RotationRotational Variables

Angular VectorsLinear and Angular Variables

Rotational Kinetic EnergyRotational Inertia

Parallel Axis TheoremNewton’s Second Law for RotationWork and Rotational Kinetic Energy

pps by C Gliniewicz

Translation is movement along a straight line while rotation is movement around an axis.A rigid body is one whose parts can rotate without the shape being changed. The axis of rotation or rotational axis is the line about which the body rotates.

Rotational position is the angle. The angle is measured in radians which is the ration of the length of the arc to the radius of the circle. One revolution is 360 degrees or 2π radians.An angular displacement in the counterclockwise direction is positive and a displacement in the clockwise direction is negative.

Angular velocity is the change in angular displacement over the time.

Angular acceleration is the time rate of change of angular velocity.

fi

sr

2 1

02 1

limav t

dt t t t dt

2 1

02 1

limav t

dt t t t dt

pps by C Gliniewicz

Angular displacement can be a vector quantity only if the displacement is very small. Angular velocity is a vector quantity. Using the right hand rule, one wraps their hand about the axis of rotation in such a way that their fingers point in the direction of rotation. Then the extended thumb points in the direction of the angular velocity vector. The same is done for the angular acceleration.

The equations of angular motion are the same as the equations for translational motion except that the symbols for angular motion are substituted.

The angular motion variables can be related to the tangential (translational) quantities.

2 2 2

2

12

21 12 2

fi fi i fi fi

fi i ffi f

t t t

t t t

22

tan tan

2 2r c

r vs r v r a r T a a r

v r

pps by C Gliniewicz

Although one knows that a rotating object has kinetic energy, the equation for kinetic energy cannot be used. Since the center of mass has a velocity of zero, the translational kinetic energy will also be zero. One can, however sum the kinetic energy of all the particles.

The new quantity inside the parentheses is the rotational inertia, I. This quantity is a measure of the inertia of the mass taking into account its position relative to the axis.

For regular shaped objects, this calculation has been done.

22 2 2

1 1 1

1 1 12 2 2

n n n

i i i i i ii i i

K mv m r m r

2 2 2

1

12

n

i ii

m r K r dm

2 2 2 2pointmass thinhoop thick hoop inner outer

2 2 2cylinder solid sphere hollow sphere

2 2 2thin rod solid cylinder diameter

12

1 2 22 5 31 1 1

12 4 12

mr mr m r r

mr mr mr

mL mr mL

pps by C Gliniewicz

The parallel axis theorem is a formula to determine the moment of inertia of a regular shaped object when it rotates about an axis different from the axis through the center of mass. One only needs to know the formula for the moment of inertia for the regular shaped object and the distance between the principle axis and the new axis, h.

Forces acting on a rigid body, not through the center of mass, will cause the body to rotate as well as translate (move in a straight line). This twisting force is called a torque. It is the product of the force and the lever arm or line of action, the distance from the axis to the force.

Newton’s second law for rotation is similar to Newton’s Second Law with torque in place of force and moment of inertia in place of mass and angular acceleration for acceleration.

2

2 2 2

For example a sphere rotating about a line

2 7tangent to the sphere would be =

5 5

com mh

mr mr mr

tansin

where is the angle between the force and radius vectors

r F rF r F

pps by C Gliniewicz

Net torque is similar to net force and can be added as vectors. One should recall that if the torque is a vector and it is the product of r, F, and the sin of the angle, then it must be a cross product. We will use that fact next chapter.

In translation,

In rotation,

f

i

x

x

dWW F dx P Fv

dt

f

i

x

x

dW dW d P

dt dt

pps by C Gliniewicz