User:J Hill/Sandbox

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[edit] Cross product

cross product diagram
cross product diagram

The cross product is a binary operation, an operation involving two quantities , on vectors in three-dimensional Euclidean space.

a \times b = \mathbf{n} |a| |b| \sin\theta_{a,b}

a and b are variables
θ is the measure of the angle formed by a and b. It must be above 0 and be no greater than 180.
n is a unit vector, a vector whose magnitude is 1 in normed vector space, a vector space in 2 or 3 dimension where all (except the zero vector) has positive magnitude.

[edit] Radians

Diagram of a radian
Diagram of a radian

One radian is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle. In short, one radian is the maesure of the angle formed by the rays from the center of the circle to the endpoints of the arc. In terms of a circle it can be seen as the ratio of the length of the arc subtended by two radii to the radius of the circle. 1rad is approximately 57.3 degrees.

360^\circ = 2 \pi \mbox{rad} \Longrightarrow 1 \mbox{rad} = \frac{360^\circ} {2 \pi} = \frac{180^\circ} {\pi}

[edit] Angular momentum

Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system
Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system

Angular momentum is a vector quantity \overrightarrow{L} that measures the intensity of rotational motion. Angular momentum is a conserved quantity: The angular momentum of a system remains constant unless acted upon by external Torque. Angular momentum is measured in Joule-seconds (J•s).

L=r\times p = |r||p|\sin\theta_{r,p}

L=I\mathbf{\omega}

L is angular momentum
r is the magnitude of the displacement vector \overrightarrow{r}
p is the linear momentum
I is the rotational inertia

p = m \mathbf{v}

m is the mass of the object
\mathbf{v} is the magnitude of the velocity vector \overrightarrow{v}

[edit] Rotational Inertia

Rotational inertia maesures the resistance in change in rotation. It is maesured in kg•m².

I \equiv \sum_{i=1}^n m_i r_i^2

m is the mass of the body
r is the magnitude of the displacement vector \overrightarrow{r}

[edit] Angular velocity and Angular frequency

Angular velocity maesures the change of an angle over time, similar to linear velocity, which maesures change in position over time. It is maesured in rad•s-1, or radians per second.

\mathbf{\omega} = \frac{\Delta \theta} {\Delta t}

Angular frequency is similar to angular velocity, and maesures the rate at which a body rotates. It is maesured in the same units as angular velocity.

\omega = \frac{2 \pi} {T} = 2 \pi f

T = \frac{2 \pi R} {v} \Longrightarrow T = \frac{C} {v}

T is the period of rotation, in seconds (s)
f is the frequency, in Herz (Hz)
R is is the magnitude of the displacement vector \overrightarrow{r}, or the radius of the circle, in meters (m)
C is the circumfrence of the circle, in meters

[edit] Torque

Torque is the tendency to change in rotational motion, often thought of as agnular force. It is measured in Newton-metres (N•m). Because torque can be thought of as change in angular momentum, it can be calculated by taking the change in angular momentum divided by the change in time.

\tau = r \times F = |r||F|\sin\theta_{r,F} \Longrightarrow \frac{dL} {dt}

F is the force acting upon the mass
r is the magnitude of the displacement vector \overrightarrow{r}

[edit] Angular acceleration

Angular acceleration, like linear acceleration, is the change in angular velocity. It is maesured in rad•s-2.

\alpha = \frac{\Delta \mathbf\omega} {\Delta t} = \frac{\tau} {I} \Longrightarrow \tau = I \alpha

[edit] Centripetal force and acceleration

this is how centripetal force is derived.
Enlarge
this is how centripetal force is derived.

Centripetal force is the external force required to cause a particle to move in a circular motion with uniform velocity toward the center of that circle.

F_c = - m \omega^2 r = \frac{m v^2} {r}

Centripetal acceleration is derived from the fact that velocity is a vector (possesing magnitude and direction), and circular motion implies change in direction, ergo there is a change in velocity. This is centripetal acceleration.

a_c = - \omega^2 r = \frac{v^2} {r}

[edit] Rotational kinetic energy

Kinetic energy is the energy a body has because it is in motion. Rotational kinetic energy is the energy that a body has because it is rotating.

E_k = \frac{1} {2} m r^2 = \frac{1} {2} I \omega^2 \Longrightarrow E = \tau \theta

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