#  Angular speed, acceleration  Rotational kinematics  Relation between rotational and translational quantities  Rotational kinetic energy  Torque 

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 Angular speed, acceleration  Rotational kinematics  Relation between rotational and translational quantities  Rotational kinetic energy  Torque  Rigid body in equilibrium, under a net torque  Angular momentum and its conservation  Precession  Rolling of rigid bodies Rotational motion (chapter ten)

Angular speed Relation between linear and angular displacement The angle here must be measured in radians (conversion:  radians = 180°) The average angular speed can be defined from the angular displacement in some time interval as And the instantaneous angular speed as Units: rad/s (rad is not a “real” unit) r  s

Angular acceleration Likewise, we can define average and instantaneous accelerations as For a rigid body (points remain fixed with respect to each other) all parts have same angular speed and acceleration These quantities are vectors – their direction is found using the right hand rule

Rotational kinematics Using the same derivations as we used to find the equations of linear kinematics, we find For constant angular acceleration

Relation between rotational and translational quantities The angular kinematic variables will be related to the linear ones for points on a rigid body according to: We also know the expression for the centripetal acceleration Note this is for the tangential acceleration

Therefore, the total acceleration vector at any instant is Note: rotational variables depend on choice of axes (location)  ConcepTest  Examples

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Rotational kinetic energy The total kinetic energy of an extended object can be found by adding up the energy of its constituents. For a collection of discrete particles: Where I is a quantity named the moment of inertia

Rotational kinetic energy The resulting equation has a similar form to the one for the linear (translational) kinetic energy Note that this is not a new type of kinetic energy – only an expression for the translation KE for a rotating rigid body For a continuous object where  is the density and V the volume, i.e., the second integral is over the spatial dimensions of the object.  ConcepTest  Examples

Moments of inertia of common objects Solid cylinder, radius R: I=½MR 2 Hollow cylinder, radii R 1 and R 2 : I=½M(R 1 2 +R 2 2 ) Thin cylindrical shell, radius R: I=½MR 2 Solid sphere: (2/5)MR 2 Thin spherical shell: (2/3) MR 2 Thin rod, length L, about center: ML 2 /12 Thin rod, length L, about end: ML 2 /3

Torque Torque is the rotational quantity corresponding to force – it represents the tendency of a force to cause an object to rotate. The expression for torque is: This is a vector product. The magnitude of the vector product is The direction of the vector product is perpendicular to the plane formed by the two vectors forming the product, in a direction given using the right hand rule.

Torque We can think about the magnitude in two ways. One is that Fsin  is the component of the force perpendicular to the position r. The second is that rsin  (the “moment arm”) is the perpendicular distance from the rotation axis to the line of action of the force. For extended objects, equilibrium implies both zero acceleration and zero angular acceleration. Since torque is the tendency of a force to cause rotation, the net torque on an object in equilibrium must be zero about any axis!!! ConcepTest, Example

Rigid body under a net torque If the torque on a rigid object about any axis is not zero, the points making up the object will be experiencing a net force, and so accelerate. The tangential acceleration of the i th mass will be governed by Multiplying by the position gives Summing over all particles gives

Rigid body under a net torque Example: Atwood machine with a massive frictionless pulley m1m1 m2m2 +y m1m1 m2m2 I 3 unknowns: T 1, T 2, and a

Rigid body under a net torque

Work and energy in rotational motion The work done by an external force on a point P on a rigid rotating body is Where  is the angle between the direction of the force and the position of its application, and d  is the infinitesimal angle over which the object rotates. Examples

Work and energy in rotational motion If we rewrite the expression for torque as Integrating over  between an initial and final angular speeds, we find the rotational version of the work-kinetic energy theorem

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Angular momentum Each point on a rotating object has a velocity and a mass, and so even if the object does not move (translate) it has a momentum associated with it. We call this the angular momentum, defined as The vector angular momentum is a cross product of the radius and the linear momentum Note that it depends on the choice of axis! Also note that the cross product is not commutative:

Angular momentum We can see that the angular momentum is the analog of the linear momentum in equations of dynamics This the rotational version of Newton’s 2 nd law Rewriting the total angular momentum of a system of particles as

Angular momentum Likewise, for a system of particles This means that the angular momentum of a system about a specific axis will be constant if the external torque about that axis is zero For an isolated system, then ConcepTest Examples

Precession A top spinning on a horizontal surface with angular speed  will have an angular momentum L with magnitude I , pointing in a direction along the axis of rotation If it makes some angle with the vertical, the gravitational force will give a torque about the point of contact with the surface. This torque will cause the angular momentum to change direction, tracing out a circle perpendicular to the vertical direction – this behavior is called precession. From Serway & Jewett, Principles of Physics, 3 rd ed., Harcourt 2002

Rolling motion As a cylinder or sphere or radius R rolls through an angle , its center of mass moves s=R , so But note that the object is not rotating about its center – the bottom is stationary at any point, while the top moves at a speed 2v COM

Rolling motion We can write the total kinetic energy of the object as Parallel axis theorem: The moment of inertia through any axis parallel to the axis through the COM follows the following relation: where D is the distance between the axes

Rolling motion Using the parallel axis theorem for a cylinder, we see that In other words, the kinetic energy is equal to that of a purely rotating cylinder about the point of contact with the ground. where D is the distance between the axes

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