a) The acceleration of the objects is [tex]3.56 m/s^2[/tex]
b) The tension in the string is 280.8 N
Explanation:
a)
We start by writing the equations of motion for the two masses attached to the pulley.
For the heavier mass, we have:
[tex]m_1 g - T = m_1 a[/tex] (1)
where
[tex]m_1 = 45.0 kg[/tex] is the mass
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
T is the tension in the string
a is the acceleration of the system (here we assumed that the heavier mass accelerates downward)
For the lighter mass, we have
[tex]T-m_2 g = m_2 a[/tex] (2)
where
T is the tension in the string
[tex]m_2 = 21.0 kg[/tex] is the mass
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
a is the acceleration of the system (here we assumed that the lighter mass accelerates upward)
From (1) we get
[tex]T=m_1g - m_1 a[/tex]
And substituting into (2),
[tex](m_1 g - m_1 a)-m_2 g = m_2 a\\(m_1 -m_2)g  = (m_1+m_2)a\\a=\frac{m_1 - m_2}{m_1+m_2}g=\frac{45-21}{45+21}(9.8)=3.56 m/s^2[/tex]
b)
From the previous part of the problem we got an expression for the tension in the string:
[tex]T=m_1g - m_1 a[/tex]
Where we have
[tex]m_1 = 45.0 kg[/tex]
[tex]g=9.8 m/s^2[/tex]
[tex]a=3.56 m/s^2[/tex] is the acceleration, found in part a)
Susbtituting, we find
[tex]T=(45.0)(9.8)-(45.0)(3.56)=280.8 N[/tex]
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