Angular Velocity: Sect. 1.15 Overview only. For details, see text!
description
Transcript of Angular Velocity: Sect. 1.15 Overview only. For details, see text!
![Page 1: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/1.jpg)
![Page 2: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/2.jpg)
Angular Velocity: Sect. 1.15 Overview only. For details, see text!
• Consider a particle moving on arbitrary path in space: – At a given instant, it can be considered as moving in a plane,
circular path about an axis
Instantaneous Rotation Axis. In an infinitesimal time dt, the path can be represented as infinitesimal circular arc.
• As the particle moves in circular path, it has angular velocity:
ω (dθ/dt) θ
![Page 3: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/3.jpg)
• Consider a particle moving in an instantaneously circular path of radius R. (See Fig.): – Magnitude of Particle
Angular Velocity:
ω (dθ/dt) θ– Magnitude of
Linear Velocity
(linear speed):
v = R(dθ/dt)
= Rθ = Rω
![Page 4: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/4.jpg)
• Particle moving in circular path, radius R. (Fig.):
Angular Velocity: ω θ
Linear Speed: v = Rω (1)
• Vector direction of ω
normal to the plane of motion,
in the direction of a right hand
screw. (Fig.). Clearly:
R = r sin(α) (2)
(1) & (2) v = rωsin(α)
So (for detailed proof, see text!):
v = ω r
![Page 5: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/5.jpg)
Gradient (Del) Operator: Sect. 1.16 Overview only. For details, see text!
• The most important vector differential operator: grad
A Vector which has components which are differential operators. Gradient operator.
• In Cartesian (rectangular) coordinates:
∑i ei (∂/∂xi) (1)
• NOTE! (For future use!) is much more complicated in cylindrical & spherical coordinates (see Appendix F)!!
![Page 6: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/6.jpg)
can operate directly on a scalar function
( gradient of Old Notation: = grad):
= ∑i ei (∂/∂xi) A VECTOR!
can operate in a scalar product with a vector A ( divergence of A; Old: A = div A):
A = ∑i (∂Ai/∂xi) A SCALAR!
can operate in a vector product with a vector A ( curl of A; Old: A = curl A):
(A)i = ∑j,k εijk(∂Ak/∂xj) A VECTOR!
(Older: A = rot A) Obviously, A = A(x,y,z)
![Page 7: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/7.jpg)
• Physical interpretation of the gradient : (Fig)
• The text shows that has the properties:
1. It is surfaces of constant 2. It is in the direction of max change in 3. The directional derivative of for any direction n is n = (∂/∂n)
(x,y)
Contour plot of (x,y)
![Page 8: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/8.jpg)
The Laplacian Operator• The Laplacian is the dot product of with itself:
2 ; 2 ∑i (∂2/∂xi2)
A SCALAR!
• The Laplacian of a scalar function 2 ∑i (∂2/∂xi
2)
![Page 9: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/9.jpg)
Integration of Vectors: Sect. 1.17 Overview only. For details, see text!
• Types of integrals of vector functions:
A = A(x,y,z) = A(x1,x2,x3) = (A1,A2,A3)
• Volume Integral (volume V, differential volume element dv) (Fig.):
∫V A dv (∫V A1dv, ∫V A2dv, ∫V A3dv)
![Page 10: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/10.jpg)
• Surface Integral (surface S, differential surface element da) (Fig.)
∫S An da, n Normal to surface S
![Page 11: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/11.jpg)
• Line integral (path in space, differential path element ds) (Fig.):
∫BC Ads ∫BC ∑i Ai dxi
![Page 12: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/12.jpg)
• Gauss’s Theorem or Divergence Theorem (for a closed surface S surrounding a volume V)
See figure; n Normal to surface S
∫S An da = ∫V A dv
Physical Interpretation of AThe net “amount” of A “flowing” in & out of closed surface S
![Page 13: Angular Velocity: Sect. 1.15 Overview only. For details, see text!](https://reader033.fdocuments.us/reader033/viewer/2022051622/56815144550346895dbf6462/html5/thumbnails/13.jpg)
• Stoke’s Theorem (for a closed loop C surrounding a surface S) See Figure; n Normal to surface S
∫C Ads = ∫S (A)n da
Physical Interpretation of AThe net “amount” of “rotation” of A