CP1: Atwood\'s machine with additional mass Two masses (m 1 and m 2 ) are hangin

Question

CP1: Atwood's machine with additional mass

Two masses (m1 and m2) are hanging from a cord, threaded through a massless pulley. The pulley is hanging from the ceiling by cord C.

In addition, an additional third mass (m3) is hung from mass 1 by an additional cord.

a. Derive an expression for the acceleration of mass 1 in terms of m1, m2, m3, and g (gravity). Make sure to start with the FBD for each of the three masses plus the pulley, and then start from first principles.

b. If m1=1.0kg, m2=3.0kg, and m3=7.0kg, calculate the tension in cord C.

c. Re-derive the acceleration, and the tension in cord C if mass 3 is instead hung from mass 2. You may start from first principles, or you may use symmetry arguments to compare with parts a and b.

Make sure to always justify your answer by beginning with Newton's second law/force balances. Do not short-cut the problems. Make sure to draw a FBD for each box separately, identifying all the forces acting on the objects.

I appreciate your time and thank you very much. (:

Explanation / Answer

(a)
Let the acceleration be a

(m3 + m1)*g - T = (m3 + m1)*a
T = m2* a + m2*g
Substituing Values,

(m3 + m1)*g - (m2* a + m2*g) = (m3 + m1)*a
(m1 - m2 + m3) * g = (m1 + m2 + m3) * a
a = [(m1 - m2 + m3)/(m1 + m2 + m3)] * g

(b)
m1=1.0 kg, m2=3.0 kg, and m3=7.0 kg
Tc = 2*T
Tc = 2 * ( m2* a + m2*g)
Tc = 2 * m2 * (a+g)
Tc = 2 * 3.0 * ( [(1 - 3 + 7)/(1 + 3 + 7)] * 9.8  + 9.8)
Tc = 85.5 N

(c)
Let the acceleration be a

(m3 + m2)*g - T = (m3 + m2)*a
T = m1* a + m1*g
Substituing Values,

a = [(- m1 + m2 + m3)/(m1 + m2 + m3)] * g

Tension in the cord,

Tc = 2*T
Tc = 2 * ( m1* a + m1*g)
Tc = 2 * m1 * (a+g)
Tc = 2 * 1.0 * ( [(-1 + 3 + 7)/(1 + 3 + 7)] * 9.8  + 9.8)
Tc = 35.64 N

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