A stick is resting on a concrete step with 15 of its total length 𝐿 hanging over the edge. ?

Question

A stick is resting on a concrete step with  15  of its total length  𝐿  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 at  𝜃=62.1∘  with respect to the horizontal, as shown in the figure. 
 
If the mass of each bug is  3.43  times the mass of the stick and the stick is  17.5 cm  long, what is the magnitude of the angular acceleration  𝛼  of the stick at the instant shown? 
 
Thank you,

NCS · Accepted Answer

These would be some frighteningly large ladybugs.

This problem is really just an exercise in calculating moments of inertia.

The bugs are easy -- they can be treated as point masses.
Bug 1: I1 = mr² = 3.43M * (0.2*0.175m)² = M * 0.0042m²
Bug 2: I2 = 3.43M * (0.8*0.175m)² = M * 0.0672m²

For the stick we have to use the parallel axis theorem. The CoM of the stick is 0.3*0.175m from the fulcrum, so
Is = mL²/12 + md² = M * [(0.175m)²/12 + (0.3*0.175m)²] = M * 0.0053m²

so the MoI for the total assembly is
I = I1 + I2 + Is = M * 0.076738m²
(I carried more precision in my calculations than I showed.)

The net torque produced by the bugs is
τ_b = (0.8 - 0.2)*0.175m * 3.43M * 9.8m/s² * cos62.1º = M * 1.65154m²/s²
and by the stick is
τ_s = M * 0.3*0.175m * 9.8m/s² * cos62.1º = M * 0.24075m²/s²
in the same direction, for a total net torque of
τ = M * 1.8923m²/s²

and now the payoff!
α = τ / I = M*1.8923m²/s² / M*0.076738m² = 24.7 rad/s²
barring computational error.

Hope this helps!

oldschool · Answer

If the stick is 17.5cm long and 15cm is hanging over the edge, it would have fallen of the step.  Doesn't make sense. No figure given either.

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A stick is resting on a concrete step with 15 of its total length 𝐿 hanging over the edge. ?

2 Answers