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Question
Two masses 8 kg and 12 kg are connected at the two ends of a light, inextensible string that goes over a frictionless pulley. Find the acceleration of the masses, and the tension in the string when the masses are released.
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Solution
The given system of two masses and a pulley can be represented as shown in the following figure:
Smaller mass, m1 = 8 kg
Larger mass, m2 = 12 kg
Tension in the string = T
Mass m2, owing to its weight, moves downward with acceleration a,and mass m1moves upward.
Applying Newton’s second law of motion to the system of each mass:
For mass m1:
The equation of motion can be written as:
T – m1g = ma … (i)
For mass m2:
The equation of motion can be written as:
m2g – T = m2a … (ii)
Adding equations (i) and (ii), we get:
(m_2-m_1)g = (m_1+ m_2)a
`:.a = ((m_2-m_1)/(m_1+m_2))g ...(iii)`
`=((12-8)/(12+8)) xx 10= 4/20 xx 10 = 2 "m/s"^2`
Therefore, the acceleration of the masses is 2 m/s2.
Substituting the value of a in equation (ii), we get:
`m_2g - T = m_2 ( (m_2-m_1)/(m_1+m_2))g`
`=((2m_1m_2)/(m_1+m_2))g`
`=(2xx12xx8)/(12+8)xx10`
`=(2xx12xx8)/20 xx 10 = 96 N`
Therefore, the tension in the string is 96 N.
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