SPH4U: Lecture 8 Notes - The Burns Home Page

How will the bodies move?From the free body diagrams for each body, and the chosencoordinate system for each block, we can apply Newton’sSecond Law:Taking “x” components:1) T 1 - m 1 g sin q 1 = m 1a 1X2) T 2 - m 2 g sin q 2 = m 2 a 2XBut T 1 = T 2 = Tand -a 1X = a 2X = a(constraints)yNm 1m 1 gq 1xT 1T 2xm 2 gyNm 2Solving the equationsUsing the constraints, we get 2 eqn and 2 unks,solve the equations.T - m 1 gsin q 1 = -m 1 a(a)T - m 2 gsin q 2 = m 2 a(b)Subtracting (a) from (b) gives:m 1 gsin q 1 - m 2 gsin q 2 = (m 1 +m 2 )aSo:m ma 1 sin q1 2 sin q2gm1 m2q 2SPH4U: Lecture 8, Pg 18SPH4U: Lecture 8, Pg 17m 1m 2q 1 q 2m ma 1 sin q1 2 sin q2gm1 m2m m 1 2m ma 1 sin q1 2 sin q2gm1 m2q 1 q 2Special Case 1:Special Case 2:BoringTTAtwood’s Machinem 1m 2If q 1 = 0 and q 2 = 0, a = 0.m1m2If q 1 = 90 and q 2 = 90,m ma ( 1 2 )( m m )g1 2SPH4U: Lecture 8, Pg 19SPH4U: Lecture 8, Pg 20Page 5