A paddle for a boat is constructed of a stick of length L and a parabolic end piece. The end can be thought of as a uniform flat board of mass m that has been cut to form a parabola given by y = x^2, so that the length from the trough (i.e., the y = 0 point of the parabola) to the end of the parabola is H. To simplify this a little, assume that the stick is massless. This paddle is stood vertically upward on the stick end and attached to a hinge on the floor so that it is free to rotate in the in/out of the page direction. When it is standing straight up that is an unstable equilibrium. A very small infinitesimal push is given to get the paddle moving and it swings downward under gravity (acceleration g). NOTE: the diagram for this question is included in the image attached! (a) When the paddle is standing vertically upward, what is the position of the center of mass of the paddle? (b) What is the moment of inertia of the paddle for rotations in/out of the page about an axis through the center of mass? (c) What is the angular velocity of the paddle just before it hits the ground?

A paddle for a boat is constructed of a stick of length L and a parabolic end piece. The end can be thought of as a uniform flat board of mass m that has been cut to form a parabola given by y = x^2, so that the length from the trough (i.e., the y = 0 point of the parabola) to the end of the parabola is H. To simplify this a little, assume that the stick is massless. This paddle is stood vertically upward on the stick end and attached to a hinge on the floor so that it is free to rotate in the in/out of the page direction. When it is standing straight up that is an unstable equilibrium. A very small infinitesimal push is given to get the paddle moving and it swings downward under gravity (acceleration g). NOTE: the diagram for this question is included in the image attached! (a) When the paddle is standing vertically upward, what is the position of the center of mass of the paddle? (b) What is the moment of inertia of the paddle for rotations in/out of the page about an axis through the center of mass? (c) What is the angular velocity of the paddle just before it hits the ground?

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