4 Rotational Dynamics Rotational Motion and Astrophysics Advanced

4. Rotational Dynamics Rotational Motion and Astrophysics Advanced Higher

Moment of a Force The moment of a force is the turning effect it can produce – for example a long handled screwdriver. A force F is applied on the screwdriver and it is also applied perpendicularly to the turning point. The force is applied a distance d from the turning point. Moment = F x d

Torque When a force is applied and this causes a rotation about an axis, this moment of force is then known as the Torque. The force is applied to the object at a certain distance from the pivot point. Perpendicular Not Perpendicular Unit (Nm)

Moment of Inertia The moment of inertia, I, of an object is described as its “resistance to change in its angular motion” The moment of inertia is dependant on the mass of the object and the distribution of mass about the axis of rotation. It can also depend on what type of object it is.

Moment of Inertia Formulae Point Mass Rod about centre Rod about end Disk about centre Sphere about centre

Example 1. A wheel has very light spokes. The mass of the rim and tyre is 2 kg and the radius of the wheel is 0. 8 m. Calculate the moment of inertia of the wheel.

Unbalanced Torque When an unbalanced torque is applied to an object, this causes an angular acceleration. The angular acceleration produced depends on the unbalanced torque and the moment of inertia.

Example 2. A cylindrical drum is free to rotate about an axis AB as shown below. The radius of the drum is 0. 3 m and the moment of inertia is 0. 4 kgm-2. A rope of length 5 m is wound round the drum and pulled with a constant force of 8 N. (a) Calculate the torque on the drum (b) Determine the angular acceleration (c) Calculate the angular velocity of the drum just as the rope leaves the drum. Assume is starts from rest.

(a) (c) Solutions (b)

Angular Momentum Similar to linear momentum, but the object must be rotating. The angular momentum L of a particle about an axis is defined as the moment of momentum

Conservation of Angular Momentum Just like with linear momentum, memorise this statement! “The total angular momentum before and impact will equal the total angular momentum after an impact providing there are no external torques acting!

Rotational Kinetic Energy When an object rolls down a hill, it moves with both translational (linear) kinetic energy and rotational (angular) kinetic energy. For conservation of energy, we must add these values together to get the total kinetic energy

Example 3. A shaft has a moment of inertia of 20 kgm 2 about its central axis. The shaft is rotating at 10 rpm. The shaft is locked onto another shaft which is initially stationery. The second shaft has a moment of inertia of 30 kgm 2 (a) Find the angular momentum of the combination after the shafts are locked together. (b)What is the angular momentum after the shafts are locked together.