Rotational Equilibrium and Dynamics Rotation and Inertia

Center of Mass There is a point about which an object will rotate if the only force acting on the object is gravity Center of mass – the point at which all the mass of the body can be considered to be when analyzing translational motion Translational motion is any motion that’s not rotational

Center of Mass Center of gravity – the position at which the gravitational force acts on the extended object as if it were a point mass We will treat center of mass and center of gravity as the same point

Moment of Inertia Some objects have several axes around which they can rotate. Some axes are easier to rotate around than others This is because of the object’s moment of inertia Moment of inertia – the tendency of a body rotating about a fixed axis to resist a change in rotational motion

Moment of Inertia Moment of inertia is rotational analog of mass Anytime we would use mass for a rotating object, we use moment of inertia The main difference is that mass remains constant for an object, assuming the object doesn’t change Moment of inertia changes depending on which axis the object is rotating around

Moment of Inertia The farther an object’s mass is from the axis of rotation, the larger the object’s moment of inertia The larger the object’s moment of inertia, the harder it is to rotate Therefore, solid objects have smaller moments of inertia than hollow objects with the same mass and radius Solid objects rotate faster than hollow ones

Moment of Inertia When a net torque acts on an object, the resulting change in rotational motion of the object depends on the object’s moment of inertia Moment of inertia is measured in kilogram meters2 (kgm2) Calculating moment of inertia depends of the object’s shape and the axis of rotation It will involve multiplying the object’s mass times its radius squared for circles and spheres (MR2) or length squared for cylinders (Ml 2) This is then multiplied by a fraction (½, 2/5, etc.)

Moment of Inertia

Rotational Equilibrium For an object to be in equilibrium, the object must experience zero net force and zero net torque Zero net force gives the object translational equilibrium Called the first condition of equilibrium Just add all of the forces up

Rotational Equilibrium Zero net torque gives the object rotational equilibrium Called the second condition of equilibrium Pick an axis of rotation and add all of the torques A force that acts along the axis of rotation will not produce a torque So you should try to position unknown forces along the axis of rotation

Rotational Equilibrium