Rotational Motion – Part I AP Physics C

The radian  There are 2 types of pure unmixed motion:  Translational - linear motion  Rotational - motion involving a rotation or revolution around a fixed chosen axis( an axis which does not move).  We need a system that defines BOTH types of motion working together on a system. Rotational quantities are usually defined with units involving a radian measure. If we take the radius of a circle and LAY IT DOWN on the circumference, it will create an angle whose arc length is equal to R. In other words, one radian angle is subtends an arc length  s equal to the radius of the circle (R)

The radian Half a radian would subtend an arc length equal to half the radius and 2 radians would subtend an arc length equal to two times the radius. A general Radian Angle (  ) subtends an arc length (  s) equal to R. The theta in this case represents ANGULAR DISPLACEMENT.

Since velocity is defined as the rate of change of displacement. ANGULAR VELOCITY is defined as the rate of change of ANGULAR DISPLACEMENT. Angular Velocity

Angular Acceleration Once again, following the same lines of logic. Since acceleration is defined as the rate of change of velocity. We can say the ANGULAR ACCELERATION is defined as the rate of change of the angular velocity. Also, we can say that the ANGULAR ACCELERATION is the TIME DERIVATIVE OF THE ANGULAR VELOCITY. All the rules for integration apply as well.

Linear Angular Relationships  We often want to be able to compare the linear quantity of something to its angular counterpart.  We are going to use our arc length-angle relationship to derive a relationship between linear and angular velocity.  This relationship shows that on a rotating rigid body, points farther away from the rotational axis move faster.  We can also use this idea to compare linear and angular acceleration.

Rotational Kinematics  Using these new linear-angular relationships, we can take our linear kinematic equations and translate them into angular kinematics.  These kinematics are used in the same way we used our linear equations. They allow us to describe the rotational motion of an object using our basic definitions of motion and time.

Example A car engine is idling at 500 RPM. When the light turns green, the crankshaft rotation speeds up at a constant rate to 2500 RPM over and interval of 3.0 s. How many revolutions does the crankshaft go through during this interval? 75 Revolutions

Rotational Kinetic Energy  A rotating object has kinetic energy because all particles in the object are in motion.  The kinetic energy due to rotation is called rotational kinetic energy.  We add up the individual kinetic energies of each particle to find the rotational kinetic energy of the object.

Rotational Kinetic Energy and Moment of Inertia  Looking at our formula for rotational kinetic energy, there is an interesting term:  We define this term as our moment of inertia, which is the rotational analog to inertia.  We use the letter I for moment of inertia, so we will substitute that into our formula for rotational kinetic energy.  This is not a new form of energy, merely the familiar kinetic energy of motion written in a new way.

Moment of Inertia  Moment of Inertia represents an object’s resistance to change in rotational motion. This is a fancy way of saying it represents and object’s resistance to angular acceleration.  The units of moment of inertia are kg m 2.  A system consisting of discrete point masses has a moment of inertia defined using the summation we discussed earlier:  Moment of inertia depends on the axis of rotation.  Mass farther from the rotation axis contributes more to the moment of inertia than mass nearer the axis.

Moment of Inertia The summation equation for moment of inertia worked fine for a collection of point masses, but what about more continuous masses like disks, rods, or sphere where the mass is extended over a volume or area? In this case, calculus is needed. This suggests that we will take small discrete amounts of mass and add them up over a set of limits. Indeed, that is what we will do. Let’s look at a few example we “MIGHT” encounter. Consider a solid rod rotating about its CM.

Example – Thin rod rotated about the middle dr We begin by using the same technique used to derive the center of mass of a continuous body.

Example – Thin rod rotated about one end dr What if the rod were rotating on one of its ENDS?

Well that was fun, but…  Will you be asked to derive the moment of inertia of an object? Maybe. Most of the time the moment of inertia is given within the free response question.  Moments of Inertia for various geometric objects can be found on tables such as the one below.

Parallel Axis Theorem This theorem will allow us to calculate the moment of inertia of any rotating body around any axis, provided we know the moment of inertia about the center of mass. It basically states that the Moment of Inertia (I p ) around any axis "P" is equal to the known moment of inertia (I cm ) about some center of mass plus M (the total mass of the system) times the square of "d" (the distance between the two parallel axes) Let’s use the thin rod example to prove this.

Parallel Axis Theorem dr

Moment of Inertia of Compound Objects  If you have two geometric objects attached to each other, you can find the moment of inertia of the system by adding the moment of inertia of the two objects (assuming they rotate about the same point).  This means that moment of inertia obeys the principle of superposition.  Notice that if the sphere is very small (L>>R), then the sphere can be approximated as a point mass, making the moment of inertia of this system:

Rolling Motion  Rolling is a combination of rotation and translation. For an object that rolls without slipping, the translation of the center of mass is related to the angular velocity by:  We also find that this is the velocity at both the top and bottom of the object.

 The figure below shows how the velocity vectors at the top, center, and bottom of a rolling wheel are found.  v top = 2v cm.  v bottom = 0.  The point on the bottom of a rolling object is instantaneously at rest. Rolling Motion

Rolling Motion and Energy  To describe the kinetic energy of a rolling object, we must take into account the linear velocity and the rotational velocity of the object.  This means that the total kinetic energy of a rolling object can be calculated by adding the kinetic energy of the center of mass and the rotational kinetic energy about the center of mass.

Example A very common problem is to find the velocity of a ball rolling down an inclined plane. It is important to realize that you cannot work out this problem the way you used to. In the past, everything was SLIDING. Now the object is rolling and thus has MORE energy than normal. So let’s assume the ball is like a thin spherical shell and was released from a position 5 m above the ground. Calculate the velocity at the bottom of the incline m/s If you HAD NOT included the rotational kinetic energy, you see the answer is very much different.