Physics Torque and Rotation Question

Question

Beam rotating from a rotation point, starting 45 degrees above the horizontal and swinging to horizontal. Find angular velocity at the horizontal point.

Answer 1

Use energy conservation.  I assume the pivot point is at the end of the beam.  Assuming the beam is initially at rest, set the initial gravitational potential energy (when at 45°) equal to the final rotational kinetic energy (when at 0°), and solve for the angular speed.

 

θ = initial angle above horizontal = 45°

m = mass of beam

L = length of beam = ?

I = moment of inertia of beam = mL²/3

ω = final angular speed

 

mg(L/2)sinθ = Iω²/2

 

mg(L/2)sinθ = (mL²/3)ω²/2

 

g sinθ = Lω²/3

 

ω = √(3g sinθ/L)

 

Plug in g, θ, and the length L, which was not given in the problem, and get ω in radians per second.