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Causes of Rotation AP Physics SUM THE TORQUES Newton’s Second Law (with rotation) How much torque is needed to rotate? You can sum the forces acting on an object and apply Newton’s Second Law for linear motion. Newton’s Second Law can be applied to rotational motion as well, using rotational quantities. I Sum the torques Torque replaces force Inertia replaces mass Angular acceleration replaces linear acceleration Inertia Rotational Inertia or Moment of Inertia Similar to mass r – radial distance from axis of rotation Unit: kg m² I mr 2 Two weights on a bar Two ‘weights’ of mass 5.0 kg and 7.0 kg are mounted 4.0 m apart on a light rod (ignore mass). Calculate the moment of inertia of the system when rotated about an axis halfway between the weights. 48 kg m² What about angular acceleration? Continue… Using the same system, calculate the moment of inertia of the system when rotated about an axis 0.5 m to the left of the 5.0 kg ‘weight’. 143 kg m² Inertia and Rolling Inertia and angular acceleration are inversely related (mass and acceleration are inversely related). I Objects rolling with low inertia will have high angular acceleration (low mass requires less force to cause acceleration). Easier to change speed. Objects rolling with high inertia will have low angular acceleration (high mass requires more force to cause acceleration). Harder to change speed. Rolling Ring and Disk A disk and ring of equal mass and radius rolling down a ramp from rest: Disk gets to bottom faster. Disk has greater acceleration. A disk and ring or equal mass and radius rolling up a ramp with same initial speed: Ring will roll higher. Ring has lower acceleration, and slows at a lower rate. I ring mr I disk 2 1 2 mr 2 Various Moments of Inertia Parallel Axis Theorem The moments of inertia of rigid objects with simple geometry are relatively easy to calculate provided the rotation axis coincides with an axis of symmetry. The calculation about an arbitrary axis can be found by adding the rotational inertia about the center of mass and the rotational inertia of the center of mass about the arbitrary axis. The arbitrary axis must be parallel to the center of mass axis of rotation. I I cm mr 2