of the spool (i.e., relationship between displacement of each block and the rotation of the spool) (b) Write the sum of the kinetic energies of each sub-system (c) Write the sum of the potential energies of each sub-system (d) Write the conservation of mechanical energy (e) Write the equation of motion of the block (m_1) in terms of its linear displacement.Explanation / Answer
(A) after m1 is displaced x distance then velocity of m1 is v1 then,
w = v1 / R1 ....angular velocity of spool at that moment
velocity of block m2, v2 = w R2 = v1 R2 / R1
m1 displaced by x distance then pulley will revove by theta
theta = x / R1
then displacement of m2 , x2 = x R2 / R1
(B) initially:
K_m1 = 0 , K_m2 = 0 and K_pulley = 0
total initial KE = 0
finally,
k_m1 = m1 v1^2 /2
k_m2 = m2 v2^2 /2 = m2 (v1 R2 / R1)^2 /2 = m2 v1^2 R2^2 / 2 R1^2
K_Pulley = Io w^2 /2 = Io v1^2 / 2 R1^2
total KE = m1 v1^2 /2 + m2 v1^2 R2^2 / 2R1^2 + Io v1 / 2 R1^2
= ( v1^2 / 2 R1^2 ) [ m1 R1^2 + m2 R2^2 + Io ]
(c) initially, PE = 0
finally, PE = m1 g x - m2 g x2
= m1 g x - m2 g (x R2 / R1)
= (g x / R1) [ m1 R1 - m2 R2 ]
(d) initial PE + KE = final PE + KE
0 + 0 = (g x / R1) [ m1 R1 - m2 R2 ] + ( v1^2 / 2 R1^2 ) [ m1 R1^2 + m2 R2^2 + Io ]
(e) ( v1^2 / 2 R1^2 ) [ m1 R1^2 + m2 R2^2 + Io ] = (g x / R1) [ m2 R2 - m1 R1]
v1^2 = 2 g x R1 [ m2 R2 - m2 R1] / (m1 R1^2 + m2 R2^2 + Io)
v1 = sqrt[ 2 g x R1 [ m2 R2 - m2 R1] / (m1 R1^2 + m2 R2^2 + Io)]
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