Lesson_Text_4.2 - 1 Lesson 4.2 Torque and Rotation 1 Torque...

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1 Lesson 4.2 Torque and Rotation. 1. Torque of a force. A rotating rigid body always has an axis of rotation. A tangential force applied on a disc results in the disc rotating about an axis passing through its center. The rotation effect produced by a force depends on the distance of direciton of rotation axis of rotaiton Fig. 1 A rigid body rotates about an axis . . F r Fig. 2 A tangential force produces rotation about an axis through the center of the disc O the force from the axis of rotation. For example a force applied on the door handle rotates the door about an axis passing through the hinges. If the door handle is placed closer to the hinges, you will have to apply a much larger force to open or close the door. The rotation effect can be increased by moving the force farther away from the axis of rotation. The rotation effect produced by a force on an object about an axis is called the torque of the force or the moment of the
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2 force. We use the letter τ to represent torque. Torque is measured as the product of the force and the perpendicular distance between the force and the axis of rotation. . 0 F r F θ r F sin θ Fig. 3a The force is perpendicular to the torque arm Fig. 3b The force makes an angle θ with the torque arm O Consider the situation shown in fig. 3 above. A crank shaft of length r is fixed at one end O and a force F is applied at the other end to rotate the crank shaft about the point O. Here r is called the torque arm. In 3a, the force is applied at right angles to the length r and therefore, the perpendicular distance between the force and the axis of rotation is r, the torque arm. In this case, the torque of F about O can be written as: τ = r × F …2 In 3b, the force is applied at an angle θ with r so that the component of the force perpendicular to the torque arm is F sin θ . Therefore, the torque of F about O in this case is given by: τ = r × F sin θ …3 Notice here that equation (3) will reduce to equation (2) when θ = 90 o . Therefore, equation (3) is more general and can be used in any situation. 2. Cross Product of two vectors. Unlike the dot product of two vectors, the cross product of two vectors is a vector. The magnitude of the cross product of two vectors A and B is given by: sin A B A B θ × = combarrowextenderarrowrightnosp combarrowextenderarrowrightnosp ….4
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3 Since the cross product is a vector, it has a direction. If vectors A and B lie in the xy plane, their cross product will be directed along the z-axis. If A × B is directed along the positive z-axis, B × A will be directed along the negative z-axis. This means that: A × B = - B × A If u and v are two vectors defined in the component form as: u = 2i – 3j + 5k and v = i + 2j + 4k, then we use the following method to find u × v.
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