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fredd [130]
3 years ago
10

A thin, uniform stick of mass M and length L is at rest on a flat, frictionless surface to which one end of it is pinned. A smal

l mass m traveling at speed v collides with and attaches to the stick at a distance 2L/3 away from the end through which it is pinned to the surface. (a) Find an expression for the moment of inertia of the stick mass object after the collision. (b) Find an expression for the final angular speed of the combined object
Physics
1 answer:
labwork [276]3 years ago
6 0

Answer:

a)  I = (\frac{M}{3} + \frac{4m}{9}) L²  ,   b)     w = (\frac{27 M}{18 m} + 2)⁻¹  Lv₀

Explanation:

a) The moment of inertia is a scalar that represents the inertia in circular motion, therefore it is an additive quantity.

The moment of inertia of a rod held at one end is

         I₁ = 1/3 M L²

The moment of inertia of the mass at y = L

        I₂ = m y²

 

The total inertia method

        I = I₁ + I₂

        I = \frac{1}{3} M L² + m (\frac{2}{3} L)²

        I = (\frac{M}{3} +\frac{4m}{9} ) L²

   

b) The conservation of angular momentum, where the system is formed by the masses and the bar, in such a way that all the forces during the collision are internal.    

Initial instant. Before the crash  

       L₀ = I₂ w₀  

angular and linear velocity are related  

       w₀ = y v₀  

      w₀ = \frac{2}{3}L v₀  

      L₀ = I₂ y v₀  

Final moment. After the crash  

      L_{f} = I w

 

how angular momentum is conserved  

      L₀ = L_{f}

      I₂ y v₀ = I w

substitute

      m (\frac{2L}{3})² (\frac{2L}{3} v₀ =  (\frac{M}{3} +\frac{4m}{9} ) L²  w

      \frac{6}{27}  m L³ v₀ = (\frac{M}{3} +\frac{4m}{9} ) L²  w

        \frac{6}{27}  m L v₀ = (\frac{M}{3} +\frac{4m}{9} )   w

        L v₀ = (\frac{27 M}{18 m} + 2)  w     

       w = (\frac{27 M}{18 m} + 2)⁻¹  Lv₀

 

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Answer:

The chance in distance is 25 knots

Explanation:

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\displaystyle{2s\frac{ds}{dt}= 2x_B\frac{dx_B}{dt}+2y_A\frac{dy_A}{dt}}  (3)

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\displaystyle{x_B = v_B\cdot t= 20 \cdot 2 = 40\ nmi

and applying Pythagoras:

\displaystyle{s = \sqrt{x_B^2+y_A^2}=\sqrt{30^2 + 40^2}=\sqrt{2500}=50}

Now substituting all the values in (3) and solving for  \displaystyle{\frac{ds}{dt} } we get:

\displaystyle{\frac{ds}{dt} = \frac{1}{2s}(2x_B\frac{dx_B}{dt}+2y_A\frac{dy_A}{dt})}\\\displaystyle{\frac{ds}{dt} = 25 \ knots}

4 0
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