A stick is resting on a concrete step with 1/6 of its length hanging over the ed

Question

A stick is resting on a concrete step with 1/6 of its length hanging over the edge. A single ladybug lands on the end of the stick hanging over the edge, and the stick begins to tip. A moment later, a second, identical ladybug lands on the other end of the stick, which results in the stick coming momentarily to rest 46.5° from the horizontal. If the mass of each bug is 3.60 times the mass of the stick and the stick is 17.5 cm long, what is the magnitude of the angular acceleration of the stick at the instant shown?

Explanation / Answer

Le M is the mass of stick

then, mass of each bug, m = M/3

length of stick, L = 17.5 cm = 0.175 m

moment of inertia of stick about axis of rotation, I_stick = M*L^2/12 + M*(L/2 - L/6)^2

= 0.194*M*L^2

moment of ineria of bugs, I_bugs = m*(L/6)^2 + m*(5*L/6)^2

= (M/3)*(L/6)^2 + (M/3)*(5*L/6)^2

= 0.24*M*L^2

Torque acting on the stick, T = m*g*(5*L/6)*sin(90-46.5) - m*g*(L/6)*sin(90-46.5)

T = (M/3)*g*(5*L/6)*cos(46.5) - (M/3)*g*(L/6)*cos(46.5)

= 0.13385*M*g*L

now Apply, T = I*alfa

==> alfa = T/I

= 0.13385*M*g*L/(0.24*M*L^2)

= 0.5577*g/L

= 0.5577*9.8/0.175

= 31.23 rad/s^2

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