- #1
Uchida
- 22
- 6
- Homework Statement
- Consider a determined structure of three particles, each with mass m, interconnected by rigid bars of negligible mass forming an equilateral triangle on the side a.
The structure falls from a height h, as shown in the figure and the particle C hooks on the stop.
The angular velocity of the structure immediately after the impact is:
- Relevant Equations
- Conservation of Linear Momentum
I solved it by two methods:
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First, by conservation of linear momentum, using the vector velocities of each particle:
In the imminence of the impact, the velocity of all the three particles are the same, \vec v_0 = - \sqrt{2gh} \hat j
After que impact, C will have zero velocity, and particles A and B will have velocities with the same magnitude v_f, in the perpendicular direction of the bars connected to them. Therefore:
\vec v_A = - v_fsin(30º) \hat i - v_fcos(30º) \hat j
\vec v_B = v_fsin(30º) \hat i - v_fcos(30º) \hat j
By conservation of linear momentum:
m_A\vec v_0 + m_B\vec v_0 + m_C\vec v_0 = m_A\vec v_A + m_B\vec v_B + m_B\vec v_C
Considering m_A = m_B = m_C = m and \vec v_C = 0, we have:
3m\vec v_0 = 2m(\vec v_A + \vec v_B)
-3\sqrt{2gh} \hat j = -2v_fcos(30º) \hat j
Which give us: v_f = ||v_A|| = ||v_B|| = \sqrt{6gh}
The angular speed of the rigid body can be calculated as ω = \frac{v_t}{r}. where v_t = v_f and r = a.
Finally:
ω = \frac{\sqrt{6gh}}{a}-----------------------------------------------------
The second method, by conservation of linear momentum of the center of gravity of the rigid body.
This method is more straight forward, and the calculations give the velocity of CG after the impact equal to the velocity in the imminence of the impact. v_{f_{CG}} = v_{0_{CG}} = \sqrt{2gh}
The distance of CG from the particle C is r_{CG} = a\frac{2}{3}{cos(30º)} = \frac{a \sqrt{3}}{3}
The angular speed of the rigid body can be calculated as ω = \frac{v_{f_{CG}}}{r_{CG}}.
Finally:
ω = \frac{\sqrt{6gh}}{a}
The problem is, that the correct answer to the question is:
The "correct" answer has a 2 dividing the result I found, and there is no answer key to the result I got.
Can someone please help me find where I am wrong, or validate my methods (which would mean that the answer key to the question must be wrong).