Blocks of mass m 1 and m 2 are connected by a massless string that passes over t
ID: 2240126 • Letter: B
Question
Blocks of mass m1 and m2 are connected by a massless string that passes over the pulley in the figure . The pulley turns on frictionless bearings, and mass m1 slides on a horizontal, frictionless surface. Mass m2 is released while the blocks are at rest.
Part A
Assume the pulley is massless. Find the acceleration of m1.
Express your answer in terms of the given quantities.
ANSWER:
a1 =
Part B
Find the tension in the string.
Express your answer in terms of the given quantities.
ANSWER:
T =
Part C
Suppose the pulley has mass mp and radius R. Find the acceleration of m1. Verify that your answers agree with part A if you set mp=0.
Express your answer in terms of the given quantities.
ANSWER:
a1 =
Part D
Find the tension in the upper portion of the string. Verify that your answers agree with part B if you set mp=0.
Express your answer in terms of the given quantities.
ANSWER:
Tupper =
Part E
Find the tension in the lower portions of the string. Verify that your answers agree with part B if you set mp=0.
Express your answer in terms of the given quantities.
ANSWER:
Tlower =
a1 =
Explanation / Answer
The net force accelerating m2 is = m2*g - T (with T = tension)
m2g - T = m2a --> T = m2g - m2a.
And the tension is also the force that accelerates m1:
T = m1a
set both equations equal to get
m2g - m2a = m1a --> solve for a:
a(m1 + m2) = m2g
a = m2g/(m1+m2)
T = m1*m2*g/(m1+m2)
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T1 is the tension between m1 and the pulley
T2 is the tension between m2 and the pulley
the net force on m1 is T1 = m1a
the net force on m2 is m2g - T2 = m2a --> T2 = m2g - m2a
the net force on the pulley is T2 - T1
and the torque ? on the pulley is then = F*R = (T2 - T1)R
and the torque ? on the pulley is also ? = I*? = I*a/R (with I = moment of inertia of pulley, ? = angular acceleration of pulley)
--> (T2-T1)R = I*a/R --> divide by R
(T2 - T1) = Ia/R^2 plug in the expressions for T1 and T2:
(m2g - m2a) - m1a = Ia/R^2 --> solve for a:
a(-m2 - m1 - I/R^2) = - m2g
a = m2g/(m1+m2+I/R^2) with I = 1/2*mp*R^2
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plug a into the two equations for T1 and T2:
T1 = m1a
T2 = m2g-m2a