What you have to know Dynamics of rotational motion Torque Equation of motion for a rotational motion Rotation about a moving axis Work & power Angular momentum and its conservation
Torque Translational motion: governed by forces Rotational motion: governed by torques When you define a rotational motion, you should specify the axis of rotation. Therefore, physical quantities in rotational motion must include information about the location of the axis of rotation. This makes the definitions of those physical quantities include x or r. For example, even if the same force (same magnitude & same direction) is applied, the rotational motion depends on where the force exerts on, i.e., the distance between the axis of ration and the point where the force exerts on. When you have a rotational motion, determine whether the rotation is clockwise or counterclockwise.
Torque definition This force causes counterclockwise rotation. (So, is positive) Axis of rotation
Torque Line of action of the force F Lever arm Only the tangential force can give a torque.
Torque for a rigid body Note the similarity of the two equations. Rotational analog of Newton’s 2 nd law for a rigid body
Ex 10.2 Same situation as in Ex 9.8 Free-body diagram Direction of the rotation & sing of the torque Reproduce the result of Ex 9.8 as explained in the class
Ex 10.3 Same situation as in Ex 9.9 Free-body diagram cylinderblock Obtain v !
Rigid-body rotation about a moving axis Extend the analysis to the case where the axis of rotation moves Translation + Rotation cm
Rolling without slipping
Translation + Rotation For translation, concentrate the mass of the object on its CM same as the motion of a point-like particle For rotation, consider the rotation about CM same as the rotational motion when the CM is fixed. The second equation is valid, if 1.The axis through the CM must be an axis of symmetry 2.The axis must not change direction
Ex 10.4 Speed of a primitive yo-yo initialfinal
Ex 10.5 Race of the rolling bodies
Ex 10.6 Acceleration of a primitive yo-yo (cf. Ex 10.4) Rolling without slipping Try to reproduce the result of Ex 10.4
Ex 10.7 Solid bowling ball (rolling without slipping) Free-body diagram If is small, the friction force which is required to have the rolling is small. If is large, the ball easily slips, then we need a large friction force to make it roll.
Work & Power in rotational motion Work To rotate a body, a force should be applied. Then this force do work on it.
Work –energy theorem
Angular momentum (linear) momentum p Similarly, we define angular momentum L This is defined with r, so it depends on the choice of the origin. This is a vector product. Therefore, the angular momentum is perpendicular to the plane spanned by r and p. If r and p lie on the xy-plane, L is in the z-direction. This is always true.
Note the similarity.
Angular momentum of a rigid-body This is valid if the rigid-body lies on the xy-plane.
Angular momentum of a rigid-body
Conservation of angular momentum
Ex d Collision problem including rotational motion. use angular momentum conservation before after Use the numbers given in the textbook to verify the result. Also check the change in kinetic energies.
Gyroscopes & precession The motion of gyroscope was discussed in the class. Here I do not repeat it. But you should understand that the gyroscope has precession and this is due to the vector nature of angular momentum.