Amplitude of oscillation of a mass which is the pivot of a pendulum

AI Thread Summary
The discussion centers on the calculation of the amplitude of oscillation for two masses in a pendulum system. Using conservation of mechanical energy and linear momentum, the velocities of the masses are derived, leading to the expression for the amplitude of mass m2 as A2=2l sin(α). There is confusion regarding the amplitude of mass m1, with calculations suggesting it should be A=m2/(m1+m2) l sin(α), which is half of what some texts indicate. Clarification is sought on whether the term "amplitude" refers to the total distance between turning points or a different measurement. The consensus is that the text may be using a broader definition of amplitude.
lorenz0
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Homework Statement
Two bodies (see figure) are connected by an ideal rope of length ##l = 0.9 m##. The body with mass ##m_1 = 2 kg## is free to slide without friction along a rigid horizontal rod. The second body has mass ##m_2 = 3 kg##. The bodies are both initially stationary in the indicated position (##\alpha = 60 °##) when they are left free to move. Find:
a) the velocities ##v_1## and ##v_2## of the two bodies when they are vertically aligned;
b) the amplitude ##A## of the motion of ##m_1##.
Relevant Equations
##U_g=mgh, E_i=E_f, E=U+K, P_i=P_f##
1) By conservation of mechanical energy we have ##m_2gl(1-\cos(\alpha))+m_1gl=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2+m_1gl## and by conservation of linear momentum along the x-axis we have ##m_1v_1+m_2v_2=0## which gives us ##v_2=\sqrt{\frac{2m_1gl(1-\cos(\theta))}{m_1+m_2}}## and ##v_1=-\frac{m_2}{m_1}v_2##

2) For ##m_2## I think that the amplitude should be ##A_2=2l\sin(\alpha)## but I don't see how find out the amplitude of ##m_1## so I would appreciate an hint about how to find it, thanks.
 

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lorenz0 said:
2) For ##m_2## I think that the amplitude should be ##A_2=2l\sin(\alpha)##
Wouldn't that be the amplitude relative to ##m_1##?
Where is the mass centre when the rope is vertical?
 
haruspex said:
Wouldn't that be the amplitude relative to ##m_1##?
yes, and my problem is that I don't see how I can go from that to the "general" amplitude of ##m_1##
 
haruspex said:
Wouldn't that be the amplitude relative to ##m_1##?
Where is the mass centre when the rope is vertical?
vertically along the rope, at position ##\frac{m_1}{m_1+m_2}\cdot l## wrt mass ##m_2## or at position ##\frac{m_2}{m_1+m_2}\cdot l## with respect to ##m_1##. Perhaps since there are no horizontal external forces the x-position of the CM is constant
 
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haruspex said:
Wouldn't that be the amplitude relative to ##m_1##?
Where is the mass centre when the rope is vertical?
By imposing that the x-position of the center of mass (calculated wrt a frame centered on ##m_1##) is constant I get that the amplitude of the motion of ##m_1## is ##\frac{m_2}{m_1+m_2}l\sin(\alpha)=A## but according to the text I am using it should be twice as much. Any thoughts?
 
lorenz0 said:
By imposing that the x-position of the center of mass (calculated wrt a frame centered on ##m_1##) is constant I get that the amplitude of the motion of ##m_1## is ##\frac{m_2}{m_1+m_2}l\sin(\alpha)=A## but according to the text I am using it should be twice as much. Any thoughts?
This looks correct to me. Perhaps the text is using "amplitude" to mean the total distance between the left and right turning points of ##m_1##. Your interpretation is the same as what I would have assumed.
 
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