A pivoted beam has mass m_1 suspended from one end and an Atwood\'s machine sus-

Question

A pivoted beam has mass m_1 suspended from one end and an Atwood's machine sus-pended from the other with masses m_2 and m_3 suspended on either side. The frictionless pulley has negligible mass and size. Find the relation between. (a) Find the relation between m_1. m_2, m_3, l_1. and l_2 which will ensure that the beam has no tendency to rotate just after the masses are released. (b) What would you predict the relation would be in the case that all three masses are equal? Does your answer from part (a)agree with that prediction?

Explanation / Answer

a)

Using Newton's 2nd law ( Fnet = ma )

on m2:

m2g - T = m2a

a = g - (T / m2)
on m3:

T - m3 g = m3a

T - m3 g = m3 (g - (T / m2))

T - m3 g = m3g - T(m3/m2)

T (1 + m3/m2) = 2m3g

T = 2 m3g / (1 + m3/m2)

T = 2 m2 m3 g / (m2 + m3)

so force on right end;

Fr = 2T = 4 m2 m3 g / (m2 + m3)

force on left side end/

FL = m1g

balancing torque Or moment (r x F) about pivot point,

(L1 x m1g) - (L2 x 4 m2 m3 g / (m2 + m3) ) = 0

m1L1 = 4 L2 m2 m3 / (m2 + m3)

L1 = 4 (m2m3 / (m1 (m2 + m3))) L2

b) if m1 = m2 = m3 = m

L1 = 4 ( m^2 / m(2m)) L2

L1 = 2L2

Yes its seems correct.

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