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e2m2a
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I have a thought experiment that I cannot solve. (This is not a homework assignment. In fact, I have never seen it posted in any physics textbook, probably because of its advanced nature.) I don't know if there is an analytical solution to this problem or if only a numerical approximation is possible. Here is the thought experiment.
Suppose we have a long, thin rod with mass m1 which rotates at one end of the rod on a vertical axis which is rigidly attached to a second object of mass m2 which we will denote as the slider. Viewed from above the slider can move in a straight line along the y-axis of an x-y coordinate system if the slider is not constrained from moving. We assume there is no friction in this thought experiment. The rod rotates in a counter-clockwise direction in a plane parallel to the surface of the slider. The initial angular velocity of the rod we denote as wi. The moment of inertia of the rod is 1/3 m1 L^2, where L is the length of the rod. The slider is initially prevented from moving by a locking mechanism.
At some point in time as the rod rotates and is at an angle < 90 degrees and > 0 degrees with respect to the x-axis of the x-y coordinate system, the locking mechanism is released. The slider begins to move forward in the positive y-direction. This is due to the y-component of the centrifugal reactive force which acts on the axis of the slider. The centrifugal reactive force is equal and opposite to the centripetal force which acts on the center of mass of the rod. The angle of the rod with respect to the x-axis at which angle the locking mechanism is released and the slider begins to move, we denote as phi. At any other angle of the rod greater than phi, but less than 90 degrees, we denote this angle as theta. At any given angle theta, the slider will have a velocity denoted as vs and the angular velocity of the rotator we will denote as simply w.
What would vs and w be as a function of theta, given the initial conditions of wi, phi, m1, m2, and moment of inertia 1/3m1 L^2? What would vs and w be as a function of time, given the same initial conditions? I think the second problem cannot be solved analytically, only with a numerical approximation, and I am more interested in the solution to vs and w as a function of theta.
Also, there is one more additional problem. Suppose the slider makes an inelastic collision with a front bumper at some angle theta. What would the post-collision angular velocity of the rod be given the same initial conditions? Would anyone venture to give a solution to this problem? I think it may involve linear angular momentum, but I am not sure.
Suppose we have a long, thin rod with mass m1 which rotates at one end of the rod on a vertical axis which is rigidly attached to a second object of mass m2 which we will denote as the slider. Viewed from above the slider can move in a straight line along the y-axis of an x-y coordinate system if the slider is not constrained from moving. We assume there is no friction in this thought experiment. The rod rotates in a counter-clockwise direction in a plane parallel to the surface of the slider. The initial angular velocity of the rod we denote as wi. The moment of inertia of the rod is 1/3 m1 L^2, where L is the length of the rod. The slider is initially prevented from moving by a locking mechanism.
At some point in time as the rod rotates and is at an angle < 90 degrees and > 0 degrees with respect to the x-axis of the x-y coordinate system, the locking mechanism is released. The slider begins to move forward in the positive y-direction. This is due to the y-component of the centrifugal reactive force which acts on the axis of the slider. The centrifugal reactive force is equal and opposite to the centripetal force which acts on the center of mass of the rod. The angle of the rod with respect to the x-axis at which angle the locking mechanism is released and the slider begins to move, we denote as phi. At any other angle of the rod greater than phi, but less than 90 degrees, we denote this angle as theta. At any given angle theta, the slider will have a velocity denoted as vs and the angular velocity of the rotator we will denote as simply w.
What would vs and w be as a function of theta, given the initial conditions of wi, phi, m1, m2, and moment of inertia 1/3m1 L^2? What would vs and w be as a function of time, given the same initial conditions? I think the second problem cannot be solved analytically, only with a numerical approximation, and I am more interested in the solution to vs and w as a function of theta.
Also, there is one more additional problem. Suppose the slider makes an inelastic collision with a front bumper at some angle theta. What would the post-collision angular velocity of the rod be given the same initial conditions? Would anyone venture to give a solution to this problem? I think it may involve linear angular momentum, but I am not sure.
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