A solid disk and a ring roll down an incline. The ring is slower than the disk if

1. mring= mdisk, where m is the inertial mass.2. rring = rdisk, where r is the radius.3. mring = mdisk and rring = rdisk.4. The ring is always slower regardless of the relative values of m and r.

Example: Determine the rotational inertia of a cylinder about its central axis.

R

ML

x

y

z

dmrI 2 dVdm

LdAdV drrL 2

R

drrLrI0

2 2

R

drrLI0

32

Rr

LI0

4

42

42

4RL 22

2

1RRL

Total volume of cylinder

2

2

1VR

Total mass of cylinder

2

2

1MRI Rotational Inertia of a solid cylinder

rotating about its longitudinal axis.

Parallel-Axis TheoremThe rotational inertial of different objects through an axis of symmetry was shown for several objects. If the rotation axis is shifted away from an axis of symmetry the calculation becomes more difficult. A simple method, called the parallel axis theorem, was devised for situations where the rotation axis was shifted some distance from the symmetry axis.

The symmetry axis is any axis that passes through the center of mass.

2MDII CM

I – rotational inertiaICM – Rotational inertia for a rotation axis that passes through the center of massM – Total mass of the objectD – Distance the axis has been shifted by

The new rotation axis must be parallel to the symmetry axis that is being used to define ICM.

Vector Product (Cross-product)

BAC

sinBAC

sinBABA

There are two different methods for determining the vector product between any two vectors: The Determinant method and the Cyclic method

Determinant Method

zyx

zyx

BBB

AAA

kji

BA

ˆˆˆ

yzzy BABAi ˆ xyyx BABAk ˆ xzzx BABAj ˆ

Cyclic Method 0ˆˆˆˆˆˆ kkjjii

i

j k

kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ

iBAjBAiBAkBAjBAkBA yzxzzyxyzxyxˆˆˆˆˆˆ

xyyxzxxzyzzy BABAkBABAjBABAi ˆˆˆ

xzzx BABAj ˆRewrite the j term as to get an identical expression

BAC

The direction of the resultant vector, for a right-handed coordinate system, for a vector product can be determined using the right-hand rule.

Thumb points in the direction of first vector.

Fingers point in the direction of the second vector

Palm points in the direction of resultant vector.

The vector product looks at the product of two vectors which are perpendicular to each other, and who are also perpendicular to the resultant vector.

Example: Determine the magnitude and direction of the area of a parallelogram described by the vectors r1 and r2 which are used to describe the length of the two sides.

r1r2

y

x

cmkjir ˆ0ˆ0.4ˆ0.21 cmkjir ˆ0ˆ0.3ˆ0.22

21 rrA

00.30.2

00.40.2

ˆˆˆ

kji

kji ˆ0.80.6ˆ0ˆ0

kcmA ˆ14 2

214cm Out of the page

TorqueWhen a force is applied to an object it will cause it to accelerate. This acceleration would correspond to a change in the velocity of the object. The velocity could be a translational velocity or a rotational velocity. A torque is a force that causes a change in the angular velocity of the object.

The force F1 will cause a counterclockwise rotation about an axis that passes through O.The force F2 will cause a clockwise rotation about an axis that passes through O.r is the distance from the origin O to the point of application of F.

d is the lever arm for F which is the distance from O to F such that d is perpendicular to F.

Fr

t – Torque [Nm]r – distance from axis of rotation [m]F – Force [N]

sinFr

q – angle measured from r to F

d = rsinq this is the lever arm, the component of r perpendicular to FThe direction of the torque vector is perpendicular to both r and F. The directions clockwise and counterclockwise are used to describe the direction the torque causes the object to rotate.

r1

r2

q1

q2