R , A coin with a center hole is wobbling about a frictionless pole. A torque...

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R , A coin with a center hole is wobbling about a frictionless pole. A torque T is applied to the coin and is always along the central axis (blue) of the coin. At the contact point p between the coin and the ground, the coin slides with no friction in the radial direction, as the blue arrow shows. In the tangent direction at point p, however, the coin does not slide and maintains a gear-like contact, rolling in the tangent direction. Gravity g exist is the -R (red) direction. The coin has mass m, moment of inertia 12 in the direction, and moment of inertia I when measured from the side ( and I directions) Derive the dynamic equations of this system using angles 0 and 1 by answering the following problems: (1) Describe unit vector R (red), which is fixed in the universe, by means of coordinate system , , k (blue), which is fixed to the wobbling coin. Note that is perpendicular to the face of the coin. (2) Use , , (blue) to describe vector ' and ' (green), which is transformed from I,, R (red) through one rotation of angle & about the axis R. (3) Use , , (blue) to describe vector R" (orange), which is transformed from ', j',' (green) through one rotation of angle 0 about the axis i'. (4) Calculate the velocity of the contact point p at the ground, up, which is only in the ' (green) direction, using the coordinate system , , k (blue). (5) Calculate the angular velocity of the coin using the coordinate system , , k (blue). Prove that the answer is W = (- sin cos 0 + cos ) + ( sin 0 + 0)) + (1) cos 0 cos 0 + sin ) k K R , A coin with a center hole is wobbling about a frictionless pole. A torque T is applied to the coin and is always along the central axis (blue) of the coin. At the contact point p between the coin and the ground, the coin slides with no friction in the radial direction, as the blue arrow shows. In the tangent direction at point p, however, the coin does not slide and maintains a gear-like contact, rolling in the tangent direction. Gravity g exist is the -R (red) direction. The coin has mass m, moment of inertia 12 in the direction, and moment of inertia I when measured from the side ( and I directions) Derive the dynamic equations of this system using angles 0 and 1 by answering the following problems: (1) Describe unit vector R (red), which is fixed in the universe, by means of coordinate system , , k (blue), which is fixed to the wobbling coin. Note that is perpendicular to the face of the coin. (2) Use , , (blue) to describe vector ' and ' (green), which is transformed from I,, R (red) through one rotation of angle & about the axis R. (3) Use , , (blue) to describe vector R" (orange), which is transformed from ', j',' (green) through one rotation of angle 0 about the axis i'. (4) Calculate the velocity of the contact point p at the ground, up, which is only in the ' (green) direction, using the coordinate system , , k (blue). (5) Calculate the angular velocity of the coin using the coordinate system , , k (blue). Prove that the answer is W = (- sin cos 0 + cos ) + ( sin 0 + 0)) + (1) cos 0 cos 0 + sin ) k K