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Unformatted text preview: The yo-yo can rotate freely about a horizontal axis through its centre. It has a
mass of 2 kg, a moment of inertia about the axis of 0.03 kg-mz. and inner and
outer radii of n = 0.1m and r2 = 0.2m. Masses m1 = 1.0kg and m2 = 1.5kg
hang from cords which are wrapped round the inner and outer radii,
respectively. (a) How are the velocities of In] and m; related? (b) Write down an expression for the kinetic energy of the system, and hence
determine the acceleration of mg. (a) v; = (OR, v1= (or ifvz = v, v1= v12 (v2 down, V] pp) (b) K = m1v12/2 + m2V22/2 + 1032/2 with m = v/R
= 0.5(1.5v2’ + 0.5(v/2)2 + 0.5(0.03)(v/R)2
= 0.75v2 + 0.12512 + 0.375v2 = 1.25v2 Let 1112 fall a distance h, then 1111 rises h/2.
Work done by gravity Wg = ngh - m1g(h/2) = gh
By work-energy theorem K = Wg 1.25va2 = gh 2.5 v dv/dt = g dh/dt
But dh/dt = v
dv/dt = a2 = g/2.5 = 0.4g
= 3.92 m/s2 The question clearly intended you to use energy methods to solve the
problem, but if you insisted on using Newtonian methods the solution is as follows. T2 1|- ”:‘3 “I3
1.5g- T. = 1.5212 (1)
T1- 1.0g = 1.0a1= 0.5a2 .............. (2)
TzI'z - T]I‘1=IOL : lag/['2 .................(3) From (3) T2 - 0.5T1= 0.75a2
.'. 1.5g ‘ 1.53.2- 0.2532 - 0.5g = 0.7532
2.532 = g
a; : 0.4g ...
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- Fall '10