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

In summary, the conversation discusses the conservation of mechanical energy and linear momentum in relation to two masses connected by a rope. The final equations for the velocities of the masses are given, and there is a discussion about the amplitude of the motion of one of the masses. The conversation concludes with a disagreement about the correct amplitude, which is resolved by considering the center of mass and the definition of amplitude.
  • #1
lorenz0
148
28
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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  • #2
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?
 
  • #3
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##
 
  • #4
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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  • #5
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?
 
  • #6
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.
 

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

What is the amplitude of oscillation of a pendulum?

The amplitude of oscillation of a pendulum is the maximum displacement of the pendulum from its equilibrium position. It is measured in units of length, such as meters or centimeters.

How is the amplitude of a pendulum related to its length?

The amplitude of a pendulum is directly proportional to its length. This means that as the length of the pendulum increases, the amplitude of its oscillation also increases.

What factors affect the amplitude of a pendulum?

The amplitude of a pendulum is affected by its length, mass, and the force of gravity. A longer pendulum will have a larger amplitude, while a heavier pendulum will have a smaller amplitude. The force of gravity also plays a role in determining the amplitude of a pendulum.

Can the amplitude of a pendulum be changed?

Yes, the amplitude of a pendulum can be changed by adjusting its length or mass. The amplitude can also be affected by external factors such as air resistance or friction.

Why is the amplitude of a pendulum important?

The amplitude of a pendulum is important because it affects the period of its oscillation. A larger amplitude will result in a longer period, while a smaller amplitude will result in a shorter period. This relationship is described by the law of conservation of energy.

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